From 40b27d2bf0b909188174673f2a72b7b090a87296 Mon Sep 17 00:00:00 2001 From: "George G. Vega Yon" Date: Tue, 1 Oct 2024 11:40:55 -0600 Subject: [PATCH] Uploading slides --- .../quarto-ext/fontawesome/_extension.yml | 7 + .../quarto-ext/fontawesome/assets/css/all.css | 7971 ++++++++ .../fontawesome/assets/css/latex-fontsize.css | 30 + .../FontAwesome6Brands-Regular-400.ttf | Bin 0 -> 180104 bytes .../FontAwesome6Brands-Regular-400.woff2 | Bin 0 -> 92140 bytes .../webfonts/FontAwesome6Free-Regular-400.ttf | Bin 0 -> 76072 bytes .../FontAwesome6Free-Regular-400.woff2 | Bin 0 -> 25124 bytes .../webfonts/FontAwesome6Free-Solid-900.ttf | Bin 0 -> 384736 bytes .../webfonts/FontAwesome6Free-Solid-900.woff2 | Bin 0 -> 133940 bytes .../assets/webfonts/fa-brands-400.ttf | Bin 0 -> 181852 bytes .../assets/webfonts/fa-brands-400.woff2 | Bin 0 -> 105536 bytes .../assets/webfonts/fa-regular-400.ttf | Bin 0 -> 60520 bytes .../assets/webfonts/fa-regular-400.woff2 | Bin 0 -> 23940 bytes .../assets/webfonts/fa-solid-900.ttf | Bin 0 -> 388460 bytes .../assets/webfonts/fa-solid-900.woff2 | Bin 0 -> 154228 bytes .../assets/webfonts/fa-v4compatibility.ttf | Bin 0 -> 10556 bytes .../assets/webfonts/fa-v4compatibility.woff2 | Bin 0 -> 4960 bytes .../quarto-ext/fontawesome/fontawesome.lua | 84 + 07-parallel-computing/fig/OSG_Map.png | Bin 0 -> 956432 bytes .../fig/Zoom-Background_PrideU.jpg | Bin 0 -> 1046218 bytes 07-parallel-computing/fig/cpu-slurm.png | Bin 47849 -> 0 bytes .../fig/embarrasingly-parallel.png | Bin 0 -> 82219 bytes 07-parallel-computing/fig/trojan-rlogo.svg | 184 + 07-parallel-computing/references.bib | 106 + 07-parallel-computing/slides.html | 15855 ++++++++++++++++ 07-parallel-computing/slides.qmd | 847 +- 07-parallel-computing/style.scss | 2 + 27 files changed, 24861 insertions(+), 225 deletions(-) create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/_extension.yml create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/css/all.css create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/css/latex-fontsize.css create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/FontAwesome6Brands-Regular-400.ttf create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/FontAwesome6Brands-Regular-400.woff2 create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/FontAwesome6Free-Regular-400.ttf create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/FontAwesome6Free-Regular-400.woff2 create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/FontAwesome6Free-Solid-900.ttf create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/FontAwesome6Free-Solid-900.woff2 create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/fa-brands-400.ttf create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/fa-brands-400.woff2 create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/fa-regular-400.ttf create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/fa-regular-400.woff2 create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/fa-solid-900.ttf create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/fa-solid-900.woff2 create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/fa-v4compatibility.ttf create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/webfonts/fa-v4compatibility.woff2 create mode 100644 07-parallel-computing/_extensions/quarto-ext/fontawesome/fontawesome.lua create mode 100644 07-parallel-computing/fig/OSG_Map.png create mode 100644 07-parallel-computing/fig/Zoom-Background_PrideU.jpg delete mode 100644 07-parallel-computing/fig/cpu-slurm.png create mode 100644 07-parallel-computing/fig/embarrasingly-parallel.png create mode 100644 07-parallel-computing/fig/trojan-rlogo.svg create mode 100644 07-parallel-computing/references.bib create mode 100644 07-parallel-computing/slides.html create mode 100644 07-parallel-computing/style.scss diff --git a/07-parallel-computing/_extensions/quarto-ext/fontawesome/_extension.yml b/07-parallel-computing/_extensions/quarto-ext/fontawesome/_extension.yml new file mode 100644 index 0000000..c0787a8 --- /dev/null +++ b/07-parallel-computing/_extensions/quarto-ext/fontawesome/_extension.yml @@ -0,0 +1,7 @@ +title: Font Awesome support +author: Carlos Scheidegger +version: 1.1.0 +quarto-required: ">=1.2.269" +contributes: + shortcodes: + - fontawesome.lua diff --git a/07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/css/all.css b/07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/css/all.css new file mode 100644 index 0000000..3e24980 --- /dev/null +++ b/07-parallel-computing/_extensions/quarto-ext/fontawesome/assets/css/all.css @@ -0,0 +1,7971 @@ +/*! + * Font Awesome Free 6.4.2 by @fontawesome - https://fontawesome.com + * License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) + * Copyright 2023 Fonticons, Inc. + */ +.fa { + font-family: var(--fa-style-family, "Font Awesome 6 Free"); + font-weight: var(--fa-style, 900); } + +.fa, +.fa-classic, +.fa-sharp, +.fas, +.fa-solid, +.far, +.fa-regular, +.fab, +.fa-brands, +.fal, +.fa-light, +.fat, +.fa-thin, +.fad, +.fa-duotone { + -moz-osx-font-smoothing: grayscale; + -webkit-font-smoothing: antialiased; + display: var(--fa-display, inline-block); + font-style: normal; + font-variant: normal; + line-height: 1; + text-rendering: auto; } + +.fas, +.fa-classic, +.fa-solid, +.far, +.fa-regular { + font-family: 'Font Awesome 6 Free'; } + +.fab, +.fa-brands { + font-family: 'Font Awesome 6 Brands'; } + +.fa-1x { + font-size: 1em; } + +.fa-2x { + font-size: 2em; } + +.fa-3x { + font-size: 3em; } + +.fa-4x { + font-size: 4em; } + +.fa-5x { + font-size: 5em; } + +.fa-6x { + font-size: 6em; } + +.fa-7x { + font-size: 7em; } + +.fa-8x { + font-size: 8em; } + +.fa-9x { + font-size: 9em; } + +.fa-10x { + font-size: 10em; } + +.fa-2xs { + font-size: 0.625em; + line-height: 0.1em; + vertical-align: 0.225em; } + +.fa-xs { + font-size: 0.75em; + line-height: 0.08333em; + vertical-align: 0.125em; } + +.fa-sm { + font-size: 0.875em; + line-height: 0.07143em; + vertical-align: 0.05357em; } + +.fa-lg { + font-size: 1.25em; + line-height: 0.05em; + vertical-align: -0.075em; } + +.fa-xl { + font-size: 1.5em; + line-height: 0.04167em; + vertical-align: -0.125em; } + +.fa-2xl { + font-size: 2em; + line-height: 0.03125em; + vertical-align: -0.1875em; } + +.fa-fw { + text-align: center; + width: 1.25em; } + +.fa-ul { + list-style-type: none; + margin-left: var(--fa-li-margin, 2.5em); + padding-left: 0; } + .fa-ul > li { + position: relative; } + +.fa-li { + left: calc(var(--fa-li-width, 2em) * -1); + position: absolute; + text-align: center; + width: var(--fa-li-width, 2em); + line-height: inherit; } + +.fa-border { + border-color: var(--fa-border-color, #eee); + border-radius: var(--fa-border-radius, 0.1em); + border-style: var(--fa-border-style, solid); + border-width: var(--fa-border-width, 0.08em); + padding: var(--fa-border-padding, 0.2em 0.25em 0.15em); } + +.fa-pull-left { + float: left; + margin-right: var(--fa-pull-margin, 0.3em); } + +.fa-pull-right { + float: right; + margin-left: var(--fa-pull-margin, 0.3em); } + +.fa-beat { + -webkit-animation-name: fa-beat; + animation-name: fa-beat; + -webkit-animation-delay: var(--fa-animation-delay, 0s); + animation-delay: var(--fa-animation-delay, 0s); + -webkit-animation-direction: var(--fa-animation-direction, normal); + animation-direction: var(--fa-animation-direction, normal); + -webkit-animation-duration: var(--fa-animation-duration, 1s); + animation-duration: var(--fa-animation-duration, 1s); + -webkit-animation-iteration-count: var(--fa-animation-iteration-count, infinite); + animation-iteration-count: var(--fa-animation-iteration-count, infinite); + -webkit-animation-timing-function: var(--fa-animation-timing, ease-in-out); + animation-timing-function: var(--fa-animation-timing, ease-in-out); } + +.fa-bounce { + -webkit-animation-name: fa-bounce; + animation-name: fa-bounce; + -webkit-animation-delay: var(--fa-animation-delay, 0s); + animation-delay: var(--fa-animation-delay, 0s); + -webkit-animation-direction: var(--fa-animation-direction, normal); + animation-direction: var(--fa-animation-direction, normal); + -webkit-animation-duration: var(--fa-animation-duration, 1s); + animation-duration: var(--fa-animation-duration, 1s); + -webkit-animation-iteration-count: var(--fa-animation-iteration-count, infinite); + animation-iteration-count: var(--fa-animation-iteration-count, infinite); + -webkit-animation-timing-function: var(--fa-animation-timing, cubic-bezier(0.28, 0.84, 0.42, 1)); + animation-timing-function: var(--fa-animation-timing, cubic-bezier(0.28, 0.84, 0.42, 1)); } + +.fa-fade { + -webkit-animation-name: fa-fade; + animation-name: fa-fade; + -webkit-animation-delay: var(--fa-animation-delay, 0s); + animation-delay: var(--fa-animation-delay, 0s); + -webkit-animation-direction: var(--fa-animation-direction, normal); + animation-direction: var(--fa-animation-direction, normal); + -webkit-animation-duration: var(--fa-animation-duration, 1s); + animation-duration: var(--fa-animation-duration, 1s); + -webkit-animation-iteration-count: var(--fa-animation-iteration-count, infinite); + animation-iteration-count: var(--fa-animation-iteration-count, infinite); + -webkit-animation-timing-function: var(--fa-animation-timing, cubic-bezier(0.4, 0, 0.6, 1)); + animation-timing-function: var(--fa-animation-timing, cubic-bezier(0.4, 0, 0.6, 1)); } + +.fa-beat-fade { + -webkit-animation-name: fa-beat-fade; + animation-name: fa-beat-fade; + -webkit-animation-delay: var(--fa-animation-delay, 0s); + animation-delay: var(--fa-animation-delay, 0s); + -webkit-animation-direction: var(--fa-animation-direction, normal); + animation-direction: var(--fa-animation-direction, normal); + -webkit-animation-duration: var(--fa-animation-duration, 1s); + animation-duration: var(--fa-animation-duration, 1s); + -webkit-animation-iteration-count: var(--fa-animation-iteration-count, infinite); + animation-iteration-count: var(--fa-animation-iteration-count, infinite); + -webkit-animation-timing-function: var(--fa-animation-timing, cubic-bezier(0.4, 0, 0.6, 1)); + animation-timing-function: var(--fa-animation-timing, cubic-bezier(0.4, 0, 0.6, 1)); } + +.fa-flip { + -webkit-animation-name: fa-flip; + animation-name: fa-flip; + -webkit-animation-delay: var(--fa-animation-delay, 0s); + animation-delay: var(--fa-animation-delay, 0s); + -webkit-animation-direction: var(--fa-animation-direction, normal); + animation-direction: var(--fa-animation-direction, normal); + -webkit-animation-duration: var(--fa-animation-duration, 1s); + animation-duration: var(--fa-animation-duration, 1s); + -webkit-animation-iteration-count: var(--fa-animation-iteration-count, infinite); + animation-iteration-count: var(--fa-animation-iteration-count, infinite); + -webkit-animation-timing-function: var(--fa-animation-timing, ease-in-out); + animation-timing-function: var(--fa-animation-timing, ease-in-out); } + +.fa-shake { + -webkit-animation-name: fa-shake; + animation-name: fa-shake; + -webkit-animation-delay: var(--fa-animation-delay, 0s); + animation-delay: var(--fa-animation-delay, 0s); + -webkit-animation-direction: var(--fa-animation-direction, normal); + animation-direction: var(--fa-animation-direction, normal); + -webkit-animation-duration: var(--fa-animation-duration, 1s); + animation-duration: var(--fa-animation-duration, 1s); + -webkit-animation-iteration-count: var(--fa-animation-iteration-count, infinite); + animation-iteration-count: var(--fa-animation-iteration-count, infinite); + -webkit-animation-timing-function: var(--fa-animation-timing, linear); + animation-timing-function: var(--fa-animation-timing, linear); } + +.fa-spin { + -webkit-animation-name: fa-spin; + animation-name: fa-spin; + -webkit-animation-delay: var(--fa-animation-delay, 0s); + animation-delay: var(--fa-animation-delay, 0s); + -webkit-animation-direction: var(--fa-animation-direction, normal); + animation-direction: var(--fa-animation-direction, normal); + -webkit-animation-duration: var(--fa-animation-duration, 2s); + animation-duration: var(--fa-animation-duration, 2s); + -webkit-animation-iteration-count: var(--fa-animation-iteration-count, infinite); + animation-iteration-count: var(--fa-animation-iteration-count, infinite); + -webkit-animation-timing-function: var(--fa-animation-timing, linear); + animation-timing-function: var(--fa-animation-timing, linear); } + +.fa-spin-reverse { + --fa-animation-direction: reverse; } + +.fa-pulse, +.fa-spin-pulse { + -webkit-animation-name: fa-spin; + animation-name: fa-spin; + -webkit-animation-direction: var(--fa-animation-direction, normal); + animation-direction: var(--fa-animation-direction, normal); + -webkit-animation-duration: var(--fa-animation-duration, 1s); + animation-duration: var(--fa-animation-duration, 1s); + -webkit-animation-iteration-count: var(--fa-animation-iteration-count, infinite); + animation-iteration-count: var(--fa-animation-iteration-count, infinite); + -webkit-animation-timing-function: var(--fa-animation-timing, steps(8)); + animation-timing-function: var(--fa-animation-timing, steps(8)); } + +@media (prefers-reduced-motion: reduce) { + .fa-beat, + .fa-bounce, + .fa-fade, + .fa-beat-fade, + .fa-flip, + .fa-pulse, + .fa-shake, + .fa-spin, + .fa-spin-pulse { + -webkit-animation-delay: -1ms; + animation-delay: -1ms; + -webkit-animation-duration: 1ms; + animation-duration: 1ms; + -webkit-animation-iteration-count: 1; + animation-iteration-count: 1; + -webkit-transition-delay: 0s; + transition-delay: 0s; + -webkit-transition-duration: 0s; + transition-duration: 0s; } } + +@-webkit-keyframes fa-beat { + 0%, 90% { + -webkit-transform: scale(1); + transform: scale(1); } + 45% { + -webkit-transform: scale(var(--fa-beat-scale, 1.25)); + transform: scale(var(--fa-beat-scale, 1.25)); } } + +@keyframes fa-beat { + 0%, 90% { + -webkit-transform: scale(1); + transform: scale(1); } + 45% { + -webkit-transform: scale(var(--fa-beat-scale, 1.25)); + transform: scale(var(--fa-beat-scale, 1.25)); } } + +@-webkit-keyframes fa-bounce { + 0% { + -webkit-transform: scale(1, 1) translateY(0); + transform: scale(1, 1) translateY(0); } + 10% { + -webkit-transform: scale(var(--fa-bounce-start-scale-x, 1.1), var(--fa-bounce-start-scale-y, 0.9)) translateY(0); + transform: scale(var(--fa-bounce-start-scale-x, 1.1), var(--fa-bounce-start-scale-y, 0.9)) translateY(0); } + 30% { + -webkit-transform: scale(var(--fa-bounce-jump-scale-x, 0.9), var(--fa-bounce-jump-scale-y, 1.1)) translateY(var(--fa-bounce-height, -0.5em)); + transform: scale(var(--fa-bounce-jump-scale-x, 0.9), var(--fa-bounce-jump-scale-y, 1.1)) translateY(var(--fa-bounce-height, -0.5em)); } + 50% { + -webkit-transform: scale(var(--fa-bounce-land-scale-x, 1.05), var(--fa-bounce-land-scale-y, 0.95)) translateY(0); + transform: scale(var(--fa-bounce-land-scale-x, 1.05), var(--fa-bounce-land-scale-y, 0.95)) translateY(0); } + 57% { + -webkit-transform: scale(1, 1) translateY(var(--fa-bounce-rebound, -0.125em)); + transform: scale(1, 1) translateY(var(--fa-bounce-rebound, -0.125em)); } + 64% { + -webkit-transform: scale(1, 1) translateY(0); + transform: scale(1, 1) translateY(0); } + 100% { + -webkit-transform: scale(1, 1) translateY(0); + transform: scale(1, 1) translateY(0); } } + +@keyframes fa-bounce { + 0% { + -webkit-transform: scale(1, 1) translateY(0); + transform: scale(1, 1) translateY(0); } + 10% { + -webkit-transform: scale(var(--fa-bounce-start-scale-x, 1.1), var(--fa-bounce-start-scale-y, 0.9)) translateY(0); + transform: scale(var(--fa-bounce-start-scale-x, 1.1), var(--fa-bounce-start-scale-y, 0.9)) translateY(0); } + 30% { + -webkit-transform: scale(var(--fa-bounce-jump-scale-x, 0.9), var(--fa-bounce-jump-scale-y, 1.1)) translateY(var(--fa-bounce-height, -0.5em)); + transform: scale(var(--fa-bounce-jump-scale-x, 0.9), var(--fa-bounce-jump-scale-y, 1.1)) translateY(var(--fa-bounce-height, -0.5em)); } + 50% { + -webkit-transform: scale(var(--fa-bounce-land-scale-x, 1.05), var(--fa-bounce-land-scale-y, 0.95)) translateY(0); + transform: scale(var(--fa-bounce-land-scale-x, 1.05), var(--fa-bounce-land-scale-y, 0.95)) translateY(0); } + 57% { + -webkit-transform: scale(1, 1) translateY(var(--fa-bounce-rebound, -0.125em)); + transform: scale(1, 1) translateY(var(--fa-bounce-rebound, -0.125em)); } + 64% { + -webkit-transform: scale(1, 1) translateY(0); + transform: scale(1, 1) translateY(0); } + 100% { + -webkit-transform: scale(1, 1) translateY(0); + transform: scale(1, 1) translateY(0); } } + +@-webkit-keyframes fa-fade { + 50% { + opacity: var(--fa-fade-opacity, 0.4); } } + +@keyframes fa-fade { + 50% { + opacity: var(--fa-fade-opacity, 0.4); } } + +@-webkit-keyframes fa-beat-fade { + 0%, 100% { + opacity: var(--fa-beat-fade-opacity, 0.4); + -webkit-transform: scale(1); + transform: scale(1); } + 50% { + opacity: 1; + -webkit-transform: scale(var(--fa-beat-fade-scale, 1.125)); + transform: scale(var(--fa-beat-fade-scale, 1.125)); } } + +@keyframes fa-beat-fade { + 0%, 100% { + opacity: var(--fa-beat-fade-opacity, 0.4); + -webkit-transform: scale(1); + transform: scale(1); } + 50% { + opacity: 1; + -webkit-transform: scale(var(--fa-beat-fade-scale, 1.125)); + transform: scale(var(--fa-beat-fade-scale, 1.125)); } } + +@-webkit-keyframes fa-flip { + 50% { + -webkit-transform: rotate3d(var(--fa-flip-x, 0), var(--fa-flip-y, 1), var(--fa-flip-z, 0), var(--fa-flip-angle, -180deg)); + transform: rotate3d(var(--fa-flip-x, 0), var(--fa-flip-y, 1), var(--fa-flip-z, 0), var(--fa-flip-angle, -180deg)); } } + +@keyframes fa-flip { + 50% { + -webkit-transform: rotate3d(var(--fa-flip-x, 0), var(--fa-flip-y, 1), var(--fa-flip-z, 0), var(--fa-flip-angle, -180deg)); + transform: rotate3d(var(--fa-flip-x, 0), var(--fa-flip-y, 1), var(--fa-flip-z, 0), var(--fa-flip-angle, -180deg)); } } + +@-webkit-keyframes fa-shake { + 0% { + -webkit-transform: rotate(-15deg); + transform: rotate(-15deg); } + 4% { + -webkit-transform: rotate(15deg); + transform: rotate(15deg); } + 8%, 24% { + -webkit-transform: rotate(-18deg); + transform: rotate(-18deg); } + 12%, 28% { + -webkit-transform: rotate(18deg); + transform: rotate(18deg); } + 16% { + -webkit-transform: rotate(-22deg); + transform: rotate(-22deg); } + 20% { + -webkit-transform: rotate(22deg); + transform: rotate(22deg); } + 32% { + -webkit-transform: rotate(-12deg); + transform: rotate(-12deg); } + 36% { + -webkit-transform: rotate(12deg); + transform: rotate(12deg); } + 40%, 100% { + -webkit-transform: rotate(0deg); + transform: rotate(0deg); } } + +@keyframes fa-shake { + 0% { + -webkit-transform: rotate(-15deg); + transform: rotate(-15deg); } + 4% { + -webkit-transform: rotate(15deg); + transform: rotate(15deg); } + 8%, 24% { + -webkit-transform: rotate(-18deg); + transform: rotate(-18deg); } + 12%, 28% { + -webkit-transform: rotate(18deg); + transform: rotate(18deg); } + 16% { + -webkit-transform: rotate(-22deg); + transform: rotate(-22deg); } + 20% { + -webkit-transform: rotate(22deg); + transform: rotate(22deg); } + 32% { + -webkit-transform: rotate(-12deg); + transform: rotate(-12deg); } + 36% { + -webkit-transform: rotate(12deg); + transform: rotate(12deg); } + 40%, 100% { + -webkit-transform: rotate(0deg); + transform: rotate(0deg); } } + +@-webkit-keyframes fa-spin { + 0% { + -webkit-transform: rotate(0deg); + transform: rotate(0deg); } + 100% { + -webkit-transform: rotate(360deg); + transform: rotate(360deg); } } + +@keyframes fa-spin { + 0% { + -webkit-transform: rotate(0deg); + transform: rotate(0deg); } + 100% { + -webkit-transform: rotate(360deg); + transform: rotate(360deg); } } + +.fa-rotate-90 { + -webkit-transform: rotate(90deg); + transform: rotate(90deg); } + +.fa-rotate-180 { + -webkit-transform: rotate(180deg); + transform: rotate(180deg); } + +.fa-rotate-270 { + -webkit-transform: rotate(270deg); + transform: rotate(270deg); } + +.fa-flip-horizontal { + -webkit-transform: scale(-1, 1); + transform: scale(-1, 1); } + +.fa-flip-vertical { + -webkit-transform: scale(1, -1); + transform: scale(1, -1); } + +.fa-flip-both, +.fa-flip-horizontal.fa-flip-vertical { + -webkit-transform: scale(-1, -1); + transform: scale(-1, -1); } + +.fa-rotate-by { + -webkit-transform: rotate(var(--fa-rotate-angle, none)); + transform: rotate(var(--fa-rotate-angle, none)); } + +.fa-stack { + display: inline-block; + height: 2em; + line-height: 2em; + position: relative; + vertical-align: middle; + width: 2.5em; } + +.fa-stack-1x, +.fa-stack-2x { + left: 0; + position: absolute; + text-align: center; + width: 100%; + z-index: var(--fa-stack-z-index, auto); } + +.fa-stack-1x { + line-height: inherit; } + +.fa-stack-2x { + font-size: 2em; } + +.fa-inverse { + color: var(--fa-inverse, #fff); } + +/* Font Awesome uses the Unicode Private Use Area (PUA) to ensure screen +readers do not read off random characters that represent icons */ + +.fa-0::before { + content: "\30"; } + +.fa-1::before { + content: "\31"; } + +.fa-2::before { + content: "\32"; } + +.fa-3::before { + content: "\33"; } + +.fa-4::before { + content: "\34"; } + +.fa-5::before { + content: "\35"; } + +.fa-6::before { + content: "\36"; } + +.fa-7::before { + content: "\37"; } + +.fa-8::before { + content: "\38"; } + +.fa-9::before { + content: "\39"; } + +.fa-fill-drip::before { + content: "\f576"; } + +.fa-arrows-to-circle::before { + content: "\e4bd"; } + +.fa-circle-chevron-right::before { + content: "\f138"; } + +.fa-chevron-circle-right::before { + content: "\f138"; } + +.fa-at::before { + content: "\40"; } + +.fa-trash-can::before { + content: "\f2ed"; } + +.fa-trash-alt::before { + content: "\f2ed"; } + +.fa-text-height::before { + content: "\f034"; } + +.fa-user-xmark::before { + content: "\f235"; } + +.fa-user-times::before { + content: "\f235"; } + +.fa-stethoscope::before { + content: "\f0f1"; } + +.fa-message::before { + content: "\f27a"; } + +.fa-comment-alt::before { + content: "\f27a"; } + +.fa-info::before { + content: "\f129"; } + +.fa-down-left-and-up-right-to-center::before { + content: "\f422"; } + +.fa-compress-alt::before { + content: "\f422"; } + +.fa-explosion::before { + content: "\e4e9"; } + +.fa-file-lines::before { + content: "\f15c"; } + +.fa-file-alt::before { + content: "\f15c"; } + +.fa-file-text::before { + content: "\f15c"; } + +.fa-wave-square::before { + content: "\f83e"; } + +.fa-ring::before { + content: "\f70b"; } + +.fa-building-un::before { + content: "\e4d9"; } + +.fa-dice-three::before { + content: "\f527"; } + +.fa-calendar-days::before { + content: "\f073"; } + +.fa-calendar-alt::before { + content: "\f073"; } + +.fa-anchor-circle-check::before { + content: "\e4aa"; } + +.fa-building-circle-arrow-right::before { + content: "\e4d1"; } + +.fa-volleyball::before { + content: "\f45f"; } + +.fa-volleyball-ball::before { + content: "\f45f"; } + +.fa-arrows-up-to-line::before { + content: "\e4c2"; } + +.fa-sort-down::before { + content: "\f0dd"; } + +.fa-sort-desc::before { + content: "\f0dd"; } + +.fa-circle-minus::before { + content: "\f056"; } + +.fa-minus-circle::before { + content: "\f056"; } + +.fa-door-open::before { + content: "\f52b"; } + +.fa-right-from-bracket::before { + content: "\f2f5"; } + +.fa-sign-out-alt::before { + content: "\f2f5"; } + +.fa-atom::before { + content: "\f5d2"; } + +.fa-soap::before { + content: "\e06e"; } + +.fa-icons::before { + content: "\f86d"; } + +.fa-heart-music-camera-bolt::before { + content: "\f86d"; } + +.fa-microphone-lines-slash::before { + content: "\f539"; } + +.fa-microphone-alt-slash::before { + content: "\f539"; } + +.fa-bridge-circle-check::before { + content: "\e4c9"; } + +.fa-pump-medical::before { + content: "\e06a"; } + +.fa-fingerprint::before { + content: "\f577"; } + +.fa-hand-point-right::before { + content: "\f0a4"; } + +.fa-magnifying-glass-location::before { + content: "\f689"; } + +.fa-search-location::before { + content: "\f689"; } + +.fa-forward-step::before { + content: "\f051"; } + +.fa-step-forward::before { + content: "\f051"; } + +.fa-face-smile-beam::before { + content: "\f5b8"; } + +.fa-smile-beam::before { + content: "\f5b8"; } + +.fa-flag-checkered::before { + content: "\f11e"; } + +.fa-football::before { + content: "\f44e"; } + +.fa-football-ball::before { + content: "\f44e"; } + +.fa-school-circle-exclamation::before { + content: "\e56c"; } + +.fa-crop::before { + content: "\f125"; } + +.fa-angles-down::before { + content: "\f103"; } + +.fa-angle-double-down::before { + content: "\f103"; } + +.fa-users-rectangle::before { + content: "\e594"; } + +.fa-people-roof::before { + content: "\e537"; } + +.fa-people-line::before { + content: "\e534"; } + +.fa-beer-mug-empty::before { + content: "\f0fc"; } + +.fa-beer::before { + content: "\f0fc"; } + +.fa-diagram-predecessor::before { + content: "\e477"; } + +.fa-arrow-up-long::before { + content: "\f176"; } + +.fa-long-arrow-up::before { + content: "\f176"; } + +.fa-fire-flame-simple::before { + content: "\f46a"; } + +.fa-burn::before { + content: "\f46a"; } + +.fa-person::before { + content: "\f183"; } + +.fa-male::before { + content: "\f183"; } + +.fa-laptop::before { + content: "\f109"; } + +.fa-file-csv::before { + content: "\f6dd"; } + +.fa-menorah::before { + content: "\f676"; } + +.fa-truck-plane::before { + content: "\e58f"; } + +.fa-record-vinyl::before { + content: "\f8d9"; } + +.fa-face-grin-stars::before { + content: "\f587"; } + +.fa-grin-stars::before { + content: "\f587"; } + +.fa-bong::before { + content: "\f55c"; } + +.fa-spaghetti-monster-flying::before { + content: "\f67b"; } + +.fa-pastafarianism::before { + content: "\f67b"; } + +.fa-arrow-down-up-across-line::before { + content: "\e4af"; } + +.fa-spoon::before { + content: "\f2e5"; } + +.fa-utensil-spoon::before { + content: "\f2e5"; } + +.fa-jar-wheat::before { + content: "\e517"; } + +.fa-envelopes-bulk::before { + content: "\f674"; } + +.fa-mail-bulk::before { + content: "\f674"; } + +.fa-file-circle-exclamation::before { + content: "\e4eb"; } + +.fa-circle-h::before { + content: "\f47e"; } + +.fa-hospital-symbol::before { + content: "\f47e"; } + +.fa-pager::before { + content: "\f815"; } + +.fa-address-book::before { + content: "\f2b9"; } + +.fa-contact-book::before { + content: "\f2b9"; } + +.fa-strikethrough::before { + content: "\f0cc"; } + +.fa-k::before { + content: "\4b"; } + +.fa-landmark-flag::before { + content: "\e51c"; } + +.fa-pencil::before { + content: "\f303"; } + +.fa-pencil-alt::before { + content: "\f303"; } + +.fa-backward::before { + content: "\f04a"; } + +.fa-caret-right::before { + content: "\f0da"; } + +.fa-comments::before { + content: "\f086"; } + +.fa-paste::before { + content: "\f0ea"; } + +.fa-file-clipboard::before { + content: "\f0ea"; } + +.fa-code-pull-request::before { + content: "\e13c"; } + +.fa-clipboard-list::before { + content: "\f46d"; } + +.fa-truck-ramp-box::before { + content: "\f4de"; } + +.fa-truck-loading::before { + content: "\f4de"; } + +.fa-user-check::before { + content: "\f4fc"; } + +.fa-vial-virus::before { + content: "\e597"; } + +.fa-sheet-plastic::before { + content: "\e571"; } + +.fa-blog::before { + content: "\f781"; } + +.fa-user-ninja::before { + content: "\f504"; } + +.fa-person-arrow-up-from-line::before { + content: "\e539"; } + +.fa-scroll-torah::before { + content: "\f6a0"; } + +.fa-torah::before { + content: "\f6a0"; } + +.fa-broom-ball::before { + content: "\f458"; } + +.fa-quidditch::before { + content: "\f458"; } + +.fa-quidditch-broom-ball::before { + content: "\f458"; } + +.fa-toggle-off::before { + content: "\f204"; } + +.fa-box-archive::before { + content: "\f187"; } + +.fa-archive::before { + content: "\f187"; } + +.fa-person-drowning::before { + content: "\e545"; } + +.fa-arrow-down-9-1::before { + content: "\f886"; } + +.fa-sort-numeric-desc::before { + content: "\f886"; } + +.fa-sort-numeric-down-alt::before { + content: "\f886"; } + +.fa-face-grin-tongue-squint::before { + content: "\f58a"; } + +.fa-grin-tongue-squint::before { + content: "\f58a"; } + +.fa-spray-can::before { + content: "\f5bd"; } + +.fa-truck-monster::before { + content: "\f63b"; } + +.fa-w::before { + content: "\57"; } + +.fa-earth-africa::before { + content: "\f57c"; } + +.fa-globe-africa::before { + content: "\f57c"; } + +.fa-rainbow::before { + content: "\f75b"; } + +.fa-circle-notch::before { + content: "\f1ce"; } + +.fa-tablet-screen-button::before { + content: "\f3fa"; } + +.fa-tablet-alt::before { + content: "\f3fa"; } + +.fa-paw::before { + content: "\f1b0"; } + +.fa-cloud::before { + content: "\f0c2"; } + +.fa-trowel-bricks::before { + content: "\e58a"; } + +.fa-face-flushed::before { + content: "\f579"; } + +.fa-flushed::before { + content: "\f579"; } + +.fa-hospital-user::before { + content: "\f80d"; } + +.fa-tent-arrow-left-right::before { + content: "\e57f"; } + +.fa-gavel::before { + content: "\f0e3"; } + +.fa-legal::before { + content: "\f0e3"; } + +.fa-binoculars::before { + content: "\f1e5"; } + +.fa-microphone-slash::before { + content: "\f131"; } + +.fa-box-tissue::before { + content: "\e05b"; } + +.fa-motorcycle::before { + content: "\f21c"; } + +.fa-bell-concierge::before { + content: "\f562"; } + +.fa-concierge-bell::before { + content: "\f562"; } + +.fa-pen-ruler::before { + content: "\f5ae"; } + +.fa-pencil-ruler::before { + content: "\f5ae"; } + +.fa-people-arrows::before { + content: "\e068"; } + +.fa-people-arrows-left-right::before { + content: "\e068"; } + +.fa-mars-and-venus-burst::before { + content: "\e523"; } + +.fa-square-caret-right::before { + content: "\f152"; } + +.fa-caret-square-right::before { + content: "\f152"; } + +.fa-scissors::before { + content: "\f0c4"; } + +.fa-cut::before { + content: "\f0c4"; } + +.fa-sun-plant-wilt::before { + content: "\e57a"; } + +.fa-toilets-portable::before { + content: "\e584"; } + +.fa-hockey-puck::before { + content: "\f453"; } + +.fa-table::before { + content: "\f0ce"; } + +.fa-magnifying-glass-arrow-right::before { + content: "\e521"; } + +.fa-tachograph-digital::before { + content: "\f566"; } + +.fa-digital-tachograph::before { + content: "\f566"; } + +.fa-users-slash::before { + content: "\e073"; } + +.fa-clover::before { + content: "\e139"; } + +.fa-reply::before { + content: "\f3e5"; } + +.fa-mail-reply::before { + content: "\f3e5"; } + +.fa-star-and-crescent::before { + content: "\f699"; } + +.fa-house-fire::before { + content: "\e50c"; } + +.fa-square-minus::before { + content: "\f146"; } + +.fa-minus-square::before { + content: "\f146"; } + +.fa-helicopter::before { + content: "\f533"; } + +.fa-compass::before { + content: "\f14e"; } + +.fa-square-caret-down::before { + content: "\f150"; } + +.fa-caret-square-down::before { + content: "\f150"; } + +.fa-file-circle-question::before { + content: "\e4ef"; } + +.fa-laptop-code::before { + content: "\f5fc"; } + +.fa-swatchbook::before { + content: "\f5c3"; } + +.fa-prescription-bottle::before { + content: "\f485"; } + +.fa-bars::before { + content: "\f0c9"; } + +.fa-navicon::before { + content: "\f0c9"; } + +.fa-people-group::before { + content: "\e533"; } + +.fa-hourglass-end::before { + content: "\f253"; } + +.fa-hourglass-3::before { + content: "\f253"; } + +.fa-heart-crack::before { + content: "\f7a9"; } + +.fa-heart-broken::before { + content: "\f7a9"; } + +.fa-square-up-right::before { + content: "\f360"; } + +.fa-external-link-square-alt::before { + content: "\f360"; } + +.fa-face-kiss-beam::before { + content: "\f597"; } + +.fa-kiss-beam::before { + content: "\f597"; } + +.fa-film::before { + content: "\f008"; } + +.fa-ruler-horizontal::before { + content: "\f547"; } + +.fa-people-robbery::before { + content: "\e536"; } + +.fa-lightbulb::before { + content: "\f0eb"; } + +.fa-caret-left::before { + content: "\f0d9"; } + +.fa-circle-exclamation::before { + content: "\f06a"; } + +.fa-exclamation-circle::before { + content: "\f06a"; } + +.fa-school-circle-xmark::before { + content: "\e56d"; } + +.fa-arrow-right-from-bracket::before { + content: "\f08b"; } + +.fa-sign-out::before { + content: "\f08b"; } + +.fa-circle-chevron-down::before { + content: "\f13a"; } + +.fa-chevron-circle-down::before { + content: "\f13a"; } + +.fa-unlock-keyhole::before { + content: "\f13e"; } + +.fa-unlock-alt::before { + content: "\f13e"; } + +.fa-cloud-showers-heavy::before { + content: "\f740"; } + +.fa-headphones-simple::before { + content: "\f58f"; } + +.fa-headphones-alt::before { + content: "\f58f"; } + +.fa-sitemap::before { + content: "\f0e8"; } + +.fa-circle-dollar-to-slot::before { + content: "\f4b9"; } + +.fa-donate::before { + content: "\f4b9"; } + +.fa-memory::before { + content: "\f538"; } + +.fa-road-spikes::before { + content: "\e568"; } + +.fa-fire-burner::before { + content: "\e4f1"; } + +.fa-flag::before { + content: "\f024"; } + +.fa-hanukiah::before { + content: "\f6e6"; } + +.fa-feather::before { + content: "\f52d"; } + +.fa-volume-low::before { + content: "\f027"; } + +.fa-volume-down::before { + content: "\f027"; } + +.fa-comment-slash::before { + content: "\f4b3"; } + +.fa-cloud-sun-rain::before { + content: "\f743"; } + +.fa-compress::before { + content: "\f066"; } + +.fa-wheat-awn::before { + content: "\e2cd"; } + +.fa-wheat-alt::before { + content: "\e2cd"; } + +.fa-ankh::before { + content: "\f644"; } + +.fa-hands-holding-child::before { + content: "\e4fa"; } + +.fa-asterisk::before { + content: "\2a"; } + +.fa-square-check::before { + content: "\f14a"; } + +.fa-check-square::before { + content: "\f14a"; } + +.fa-peseta-sign::before { + content: "\e221"; } + +.fa-heading::before { + content: "\f1dc"; } + +.fa-header::before { + content: "\f1dc"; } + +.fa-ghost::before { + content: "\f6e2"; } + +.fa-list::before { + content: "\f03a"; } + +.fa-list-squares::before { + content: "\f03a"; } + +.fa-square-phone-flip::before { + content: "\f87b"; } + +.fa-phone-square-alt::before { + content: "\f87b"; } + +.fa-cart-plus::before { + content: "\f217"; } + +.fa-gamepad::before { + content: "\f11b"; } + +.fa-circle-dot::before { + content: "\f192"; } + +.fa-dot-circle::before { + content: "\f192"; } + +.fa-face-dizzy::before { + content: "\f567"; } + +.fa-dizzy::before { + content: "\f567"; } + +.fa-egg::before { + content: "\f7fb"; } + +.fa-house-medical-circle-xmark::before { + content: "\e513"; } + +.fa-campground::before { + content: "\f6bb"; } + +.fa-folder-plus::before { + content: "\f65e"; } + +.fa-futbol::before { + content: "\f1e3"; } + +.fa-futbol-ball::before { + content: "\f1e3"; } + +.fa-soccer-ball::before { + content: "\f1e3"; } + +.fa-paintbrush::before { + content: "\f1fc"; } + +.fa-paint-brush::before { + content: "\f1fc"; } + +.fa-lock::before { + content: "\f023"; } + +.fa-gas-pump::before { + content: "\f52f"; } + +.fa-hot-tub-person::before { + content: "\f593"; } + +.fa-hot-tub::before { + content: "\f593"; } + +.fa-map-location::before { + content: "\f59f"; } + +.fa-map-marked::before { + content: "\f59f"; } + +.fa-house-flood-water::before { + content: "\e50e"; } + +.fa-tree::before { + content: "\f1bb"; } + +.fa-bridge-lock::before { + content: "\e4cc"; } + +.fa-sack-dollar::before { + content: "\f81d"; } + +.fa-pen-to-square::before { + content: "\f044"; } + +.fa-edit::before { + content: "\f044"; } + +.fa-car-side::before { + content: "\f5e4"; } + +.fa-share-nodes::before { + content: "\f1e0"; } + +.fa-share-alt::before { + content: "\f1e0"; } + +.fa-heart-circle-minus::before { + content: "\e4ff"; } + +.fa-hourglass-half::before { + content: "\f252"; } + +.fa-hourglass-2::before { + content: "\f252"; } + +.fa-microscope::before { + content: "\f610"; } + +.fa-sink::before { + content: "\e06d"; } + +.fa-bag-shopping::before { + content: "\f290"; } + +.fa-shopping-bag::before { + content: "\f290"; } + +.fa-arrow-down-z-a::before { + content: "\f881"; } + +.fa-sort-alpha-desc::before { + content: "\f881"; } + +.fa-sort-alpha-down-alt::before { + content: "\f881"; } + +.fa-mitten::before { + content: "\f7b5"; } + +.fa-person-rays::before { + content: "\e54d"; } + +.fa-users::before { + content: "\f0c0"; } + +.fa-eye-slash::before { + content: "\f070"; } + +.fa-flask-vial::before { + content: "\e4f3"; } + +.fa-hand::before { + content: "\f256"; } + +.fa-hand-paper::before { + content: "\f256"; } + +.fa-om::before { + content: "\f679"; } + +.fa-worm::before { + content: "\e599"; } + +.fa-house-circle-xmark::before { + content: "\e50b"; } + +.fa-plug::before { + content: "\f1e6"; } + +.fa-chevron-up::before { + content: "\f077"; } + +.fa-hand-spock::before { + content: "\f259"; } + +.fa-stopwatch::before { + content: "\f2f2"; } + +.fa-face-kiss::before { + content: "\f596"; } + +.fa-kiss::before { + content: "\f596"; } + +.fa-bridge-circle-xmark::before { + content: "\e4cb"; } + +.fa-face-grin-tongue::before { + content: "\f589"; } + +.fa-grin-tongue::before { + content: "\f589"; } + +.fa-chess-bishop::before { + content: "\f43a"; } + +.fa-face-grin-wink::before { + content: "\f58c"; } + +.fa-grin-wink::before { + content: "\f58c"; } + +.fa-ear-deaf::before { + content: "\f2a4"; } + +.fa-deaf::before { + content: "\f2a4"; } + +.fa-deafness::before { + content: "\f2a4"; } + +.fa-hard-of-hearing::before { + content: "\f2a4"; } + 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{ + content: "\e579"; } + +.fa-staff-aesculapius::before { + content: "\e579"; } + +.fa-head-side-cough-slash::before { + content: "\e062"; } + +.fa-truck-medical::before { + content: "\f0f9"; } + +.fa-ambulance::before { + content: "\f0f9"; } + +.fa-wheat-awn-circle-exclamation::before { + content: "\e598"; } + +.fa-snowman::before { + content: "\f7d0"; } + +.fa-mortar-pestle::before { + content: "\f5a7"; } + +.fa-road-barrier::before { + content: "\e562"; } + +.fa-school::before { + content: "\f549"; } + +.fa-igloo::before { + content: "\f7ae"; } + +.fa-joint::before { + content: "\f595"; } + +.fa-angle-right::before { + content: "\f105"; } + +.fa-horse::before { + content: "\f6f0"; } + +.fa-q::before { + content: "\51"; } + +.fa-g::before { + content: "\47"; } + +.fa-notes-medical::before { + content: "\f481"; } + +.fa-temperature-half::before { + content: "\f2c9"; } + +.fa-temperature-2::before { + content: "\f2c9"; } + +.fa-thermometer-2::before { + content: "\f2c9"; } + 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content: "\f24d"; } + +.fa-person-walking-arrow-loop-left::before { + content: "\e551"; } + +.fa-arrow-up-z-a::before { + content: "\f882"; } + +.fa-sort-alpha-up-alt::before { + content: "\f882"; } + +.fa-fire-flame-curved::before { + content: "\f7e4"; } + +.fa-fire-alt::before { + content: "\f7e4"; } + +.fa-tornado::before { + content: "\f76f"; } + +.fa-file-circle-plus::before { + content: "\e494"; } + +.fa-book-quran::before { + content: "\f687"; } + +.fa-quran::before { + content: "\f687"; } + +.fa-anchor::before { + content: "\f13d"; } + +.fa-border-all::before { + content: "\f84c"; } + +.fa-face-angry::before { + content: "\f556"; } + +.fa-angry::before { + content: "\f556"; } + +.fa-cookie-bite::before { + content: "\f564"; } + +.fa-arrow-trend-down::before { + content: "\e097"; } + +.fa-rss::before { + content: "\f09e"; } + +.fa-feed::before { + content: "\f09e"; } + +.fa-draw-polygon::before { + content: "\f5ee"; } + +.fa-scale-balanced::before { + content: "\f24e"; } + +.fa-balance-scale::before { + content: "\f24e"; } + +.fa-gauge-simple-high::before { + content: "\f62a"; } + +.fa-tachometer::before { + content: "\f62a"; } + +.fa-tachometer-fast::before { + content: "\f62a"; } + +.fa-shower::before { + content: "\f2cc"; } + +.fa-desktop::before { + content: "\f390"; } + +.fa-desktop-alt::before { + content: "\f390"; } + +.fa-m::before { + content: "\4d"; } + +.fa-table-list::before { + content: "\f00b"; } + +.fa-th-list::before { + content: "\f00b"; } + +.fa-comment-sms::before { + content: "\f7cd"; } + +.fa-sms::before { + content: "\f7cd"; } + +.fa-book::before { + content: "\f02d"; } + +.fa-user-plus::before { + content: "\f234"; } + +.fa-check::before { + content: "\f00c"; } + +.fa-battery-three-quarters::before { + content: "\f241"; } + +.fa-battery-4::before { + content: "\f241"; } + +.fa-house-circle-check::before { + content: "\e509"; } + +.fa-angle-left::before { + content: "\f104"; } + +.fa-diagram-successor::before { + content: "\e47a"; } + +.fa-truck-arrow-right::before { + content: "\e58b"; } + +.fa-arrows-split-up-and-left::before { + content: "\e4bc"; } + +.fa-hand-fist::before { + content: "\f6de"; } + +.fa-fist-raised::before { + content: "\f6de"; } + +.fa-cloud-moon::before { + content: "\f6c3"; } + +.fa-briefcase::before { + content: "\f0b1"; } + +.fa-person-falling::before { + content: "\e546"; } + +.fa-image-portrait::before { + content: "\f3e0"; } + +.fa-portrait::before { + content: "\f3e0"; } + +.fa-user-tag::before { + content: "\f507"; } + +.fa-rug::before { + content: "\e569"; } + +.fa-earth-europe::before { + content: "\f7a2"; } + +.fa-globe-europe::before { + content: "\f7a2"; } + +.fa-cart-flatbed-suitcase::before { + content: "\f59d"; } + +.fa-luggage-cart::before { + content: "\f59d"; } + +.fa-rectangle-xmark::before { + content: "\f410"; } + +.fa-rectangle-times::before { + content: "\f410"; } + +.fa-times-rectangle::before { + content: "\f410"; } + +.fa-window-close::before { + content: "\f410"; } + +.fa-baht-sign::before { + content: "\e0ac"; } + +.fa-book-open::before { + content: "\f518"; } + +.fa-book-journal-whills::before { + content: "\f66a"; } + +.fa-journal-whills::before { + content: "\f66a"; } + +.fa-handcuffs::before { + content: "\e4f8"; } + +.fa-triangle-exclamation::before { + content: "\f071"; } + +.fa-exclamation-triangle::before { + content: "\f071"; } + +.fa-warning::before { + content: "\f071"; } + +.fa-database::before { + content: "\f1c0"; } + +.fa-share::before { + content: "\f064"; } + +.fa-arrow-turn-right::before { + content: "\f064"; } + +.fa-mail-forward::before { + content: "\f064"; } + +.fa-bottle-droplet::before { + content: "\e4c4"; } + +.fa-mask-face::before { + content: "\e1d7"; } + +.fa-hill-rockslide::before { + content: "\e508"; } + +.fa-right-left::before { + content: "\f362"; } + +.fa-exchange-alt::before { + content: "\f362"; } + +.fa-paper-plane::before { + content: "\f1d8"; } + +.fa-road-circle-exclamation::before { + content: "\e565"; } + +.fa-dungeon::before { + content: "\f6d9"; } + +.fa-align-right::before { + content: "\f038"; } + +.fa-money-bill-1-wave::before { + content: "\f53b"; } + +.fa-money-bill-wave-alt::before { + content: "\f53b"; } + +.fa-life-ring::before { + content: "\f1cd"; } + +.fa-hands::before { + content: "\f2a7"; } + +.fa-sign-language::before { + content: "\f2a7"; } + +.fa-signing::before { + content: "\f2a7"; } + +.fa-calendar-day::before { + content: "\f783"; } + +.fa-water-ladder::before { + content: "\f5c5"; } + +.fa-ladder-water::before { + content: "\f5c5"; } + +.fa-swimming-pool::before { + content: "\f5c5"; } + +.fa-arrows-up-down::before { + content: "\f07d"; } + +.fa-arrows-v::before { + content: "\f07d"; } + +.fa-face-grimace::before { + content: "\f57f"; } + +.fa-grimace::before { + content: "\f57f"; } + +.fa-wheelchair-move::before { + content: "\e2ce"; } + +.fa-wheelchair-alt::before { + content: "\e2ce"; } + +.fa-turn-down::before { + content: "\f3be"; } + +.fa-level-down-alt::before { + content: "\f3be"; } + +.fa-person-walking-arrow-right::before { + content: "\e552"; } + +.fa-square-envelope::before { + content: "\f199"; } + +.fa-envelope-square::before { + content: "\f199"; } + +.fa-dice::before { + content: "\f522"; } + +.fa-bowling-ball::before { + content: "\f436"; } + +.fa-brain::before { + content: "\f5dc"; } + +.fa-bandage::before { + content: "\f462"; } + +.fa-band-aid::before { + content: "\f462"; } + +.fa-calendar-minus::before { + content: "\f272"; } + +.fa-circle-xmark::before { + content: "\f057"; } + +.fa-times-circle::before { + content: "\f057"; } + +.fa-xmark-circle::before { + content: "\f057"; } + +.fa-gifts::before { + content: "\f79c"; } + +.fa-hotel::before { + content: "\f594"; } + +.fa-earth-asia::before { + content: "\f57e"; } + +.fa-globe-asia::before { + content: "\f57e"; } + +.fa-id-card-clip::before { + content: "\f47f"; } + +.fa-id-card-alt::before { + content: "\f47f"; } + 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"\f48e"; } + +.fa-cloud-sun::before { + content: "\f6c4"; } + +.fa-stopwatch-20::before { + content: "\e06f"; } + +.fa-square-full::before { + content: "\f45c"; } + +.fa-magnet::before { + content: "\f076"; } + +.fa-jar::before { + content: "\e516"; } + +.fa-note-sticky::before { + content: "\f249"; } + +.fa-sticky-note::before { + content: "\f249"; } + +.fa-bug-slash::before { + content: "\e490"; } + +.fa-arrow-up-from-water-pump::before { + content: "\e4b6"; } + +.fa-bone::before { + content: "\f5d7"; } + +.fa-user-injured::before { + content: "\f728"; } + +.fa-face-sad-tear::before { + content: "\f5b4"; } + +.fa-sad-tear::before { + content: "\f5b4"; } + +.fa-plane::before { + content: "\f072"; } + +.fa-tent-arrows-down::before { + content: "\e581"; } + +.fa-exclamation::before { + content: "\21"; } + +.fa-arrows-spin::before { + content: "\e4bb"; } + +.fa-print::before { + content: "\f02f"; } + +.fa-turkish-lira-sign::before { + content: "\e2bb"; } + +.fa-try::before { + content: "\e2bb"; } + +.fa-turkish-lira::before { + content: "\e2bb"; } + +.fa-dollar-sign::before { + content: "\24"; } + +.fa-dollar::before { + content: "\24"; } + +.fa-usd::before { + content: "\24"; } + +.fa-x::before { + content: "\58"; } + +.fa-magnifying-glass-dollar::before { + content: "\f688"; } + +.fa-search-dollar::before { + content: "\f688"; } + +.fa-users-gear::before { + content: "\f509"; } + +.fa-users-cog::before { + content: "\f509"; } + +.fa-person-military-pointing::before { + content: "\e54a"; } + +.fa-building-columns::before { + content: "\f19c"; } + +.fa-bank::before { + content: "\f19c"; } + +.fa-institution::before { + content: "\f19c"; } + +.fa-museum::before { + content: "\f19c"; } + +.fa-university::before { + content: "\f19c"; } + +.fa-umbrella::before { + content: "\f0e9"; } + +.fa-trowel::before { + content: "\e589"; } + +.fa-d::before { + content: "\44"; } + +.fa-stapler::before { + content: "\e5af"; } + +.fa-masks-theater::before { + content: "\f630"; } + +.fa-theater-masks::before { + content: "\f630"; } + +.fa-kip-sign::before { + content: "\e1c4"; } + +.fa-hand-point-left::before { + content: "\f0a5"; } + +.fa-handshake-simple::before { + content: "\f4c6"; } + +.fa-handshake-alt::before { + content: "\f4c6"; } + +.fa-jet-fighter::before { + content: "\f0fb"; } + +.fa-fighter-jet::before { + content: "\f0fb"; } + +.fa-square-share-nodes::before { + content: "\f1e1"; } + +.fa-share-alt-square::before { + content: "\f1e1"; } + +.fa-barcode::before { + content: "\f02a"; } + +.fa-plus-minus::before { + content: "\e43c"; } + +.fa-video::before { + content: "\f03d"; } + +.fa-video-camera::before { + content: "\f03d"; } + +.fa-graduation-cap::before { + content: "\f19d"; } + +.fa-mortar-board::before { + content: "\f19d"; } + +.fa-hand-holding-medical::before { + content: "\e05c"; } + +.fa-person-circle-check::before { + content: "\e53e"; } + +.fa-turn-up::before { + content: "\f3bf"; } + +.fa-level-up-alt::before { + content: "\f3bf"; } + +.sr-only, +.fa-sr-only { + position: absolute; + width: 1px; + height: 1px; + padding: 0; + margin: -1px; + overflow: hidden; + clip: rect(0, 0, 0, 0); + white-space: nowrap; + border-width: 0; } + +.sr-only-focusable:not(:focus), +.fa-sr-only-focusable:not(:focus) { + position: absolute; + width: 1px; + height: 1px; + padding: 0; + margin: -1px; + overflow: hidden; + clip: rect(0, 0, 0, 0); + white-space: nowrap; + border-width: 0; } +:root, :host { + --fa-style-family-brands: 'Font Awesome 6 Brands'; + --fa-font-brands: normal 400 1em/1 'Font Awesome 6 Brands'; } + +@font-face { + font-family: 'Font Awesome 6 Brands'; + font-style: normal; + font-weight: 400; + font-display: block; + src: url("../webfonts/FontAwesome6Brands-Regular-400.woff2") format("woff2"), url("../webfonts/FontAwesome6Brands-Regular-400.ttf") format("truetype"); } + +.fab, +.fa-brands { + font-weight: 400; } + +.fa-monero:before { + content: "\f3d0"; } + +.fa-hooli:before { + content: "\f427"; } + +.fa-yelp:before { + content: "\f1e9"; } + +.fa-cc-visa:before { + content: "\f1f0"; } + +.fa-lastfm:before { + content: "\f202"; } + +.fa-shopware:before { + content: "\f5b5"; } + +.fa-creative-commons-nc:before { + content: "\f4e8"; } + +.fa-aws:before { + content: "\f375"; } + +.fa-redhat:before { + content: "\f7bc"; } + +.fa-yoast:before { + content: "\f2b1"; } + +.fa-cloudflare:before { + content: "\e07d"; } + +.fa-ups:before { + content: "\f7e0"; } + +.fa-wpexplorer:before { + content: "\f2de"; } + +.fa-dyalog:before { + content: "\f399"; } + +.fa-bity:before { + content: "\f37a"; } + +.fa-stackpath:before { + content: "\f842"; } + +.fa-buysellads:before { + content: "\f20d"; } + +.fa-first-order:before { + content: "\f2b0"; } + +.fa-modx:before { + content: "\f285"; } + +.fa-guilded:before { + content: "\e07e"; } + +.fa-vnv:before { + content: "\f40b"; } + +.fa-square-js:before { + content: "\f3b9"; } + +.fa-js-square:before { + content: "\f3b9"; } + +.fa-microsoft:before { + content: "\f3ca"; } + +.fa-qq:before { + content: "\f1d6"; } + +.fa-orcid:before { + content: "\f8d2"; } + +.fa-java:before { + content: "\f4e4"; } + +.fa-invision:before { + content: "\f7b0"; } + +.fa-creative-commons-pd-alt:before { + content: "\f4ed"; } + +.fa-centercode:before { + content: "\f380"; } + +.fa-glide-g:before { + content: "\f2a6"; } + +.fa-drupal:before { + content: "\f1a9"; } + +.fa-hire-a-helper:before { + content: "\f3b0"; } + +.fa-creative-commons-by:before { + content: "\f4e7"; } + +.fa-unity:before { + content: "\e049"; } + +.fa-whmcs:before { + content: "\f40d"; } + +.fa-rocketchat:before { + content: "\f3e8"; } + +.fa-vk:before { + content: "\f189"; } + +.fa-untappd:before { + content: "\f405"; } + +.fa-mailchimp:before { + content: "\f59e"; } + +.fa-css3-alt:before { + content: "\f38b"; } + +.fa-square-reddit:before { + content: "\f1a2"; } + +.fa-reddit-square:before { + content: "\f1a2"; } + +.fa-vimeo-v:before { + content: "\f27d"; } + +.fa-contao:before { + content: "\f26d"; } + +.fa-square-font-awesome:before { + content: "\e5ad"; } + +.fa-deskpro:before { + content: "\f38f"; } + +.fa-sistrix:before { + content: "\f3ee"; } + +.fa-square-instagram:before { + content: "\e055"; } + +.fa-instagram-square:before { + content: "\e055"; } + +.fa-battle-net:before { + content: "\f835"; } + +.fa-the-red-yeti:before { + content: "\f69d"; } + +.fa-square-hacker-news:before { + content: "\f3af"; } + +.fa-hacker-news-square:before { + content: "\f3af"; } + +.fa-edge:before { + content: "\f282"; } + +.fa-threads:before { + content: "\e618"; } + +.fa-napster:before { + content: "\f3d2"; } + +.fa-square-snapchat:before { + content: "\f2ad"; } + +.fa-snapchat-square:before { + content: "\f2ad"; } + +.fa-google-plus-g:before { + content: "\f0d5"; } + +.fa-artstation:before { + content: "\f77a"; } + +.fa-markdown:before { + content: "\f60f"; } + +.fa-sourcetree:before { + content: "\f7d3"; } + +.fa-google-plus:before { + content: "\f2b3"; } + +.fa-diaspora:before { + content: "\f791"; } + +.fa-foursquare:before { + content: "\f180"; } + +.fa-stack-overflow:before { + content: "\f16c"; } + +.fa-github-alt:before { + content: "\f113"; } + +.fa-phoenix-squadron:before { + content: "\f511"; } + +.fa-pagelines:before { + content: "\f18c"; } + +.fa-algolia:before { + content: "\f36c"; } + +.fa-red-river:before { + content: "\f3e3"; } + +.fa-creative-commons-sa:before { + content: "\f4ef"; } + +.fa-safari:before { + content: "\f267"; } + +.fa-google:before { + content: "\f1a0"; } + +.fa-square-font-awesome-stroke:before { + content: "\f35c"; } + +.fa-font-awesome-alt:before { + content: "\f35c"; } + +.fa-atlassian:before { + content: "\f77b"; } + +.fa-linkedin-in:before { + content: "\f0e1"; } + +.fa-digital-ocean:before { + content: "\f391"; } + +.fa-nimblr:before { + content: "\f5a8"; } + +.fa-chromecast:before { + content: "\f838"; } + +.fa-evernote:before { + content: "\f839"; } + +.fa-hacker-news:before { + content: "\f1d4"; } + +.fa-creative-commons-sampling:before { + content: "\f4f0"; } + +.fa-adversal:before { + content: "\f36a"; } + +.fa-creative-commons:before { + content: "\f25e"; } + +.fa-watchman-monitoring:before { + content: "\e087"; } + +.fa-fonticons:before { + content: "\f280"; } + +.fa-weixin:before { + content: "\f1d7"; } + +.fa-shirtsinbulk:before { + content: "\f214"; } + +.fa-codepen:before { + content: "\f1cb"; } + +.fa-git-alt:before { + content: "\f841"; } + +.fa-lyft:before { + content: "\f3c3"; } + +.fa-rev:before { + content: "\f5b2"; } + +.fa-windows:before { + content: "\f17a"; } + +.fa-wizards-of-the-coast:before { + content: "\f730"; } + +.fa-square-viadeo:before { + content: "\f2aa"; } + +.fa-viadeo-square:before { + content: "\f2aa"; } + +.fa-meetup:before { + content: "\f2e0"; } + +.fa-centos:before { + content: "\f789"; } + +.fa-adn:before { + content: "\f170"; } + +.fa-cloudsmith:before { + content: "\f384"; } + +.fa-pied-piper-alt:before { + content: "\f1a8"; } + 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+.fa-wordpress-simple:before { + content: "\f411"; } + +.fa-nutritionix:before { + content: "\f3d6"; } + +.fa-wodu:before { + content: "\e088"; } + +.fa-google-pay:before { + content: "\e079"; } + +.fa-intercom:before { + content: "\f7af"; } + +.fa-zhihu:before { + content: "\f63f"; } + +.fa-korvue:before { + content: "\f42f"; } + +.fa-pix:before { + content: "\e43a"; } + +.fa-steam-symbol:before { + content: "\f3f6"; } +:root, :host { + --fa-style-family-classic: 'Font Awesome 6 Free'; + --fa-font-regular: normal 400 1em/1 'Font Awesome 6 Free'; } + +@font-face { + font-family: 'Font Awesome 6 Free'; + font-style: normal; + font-weight: 400; + font-display: block; + src: url("../webfonts/FontAwesome6Free-Regular-400.woff2") format("woff2"), url("../webfonts/FontAwesome6Free-Regular-400.ttf") format("truetype"); } + +.far, +.fa-regular { + font-weight: 400; } +:root, :host { + --fa-style-family-classic: 'Font Awesome 6 Free'; + --fa-font-solid: normal 900 1em/1 'Font Awesome 6 Free'; } + +@font-face { + font-family: 'Font Awesome 6 Free'; + font-style: normal; + font-weight: 900; + font-display: block; + src: url("../webfonts/FontAwesome6Free-Solid-900.woff2") format("woff2"), url("../webfonts/FontAwesome6Free-Solid-900.ttf") format("truetype"); } + +.fas, +.fa-solid { + font-weight: 900; } +@font-face { + font-family: 'Font Awesome 6 Brands'; + font-display: block; + font-weight: 400; + src: url("../webfonts/FontAwesome6Brands-Regular-400.woff2") format("woff2"), url("../webfonts/FontAwesome6Brands-Regular-400.ttf") format("truetype"); } + +@font-face { + font-family: 'Font Awesome 6 Free'; + font-display: block; + font-weight: 900; + src: url("../webfonts/FontAwesome6Free-Solid-900.woff2") format("woff2"), url("../webfonts/FontAwesome6Free-Solid-900.ttf") format("truetype"); } + +@font-face { + font-family: 'Font Awesome 6 Free'; + font-display: block; + font-weight: 400; + src: url("../webfonts/../webfonts/FontAwesome6Free-Regular-400.woff2") format("woff2"), url("../webfonts/FontAwesome6Free-Regular-400.ttf") format("truetype"); } +@font-face { + font-family: 'Font Awesome 6 Free'; + font-display: block; + src: url("../webfonts/FontAwesome6Free-Solid-900.woff2") format("woff2"), url("../webfonts/FontAwesome6Free-Solid-900.ttf") format("truetype"); } + +@font-face { + font-family: 'Font Awesome 6 Brands'; + font-display: block; + src: url("../webfonts/FontAwesome6Brands-Regular-400.woff2") format("woff2"), url("../webfonts/FontAwesome6Brands-Regular-400.ttf") format("truetype"); } + +@font-face { + font-family: 'Font Awesome 6 Free'; + font-display: block; + src: url("../webfonts/FontAwesome6Free-Regular-400.woff2") format("woff2"), url("../../webfonts/FontAwesome6Free-Regular-400.ttf") format("truetype"); + unicode-range: 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version = '0.1.0', + stylesheets = {'assets/css/all.css', 'assets/css/latex-fontsize.css'} + }) +end + +local function isEmpty(s) + return s == nil or s == '' +end + +local function isValidSize(size) + local validSizes = { + "tiny", + "scriptsize", + "footnotesize", + "small", + "normalsize", + "large", + "Large", + "LARGE", + "huge", + "Huge" + } + for _, v in ipairs(validSizes) do + if v == size then + return size + end + end + return "" +end + +return { + ["fa"] = function(args, kwargs) + + local group = "solid" + local icon = pandoc.utils.stringify(args[1]) + if #args > 1 then + group = icon + icon = pandoc.utils.stringify(args[2]) + end + + local title = pandoc.utils.stringify(kwargs["title"]) + if not isEmpty(title) then + title = " title=\"" .. title .. "\"" + end + + local label = pandoc.utils.stringify(kwargs["label"]) + if isEmpty(label) then + label = " aria-label=\"" .. icon .. "\"" + else + label = " aria-label=\"" .. label .. "\"" + end + + local size = pandoc.utils.stringify(kwargs["size"]) + + -- detect html (excluding epub which won't handle fa) + if quarto.doc.is_format("html:js") then + ensureHtmlDeps() + if not isEmpty(size) then + size = " fa-" .. size + end + return pandoc.RawInline( + 'html', + "" + ) + -- detect pdf / beamer / latex / etc + elseif quarto.doc.is_format("pdf") then + ensureLatexDeps() + if isEmpty(isValidSize(size)) then + return pandoc.RawInline('tex', "\\faIcon{" .. icon .. "}") + else + return pandoc.RawInline('tex', "{\\" .. size .. "\\faIcon{" .. icon .. 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wWC%4XOk5W@PXF&i|GNeMdkX$PO@z+p7e*KGQg7K`j7Y#sURqhI5^5Oyf9Ae48vp + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + diff --git a/07-parallel-computing/references.bib b/07-parallel-computing/references.bib new file mode 100644 index 0000000..1d57c0c --- /dev/null +++ b/07-parallel-computing/references.bib @@ -0,0 +1,106 @@ +@article{knuth84, + author = {Knuth, Donald E.}, + title = {Literate Programming}, + year = {1984}, + issue_date = {May 1984}, + publisher = {Oxford University Press, Inc.}, + address = {USA}, + volume = {27}, + number = {2}, + issn = {0010-4620}, + url = {https://doi.org/10.1093/comjnl/27.2.97}, + doi = {10.1093/comjnl/27.2.97}, + journal = {Comput. J.}, + month = may, + pages = {97–111}, + numpages = {15} +} + +@Manual{R, + title = {R: A Language and Environment for Statistical Computing}, + author = {{R Core Team}}, + organization = {R Foundation for Statistical Computing}, + address = {Vienna, Austria}, + year = {2023}, + url = {https://www.R-project.org/}, +} + +@Article{slurmRpaper, + title = {slurmR: A lightweight wrapper for HPC with Slurm}, + author = {George {Vega Yon} and Paul Marjoram}, + journal = {The Journal of Open Source Software}, + year = {2019}, + month = {jul}, + volume = {4}, + number = {39}, + doi = {10.21105/joss.01493}, + url = {https://doi.org/10.21105/joss.01493}, +} + +@Manual{slurmRpkg, + title = {{{slurmR: A Lightweight Wrapper for 'Slurm'}}}, + author = {George {Vega Yon} and Paul Marjoram}, + year = {2022}, + note = {R package version 0.5-2}, + url = {https://github.com/USCbiostats/slurmR}, +} + + @article{Vega_Yon_Quistorff_2019, title={parallel: A command for parallel computing}, volume={19}, ISSN={1536-867X}, DOI={10.1177/1536867X19874242}, abstractNote={The parallel package allows parallel processing of tasks that are not interdependent. This allows all flavors of Stata to take advantage of multiprocessor machines. Even Stata/MP users can benefit because many community-contributed programs are not automatically parallelized but could be under our framework.}, number={3}, journal={The Stata Journal: Promoting communications on statistics and Stata}, author={Vega Yon, George G. Vega and Quistorff, Brian}, year={2019}, month={Sep}, pages={667–684} } + +@Manual{datatable, + title = {data.table: Extension of `data.frame`}, + author = {Matt Dowle and Arun Srinivasan}, + year = {2021}, + note = {R package version 1.14.2}, + url = {https://CRAN.R-project.org/package=data.table} + } + + @misc{ + lumi2023, + title={Documentation - Distribution and binding}, url={https://docs.lumi-supercomputer.eu/runjobs/scheduled-jobs/distribution-binding/}, abstractNote={Instructions, user guides and tutorials for the use of the LUMI supercomputer}, author={{LUMI consortium}}, language={en}, year=2023} + + @inbook{Yoo_Jette_Grondona_2003, address={Berlin, Heidelberg}, series={Lecture Notes in Computer Science}, title={SLURM: Simple Linux Utility for Resource Management}, volume={2862}, ISBN={978-3-540-20405-3}, url={http://link.springer.com/10.1007/10968987_3}, DOI={10.1007/10968987_3}, booktitle={Job Scheduling Strategies for Parallel Processing}, publisher={Springer Berlin Heidelberg}, author={Yoo, Andy B. and Jette, Morris A. and Grondona, Mark}, editor={Feitelson, Dror and Rudolph, Larry and Schwiegelshohn, Uwe}, year={2003}, pages={44–60}, collection={Lecture Notes in Computer Science} } + + +@Article{Eddelbuettel2011, + title = {{Rcpp}: Seamless {R} and {C++} Integration}, + author = {Dirk Eddelbuettel and Romain Fran\c{c}ois}, + journal = {Journal of Statistical Software}, + year = {2011}, + volume = {40}, + number = {8}, + pages = {1--18}, + doi = {10.18637/jss.v040.i08}, +} + +@Book{RcppBook, + title = {Seamless {R} and {C++} Integration with {Rcpp}}, + author = {Dirk Eddelbuettel}, + publisher = {Springer}, + address = {New York}, + year = {2013}, + note = {ISBN 978-1-4614-6867-7}, + doi = {10.1007/978-1-4614-6868-4}, +} + +@Article{Rcpp, + title = {{Extending extit{R} with extit{C++}: A Brief Introduction to extit{Rcpp}}}, + author = {Dirk Eddelbuettel and James Joseph Balamuta}, + journal = {The American Statistician}, + year = {2018}, + volume = {72}, + number = {1}, + pages = {28-36}, + doi = {10.1080/00031305.2017.1375990}, +} + + +@book{rue-albrechtWelcomeBioconductorPackages, + title = {Welcome | {{Bioconductor Packages}}: {{Development}}, {{Maintenance}}, and {{Peer Review}}}, + shorttitle = {Welcome | {{Bioconductor Packages}}}, + author = {{Rue-Albrecht}, Kevin and Cassol, Daniela and Rainer, Johannes and Shepherd, Lori}, + urldate = {2023-05-02}, + abstract = {This is a minimal example of using the bookdown package to write a book. The output format for this example is bookdown::gitbook.}, + langid = {english}, + file = {/home/george/Zotero/storage/RPSRJTPT/index.html} +} \ No newline at end of file diff --git a/07-parallel-computing/slides.html b/07-parallel-computing/slides.html new file mode 100644 index 0000000..12ee43a --- /dev/null +++ b/07-parallel-computing/slides.html @@ -0,0 +1,15855 @@ + + + + + + + + + + + + + + + + + A Brief Introduction to High-Performance Computing + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/07-parallel-computing/slides.qmd b/07-parallel-computing/slides.qmd index 28046a8..658c6e7 100644 --- a/07-parallel-computing/slides.qmd +++ b/07-parallel-computing/slides.qmd @@ -1,32 +1,42 @@ --- -title: 'Parallel Computing' -author: 'Jonathan Chipman with content provided by George Vega Yon' -format: - html: - toc: TRUE - toc-depth: 2 - toc-location: left - highlight: pygments - font_adjustment: -1 - css: styles.css - # code-fold: true - code-tools: true - smooth-scroll: true - embed-resources: true +title: A Brief Introduction to High-Performance Computing +subtitle: "PHS 7045: Advanced Programming" +author: George G. Vega Yon, Ph.D. +institute: University of Utah, EEUU +date: 2024-10-01 +format: + revealjs: + slide-number: true + code-copy: true + smaller: true + fig-format: svg + fig-dpi: 200 + fig-asp: 1 + css: ["default", "style.css"] + footer: George G. Vega Yon, Ph.D. -- [ggv.cl/slides/exeter-hpc-2024](https://ggv.cl/slides/hpc-exeter-sep2024) + title-slide-attributes: + data-background-image: 'fig/Zoom-Background_PrideU.jpg' + data-background-opacity: '0.2' + data-background-size: 'contain' +bibliography: references.bib +embed-resources: true --- +## Before we start -# Introduction +You can access the content here: -Today's content has been drawn from: +- Slides: +- Code: +- posit.cloud: + +# Fundamentals {background-color="#515A5A"} + +![](https://raw.githubusercontent.com/USCbiostats/hpc-with-r/df60e1cfdc0f848f4f0de5a0aa7d0833f4cfe3d5/fig/bulldog-teaches-baby-crawl.gif){fig-align="center"} - 1. [https://github.com/gvegayon/ocrug-hpc-august2019](https://github.com/gvegayon/ocrug-hpc-august2019) - - 2. Roger Peng's book: [R Programming for data science, Ch 22](https://bookdown.org/rdpeng/rprogdatascience/parallel-computation.html) - - 3. Norman Matloff's book: [The Art of R Programming, Ch 16](https://diytranscriptomics.com/Reading/files/The%20Art%20of%20R%20Programming.pdf#page=311) -# High-Performance Computing: An overview + +## High-Performance Computing: An overview Loosely, from R's perspective, we can think of HPC in terms of two, maybe three things: @@ -34,73 +44,251 @@ Loosely, from R's perspective, we can think of HPC in terms of two, maybe three 2. Parallel computing: How to take advantage of multiple core systems -3. Compiled code: Write your own low-level code (if R doesn't have it yet...) +3. Compiled code: Write your own low-level code (if R doesn't has it yet...) (Checkout [CRAN Task View on HPC](https://cran.r-project.org/web/views/HighPerformanceComputing.html)) -## Some vocabulary for HPC +## Some vocabulary for HPC -In raw terms -* Supercomputer: A **single** big machine with thousands of cores/gpus. +::: {.columns layout-align="center"} +::: {.column width="33%"} +High Throughput Computing Cluster (HTC Cluster) -* High Performance Computing (HPC): **Multiple** machines within - a **single** network. - -* High Throughput Computing (HTC): **Multiple** machines across **multiple** - networks. - -You may not have access to a supercomputer, but certainly HPC/HTC clusters are -more accessible these days, e.g. AWS provides a service to create HPC clusters -at a low cost +![Open Science Grid Consortium (OSG) ](fig/OSG_Map.png){width=80%} +::: +::: {.column width="33%"} +Supercomputer (HPC Cluster) -## What's "a core"? +![The Exascale-class HPE Cray EX Supercomputer at Oak Ridge National Laboratory (fastest as of June 2024)](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Frontier_Supercomputer_%282%29.jpg/585px-Frontier_Supercomputer_%282%29.jpg){width=80%} +::: +::: {.column width="33%"} +Embarassingly Parallel -![Taxonomy of CPUs (Downloaded from de https://slurm.schedmd.com/mc_support.html)](fig/cpu-slurm.png){width="400px"} +![Downloaded from [Manning's "Modern Fortran"](https://livebook.manning.com/concept/fortran/addition)](fig/embarrasingly-parallel.png){width=80%} +::: +::: -Now, how many cores does your computer has, the parallel package can tell you that: +**In terms of scale** -```{r 03-how-many-cores} +::: {layout-align="center"} +HTC > HPC > Single node > Socket > Core > Thread | SIMD vectorization +::: + +## What's "a core"? + +![Source: Original figure from LUMI consortium documentation [@lumi2023]](https://raw.githubusercontent.com/gvegayon/appliedhpcr/e33949f25184bed5e84b02c356656782ddd5a46b/fig/socket-core-threads.svg){width=90% height=400px} + +How many cores does your computer has? + +```{r} +#| label: 03-how-many-cores +#| echo: true parallel::detectCores() ``` -## What is parallel computing, anyway? +## What is parallel computing? +::: {.columns} +::: {.column width="40%"} ```r f <- function(n) n*2 + f(1:4) ``` -![Here we are using a single core. The function is applied one element at a time, leaving the other 3 cores without usage.](fig/pll-computing-explained-serial.svg){width="50%"} +- Using a single core. +- One element at a time. +- 3 idle cores. +::: +::: {.column width="60%"} +![](fig/pll-computing-explained-serial.svg){width=450px} +::: +::: +## What is parallel computing? + +::: {.columns} +::: {.column width="40%"} ```r f <- function(n) n*2 -f(1:4) +f_pll <- ...magic to parallelize f... +f_pll(1:4) ``` -![In this more intelligent way of computation, we are taking full advantage of our computer by using all 4 cores at the same time. This will translate in a reduced computation time which, in the case of complicated/long calculations, can be an important speed gain.](fig/pll-computing-explained-parallel.svg){width="50%"} +- Using 4 cores. +- 4 elements at a time. +- No idle cores. +- 4 times faster. +::: +::: {.column width="60%"} +![](fig/pll-computing-explained-parallel.svg){width=450px} +::: +::: -## When is it a good idea? +## {background-color="#515A5A" style="margin:auto;text-align:center;"} -Let's think before we start... +Let's think before we start... ![](https://media.giphy.com/media/Dwclsfe6Gb91m/giphy.gif){style="width:500px"} +When is it a good idea to go HPC? + +## When is it a good idea? + +::: {.columns} + +::: {.column width="70%"} +![Ask yourself these questions before jumping into HPC!](fig/when_to_parallel.svg){fig-align="center" width="100%"} +::: + +::: {.column width="30%"} +::: {.callout-tip title="Pro-tip" .fragment} +When in doubt, **profile** your code first! In R, you can use the [`profvis`](https://cran.r-project.org/package=profvis) package. which will give you a visual representation of where your code is spending most of the time. +::: +::: + +::: + +## When is it a good idea? + +::: {layout-ncol="2"} +### Things that are easily parallelizable + +- Bootstrapping. +- Cross-validation. +- Monte Carlo simulations. +- Multiple MCMC chains. -> "In almost any parallel-processing application, you encounter overhead, or “wasted” time spent on noncomputational activity." [Matloff, ch 16, pg 337](https://diytranscriptomics.com/Reading/files/The%20Art%20of%20R%20Programming.pdf#page=363). See [16.4](https://diytranscriptomics.com/Reading/files/The%20Art%20of%20R%20Programming.pdf#page=371) for sources of overhead. +### Things that are not easily parallelizable -```{r good-idea, echo=FALSE, fig.cap="Ask yourself these questions before jumping into HPC!", fig.align='center', out.width="60%"} -knitr::include_graphics("fig/when_to_parallel.svg") +- Regression models. +- Within Chain MCMC. +- Matrix Algebra (generally)[^whenmatrix]. +- Any sequential algorithm. + +[^whenmatrix]: Parallelization of matrix operations is usually done via SIMD instructions, like those in GPUs and some CPUs. Most cases it applies via implicit parallelism (not at the user level). +::: + +::: {.fragment .incremental} +Good example usage case: + +- Large simulation study running ABMs. +- It took about **30 minutes** using R + the parallel package... using 50 threads! +- It would have taken about 30 mins x 50 = **24 hours using a single thread**. +::: + +## Overhead cost + +::: {style="font-size:80%"} +- Parallelization is not free: Most cost is in sending+receiving data. + +- In R (and other flavors), you can mitigate by (i) reducing the amount of data communicated, and (ii) reducing the number of times you communicate. +::: + +```{r} +#| label: overhead-cost +#| cache: true +#| echo: false +# Simulating data for the +nvars <- c(100, 500, 1000) * 3 +n <- 10000 +set.seed(331) +x <- matrix(rnorm(n*max(nvars)), nrow=n) +y <- rnorm(n) + +# Saving for later +dat_overhead <- list(x = x, y = y, nvars = nvars) + +# Creating the cluster object +library(parallel) +cl <- makePSOCKcluster(4) +clusterExport(cl, c("x", "y", "nvars")) + +# Making room +timing_serial_lm_all <- list() +timing_parallel_lm_all <- list() +timing_serial_lm <- list() +timing_parallel_lm <- list() +timing_parallel_lm_lite <- list() + +# Runs the call and returns the time +timer <- function(x., fun, cl = NULL) { + if (is.null(cl)) { + system.time({ + apply(x., 2, fun, y = y) + }) * 1000 + } else { + system.time({ + parApply(cl, x., 2, fun, y = y) + }) * 1000 + } +} + +x2 <- cbind(1, x) +clusterExport(cl, c("x2")) + +for (i in seq_along(nvars)) { + + timing_serial_lm_all[[i]] <- timer(x[, 1:nvars[i]], function(x., y) lm(y ~ x.)) + timing_parallel_lm_all[[i]] <- timer(x[, 1:nvars[i]], function(x., y) lm(y ~ x.), cl) + + timing_serial_lm[[i]] <- timer(x[, 1:nvars[i]], function(x., y) coef(lm(y ~ x.))) + timing_parallel_lm[[i]] <- timer(x[, 1:nvars[i]], function(x., y) coef(lm(y ~ x.)), cl) + + timing_parallel_lm_lite[[i]] <- system.time({ + mclapply(1:nvars[i], function(j) coef(lm.fit(y = y, x=x2[,c(1,j),drop=FALSE])), mc.cores = 4) + }) * 1000 + +} +stopCluster(cl) + +# Preparing the data +timing_serial_lm<- do.call(rbind, timing_serial_lm) |> data.frame() +timing_parallel_lm<- do.call(rbind, timing_parallel_lm) |> data.frame() + +timing_serial_lm_all<- do.call(rbind, timing_serial_lm_all) |> data.frame() +timing_parallel_lm_all<- do.call(rbind, timing_parallel_lm_all) |> data.frame() + +timing_parallel_lm_lite<- do.call(rbind, timing_parallel_lm_lite) |> data.frame() +``` + +```{r} +#| echo: false +#| fig-asp: .6 +#| fig-align: center +#| fig-cap: "Overhead cost of parallelization: Fitting $y = \\alpha + \\beta_k X_k + \\varepsilon,\\quad k = 1, \\dots$ (more about this later)" +# Nice color palette +palette(hcl.colors(8, "viridis")) + +# Visualize +op <- par(cex=1.25, mai = par("mai") * c(1.5,1.5,0,0)) +maxy <- range(c(timing_serial_lm_all[, "elapsed"], timing_parallel_lm_all[, "elapsed"], timing_parallel_lm[, "elapsed"], timing_parallel_lm_lite[, "elapsed"])) + +plot(timing_serial_lm_all[, "elapsed"] ~ nvars, type="b", col=1, pch=19, xlab="Number of variables", ylab="Time (ms) (log-scale)", lty=1, ylim=maxy, log="y", lwd=4) +lines(timing_parallel_lm_all[, "elapsed"] ~ nvars, type="b", col=2, pch=19, lty=2, lwd = 4) +lines(timing_parallel_lm[, "elapsed"] ~ nvars, type="b", col=3, pch=19, lty=3, lwd = 4) +lines(timing_parallel_lm_lite[, "elapsed"] ~ nvars, type="b", col=4, pch=19, lty=4, lwd = 4) + +legend("bottomright", legend=c("Serial", "Naive parallel", "Good parallel", "Best parallel"), col=1:4, lty=1:4, bty="n", lwd=4, cex=1.2) +par(op) ``` -# Parallel computing in R + +# Parallel computing in R {background-color="#515A5A"} + +![](https://raw.githubusercontent.com/USCbiostats/hpc-with-r/master/fig/spider-toddler.gif){fig-align="center" fig-alt="Spiderman teaching a toddler to walk"} + +## Parallel computing in R While there are several alternatives (just take a look at the [High-Performance Computing Task View](https://cran.r-project.org/web/views/HighPerformanceComputing.html)), -we'll focus on the following R-packages for **explicit parallelism**. +we'll focus on the following R-packages for **explicit parallelism** + +Some examples: > * [**parallel**](https://cran.r-project.org/package=parallel): R package that provides '[s]upport for parallel computation, including random-number generation'. @@ -109,12 +297,17 @@ we'll focus on the following R-packages for **explicit parallelism**. > * [**future**](https://cran.r-project.org/package=future): '[A] lightweight and unified Future API for sequential and parallel processing of R - expression via futures.' (won't cover here) + expression via futures.' + +> * [**slurmR**](https://cran.r-project.org/package=slurmR): R package for working with + the Slurm Workload Manager (by yours truly). -**Implicit parallelism or hidden parallelism**, on the other hand, are out-of-the-box tools that allow the programmer not to worry about parallelization, e.g. such as +Implicit parallelism, on the other hand, are out-of-the-box tools that allow the +programmer not to worry about parallelization, e.g. such as [**gpuR**](https://cran.r-project.org/package=gpuR) for Matrix manipulation using -GPU, [**tensorflow**](https://cran.r-project.org/package=tensorflow), and [**data.table**](https://cran.r-project.org/web/packages/data.table/index.html) +GPU, [**tensorflow**](https://cran.r-project.org/package=tensorflow) +--- And there's also a more advanced set of options @@ -126,257 +319,472 @@ And there's also a more advanced set of options > * A ton of other type of resources, notably the tools for working with batch schedulers such as Slurm, HTCondor, etc. -# The parallel package - -* Based on the `snow` (simple network of workstations) and `multicore` R Packages. - -* Explicit parallelism. - -* Simple yet powerful idea: Parallel computing as multiple R sessions. - -* Clusters can be made of both local and remote sessions - -* Multiple types of cluster: `PSOCK`, `Fork`, `MPI`, etc. - - > [Differences between socket and fork](https://www.r-bloggers.com/2019/06/parallel-r-socket-or-fork/) - - > "For the fork, each parallel thread is a complete duplication of the master process with the shared environment, including objects or variables defined prior to the kickoff of parallel threads. Therefore, it runs fast. However, the major limitation is that the fork doesn’t work on the Windows system." - - > "On the other hand, the socket works on all operating systems. Each thread runs separately without sharing objects or variables, which can only be passed from the master process explicitly. As a result, it runs slower due to the communication overhead." - -
    - -# Parallel workflow +## The parallel package + +::: {.columns} +::: {.column width="50%"} +- Explicit parallelism. +- Parallel computing as multiple R sessions. +- Clusters can be made of both local and remote sessions +- Multiple types of cluster: `PSOCK`, `Fork`, `MPI`, etc. +::: +::: {.column width="50%"} +![](fig/parallel-package.svg) +::: +::: + +## Parallel workflow (Usually) We do the following: +::: {.fragment} 1. Create a `PSOCK/FORK` (or other) cluster using `makePSOCKCluster`/`makeForkCluster` (or `makeCluster`) - +::: + +::: {.fragment} 2. Copy/prepare each R session (if you are using a `PSOCK` cluster): - a. Copy objects with `parallel::clusterExport` + a. Copy objects with `clusterExport` - b. Pass expressions with `parallel::clusterEvalQ` + b. Pass expressions with `clusterEvalQ` c. Set a seed +::: + +::: {.fragment} +3. Do your call: `parApply`, `parLapply`, etc. +::: + +::: {.fragment} +4. Stop the cluster with `clusterStop` +::: -3. Do your call: `parallel::parApply`, `parallel::parLapply`, etc. +## Types of clusters -4. Stop the cluster with `parallel::clusterStop` +| Type | Description | Pros | Cons | +|------|-------------|------|------| +| `PSOCK` | Multiple machines via [socket](https://en.wikipedia.org/w/index.php?title=Network_socket&oldid=1234852476) connection | Works in all OSs | Slowest | +| `FORK` | Single machine via [forking](https://en.wikipedia.org/w/index.php?title=Fork_(system_call)&oldid=1241385768) | Avoids memory duplication | Only for Unix-based | +| `MPI`[^mpi] | Multiple machines via [Message Passage Interface](https://en.wikipedia.org/w/index.php?title=Message_Passing_Interface&oldid=1238011288) | Best alternative for HPC clusters | Sometimes hard to setup | -[parallel::clusterApply documentation](https://stat.ethz.ch/R-manual/R-devel/library/parallel/html/clusterApply.html) +Using PSOCK, the [`slurmR`](https://cran.r-project.org/package=slurmR){target="_blank"} package creates clusters featuring multiple nodes in HPC environments, think *hundreds of cores*. -[parallel documentation](https://stat.ethz.ch/R-manual/R-devel/library/parallel/doc/parallel.pdf) +[^mpi]: Requires the [`Rmpi`](https://cran.r-project.org/package=Rmpi) package +# Hands-on {background-color="#515A5A"} -# Ex 1: Hello world! +![](https://i.imgflip.com/38jiku.jpg){fig-align="center" fig-alt="Emergency broadcat: Your R code will get some seriuos speed boost" width="50%"} -```{r parallel-ex-psock, echo=TRUE, cache=TRUE} +## Ex 1: Hello world! + +```{r} +#| label: parallel-ex-psock +#| echo: true +#| code-line-numbers: "|2,3|6,7,8|11,12,13" # 1. CREATING A CLUSTER library(parallel) -cores <- parallel::detectCores() -cl <- parallel::makePSOCKcluster(cores) -x <- 20 +cl <- makePSOCKcluster(4) # 2. PREPARING THE CLUSTER -parallel::clusterSetRNGStream(cl, 123) # Equivalent to `set.seed(123)` -parallel::clusterExport(cl, "x") +clusterSetRNGStream(cl, 123) # Equivalent to `set.seed(123)` +x <- 20 +clusterExport(cl, "x") # 3. DO YOUR CALL -parallel::clusterEvalQ(cl, { - paste0("Hello from process #", Sys.getpid(), ". I see x and it is equal to ", x) +clusterEvalQ(cl, { + paste0("Hello from process #", Sys.getpid(), ". x = ", x) }) +``` +```{r} +#| code-line-numbers: "|2" +#| echo: true # 4. STOP THE CLUSTER -parallel::stopCluster(cl) +stopCluster(cl) ``` -# Ex 2: Parallel regressions - -**Problem**: Run multiple regressions for each column of a very wide dataset. Report the $\beta$ coefficients from fitting the following regression models: +## Ex 2: Regressions -$$ -y = X_{i}\beta_i + \varepsilon_{i},\quad \varepsilon_{i} \sim N(0, \sigma^2_i) -$$ +**Problem**: Run multiple regressions on a very wide dataset. We need to fit the +following model: $$ -\quad\forall \text{ columns } i \in \left\{1, \dots, 999 \right\} +y = X_i\beta_i + \varepsilon,\quad \varepsilon\sim N(0, \sigma^2_i),\quad\forall i $$ -```{r lots-of-lm-dgp, echo=TRUE} +```{r, echo=FALSE} +#| label: lots-of-lm-dgp set.seed(131) y <- rnorm(500) X <- matrix(rnorm(500*999), nrow = 500, dimnames = list(1:500, sprintf("x%03d", 1:999))) ``` -```{r lots-of-lm-print} +```{r} +#| label: lots-of-lm-dim +#| echo: true dim(X) X[1:6, 1:5] str(y) ``` -**Serial solution**: +## Ex 2: Regressions - Serial -. - -. - -. +```{r} +#| label: lots-of-lm-serial +#| echo: true +#| strip-white: false +#| code-annotations: hover +#| output-location: column +#| collapse: true -. -. +ans <- apply( # <1> + + X = X, + MARGIN = 2, # <2> + FUN = function(x, y) coef(lm(y ~ x)), # <3> + y = y # <4> + ) -. +ans[,1:3] +``` +1. Apply calls a function over rows or columns of a matrix. +2. We are applying over columns of `X`. +3. The function fits a linear model (`lm`) and returns the coefficients (`coef`). +4. Since the function also depends on `y`, we pass it as an argument. -. +## Ex 2: Regressions - Parallel -. +```{r} +#| label: lots-of-lm-parallel +#| echo: true +#| output-location: column +#| collapse: true +cl <- parallel::makePSOCKcluster(4L) # <1> +ans <- parallel::parApply( # <2> + cl = cl, # <3> + X = X, + MARGIN = 2, + FUN = function(x, y) coef(lm(y ~ x)), + y = y + ) -. +ans[,1:3] +``` +1. Creating a cluster with 4 cores. +2. Replacing `apply` with `parApply`. +3. Passing the cluster object to `parApply`. -. +Both results should be the same. -. -. +----- -. +Are we going any faster? The `microbenchmark` package can help us with that: -. +```{r} +#| label: lots-of-lm-bench +#| echo: true +library(microbenchmark) +microbenchmark( + parallel = parallel::parApply( + cl = cl, + X = X, MARGIN = 2, + FUN = function(x, y) coef(lm(y ~ x)), + y = y + ), + serial = apply( + X = X, MARGIN = 2, + FUN = function(x, y) coef(lm(y ~ x)), + y = y + ), + times = 10, + unit = "relative" +) +parallel::stopCluster(cl) +``` -Use `apply` (for loop) to solve it +## Ex 3: Bootstrap -```{r lots-of-lm-serial, cache = TRUE, strip.white=FALSE} +Problem: We want to bootstrap a logistic regression model. We need to fit the +following model: +$$ +P(Y=1) = \text{logit}^{-1}\left(X\beta\right) +$$ -ans <- apply( - X = X, - MARGIN = 2, - FUN = function(x, y) coef(lm(y ~ x)), - y = y - ) -ans[,1:5] +```{r} +#| label: parallel-ex-bootstrap +#| echo: false +# Simulating some data +n <- 100 +k <- 5 +set.seed(33) +X <- matrix(rnorm(n*k), ncol=k) +b <- cbind(rnorm(k)) +y <- rbinom(n, 1, prob=plogis(2 + X %*% b)) +# glm(y ~ X, family = binomial("logit")) |> coef() ``` -# Ex 2: Parallel regressions (cont'd 2) +```{r} +#| label: parallel-ex-bootstrap-data +#| echo: true +dim(X) +head(X) +y[1:6] +``` -. +## Ex 3: Bootstrap - Serial -. +```{r} +#| label: parallel-ex-bootstrap-serial +#| echo: true +my_boot <- function(y, X, B=1000) { -. + # Generating the indices + n <- length(y) + indices <- sample.int(n = n, size = n * B, replace = TRUE) |> + matrix(nrow = n) -. + -. -. + # Fitting the model + apply(indices, 2, function(i) { + glm(y[i] ~ X[i,], family = binomial("logit")) |> + coef() + }) |> t() -. +} -. -. +set.seed(3312) +ans <- my_boot(y, X, B=50) +head(ans) +``` -. +## Ex 3: Bootstrap - Parallel -. +```{r} +#| label: parallel-ex-bootstrap-parallel +#| echo: true +my_boot_pll <- function(y, X, cl, B=1000) { -. + # Generating the indices + n <- length(y) + indices <- sample.int(n = n, size = n * B, replace = TRUE) |> + matrix(nrow = n) -. + # Making sure y and X are available in the cluster + parallel::clusterExport(cl, c("y", "X")) -. + # Fitting the model + parallel::parApply(cl, indices, 2, function(i) { + glm(y[i] ~ X[i,], family = binomial("logit")) |> + coef() + }) |> t() +} +cl <- parallel::makeForkCluster(4) +set.seed(3312) +ans_pll <- my_boot_pll(y, X, cl, B=50) +head(ans_pll) +``` -**Parallel solution**: Use `parApply` +--- -```{r lots-of-lm-parallel, cache = TRUE} -library(parallel) -cl <- makePSOCKcluster(4L) -ans <- parApply( - cl = cl, - X = X, - MARGIN = 2, - FUN = function(x, y) coef(lm(y ~ x)), - y = y - ) +How much faster? -ans[,1:5] +```{r} +#| label: parallel-ex-bootstrap-bench +#| echo: true +#| warning: false +#| cache: true +microbenchmark::microbenchmark( + parallel = my_boot_pll(y, X, cl, B=1000), + serial = my_boot(y, X, B=1000), + times = 1, + unit = "s" +) +parallel::stopCluster(cl) ``` ------ -Are we going any faster? Compare the timing using the `bench` package. +## Ex 4: Overhead cost +Problem: Revisit of the overhead cost of parallelization. We want to fit the following model $$y = X_k\beta_k + \varepsilon,\quad k = 1, \dots$$ -. +```{r} +#| label: overhead-cost-data +# Simulating some data +n <- 1e4 +k <- 3e3 +X <- matrix(rnorm(n*k), ncol=k) +y <- rnorm(n) + +X[1:4, 1:5] +y[1:6] +``` -. +::: {.callout-important} +For this exercise only, we are excluding the time required to setup and stop the cluster. Those times are usually negligible for large computations but are also part of the overhead cost. +::: -. +## Ex 4: Overhead cost - Naive -. +Let's start with the naive approach: fitting the model and returning the full output. -. +```{r} +#| label: overhead-cost-run +#| echo: true +#| cache: true +#| code-line-numbers: "|2,7" +library(parallel) +cost_serial <- system.time(lapply(1:ncol(X), function(i) lm(y ~ X[,i]))) -. +# Running the benchmark +cl <- makePSOCKcluster(4) +clusterExport(cl, c("X", "y")) +cost_pll <- system.time(parLapply(cl, 1:ncol(X), function(i) lm(y ~ X[,i]))) -. +# Stopping the cluster +stopCluster(cl) +``` -. +```{r} +#| label: overhead-cost-output-table-naive +#| echo: false +#| code-line-numbers: "|2,7" +data.frame( + Serial = cost_serial["elapsed"], + `Parallel naive` = cost_pll["elapsed"], + row.names = "Elapsed time (s)", + check.names = FALSE + ) |> t() |> knitr::kable() +``` -. +The problem: we are returning a lot of information that we may not need: -. +```{r} +#| label: overhead-cost-lm-output +#| echo: true +# Approximate size of the output of apply/parApply +format(ncol(X) * object.size(lm(y ~ X[,1])), units="GB") +``` -. +--- -. +## Ex 4: Overhead cost - Less receiving -. +Instead of capturing the full output, we can just return the coefficients. -. +```{r} +#| label: overhead-cost-coef-only +#| echo: true +#| code-line-numbers: "4" +cl <- makePSOCKcluster(4) +clusterExport(cl, c("X", "y")) +cost_pll_coef <- system.time( + parLapply(cl, 1:ncol(X), function(i) coef(lm(y ~ X[,i]))) + ) -```{r lots-of-lm-benchmark, cache = TRUE, warning=FALSE} -library(bench) -mark( - parallel = parApply( - cl = cl, - X = X, MARGIN = 2, - FUN = function(x, y) coef(lm(y ~ x)), - y = y - ), - serial = apply( - X = X, MARGIN = 2, - FUN = function(x, y) coef(lm(y ~ x)), - y = y - ) -) +# Stopping the cluster +stopCluster(cl) +``` + +```{r} +#| label: overhead-cost-output-table-coef +#| echo: false +data.frame( + Serial = cost_serial["elapsed"], + `Parallel naive` = cost_pll["elapsed"], + `Parallel coef` = cost_pll_coef["elapsed"], + row.names = "Elapsed time (s)", + check.names = FALSE + ) |> t() |> knitr::kable() ``` -```{r lots-of-lm-stopcluster, cache = TRUE, echo=FALSE} +The coefficients are much smaller, significantly reducing the overhead cost to about `r format(ncol(X) * object.size(coef(lm(y ~ X[,1]))[1]), units="MB")`. + +## Ex 4: Overhead cost - Less doing + +Since we only get coefficients, we can use a lighter version of `lm` called `lm.fit`. + +```{r} +#| label: overhead-cost-lm-fit-lite +#| echo: true +#| code-line-numbers: "5" +cl <- makePSOCKcluster(4) +X1 <- cbind(1, X) +clusterExport(cl, c("X1", "y")) +cost_pll_lite <- system.time({ + parLapply(cl, 1:ncol(X), function(i) coef(lm.fit(y = y, x=X1[,c(1, i),drop=FALSE])) + ) +}) + +# Stopping the cluster stopCluster(cl) ``` +::: {style="font-size:70%"} +```{r} +#| label: overhead-cost-output-table-lite +#| echo: false +data.frame( + Serial = cost_serial["elapsed"], + `Parallel naive` = cost_pll["elapsed"], + `Parallel coef` = cost_pll_coef["elapsed"], + `Parallel lite` = cost_pll_lite["elapsed"], + row.names = "Elapsed time (s)", + check.names = FALSE + ) |> t() |> knitr::kable() +``` + +::: + +::: {.callout-tip title="Pro-tip"} +Using a Fork cluster instead of a PSOCK cluster can further reduce the overhead cost. Both `X` and `y` would have been automatically available in the Fork cluster at 0 cost. +::: + +## Conclusion + +::: {.incremental} + +- Parallel computing is a powerful tool to speed up your ~R~ code. -# How can you create a general purpose parallel computing package specific to OS? +- It's not always the best solution, **you have to think first!** -There may be smarter ways, though this is how I approached it: [SeqSGPV package](https://github.com/chipmanj/SeqSGPV/blob/master/R/SeqSGPVrules.R) +- In R the `parallel` package is a good starting point for explicit parallelism. +- When parallelizing, think about the overhead cost and how to "do less": + 1. Reduce the amount of data communicated (send/receive). + 2. Pass only what you need (*i.e.*, communicated data x2). + 3. Use lighter versions of functions when possible. +::: +## {style="text-align:center!important;"} +```{r thanks, out.width="300px", echo=FALSE} +knitr::include_graphics("fig/speed.gif") +``` + + +### Thanks! +

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    -# See also +## See also -* [Blog example](http://gradientdescending.com/simple-parallel-processing-in-r/) * [Package parallel](https://stat.ethz.ch/R-manual/R-devel/library/parallel/doc/parallel.pdf) * [Using the iterators package](https://cran.r-project.org/web/packages/iterators/vignettes/iterators.pdf) * [Using the foreach package](https://cran.r-project.org/web/packages/foreach/vignettes/foreach.pdf) @@ -387,7 +795,7 @@ There may be smarter ways, though this is how I approached it: [SeqSGPV package] For more, checkout the [CRAN Task View on HPC](https://cran.r-project.org/web/views/HighPerformanceComputing.html){target="_blank"} -# Bonus track: Simulating $\pi$ +## Bonus track: Simulating $\pi$ * We know that $\pi = \frac{A}{r^2}$. We approximate it by randomly adding @@ -395,14 +803,23 @@ For more, checkout the [CRAN Task View on HPC](https://cran.r-project.org/web/vi * So, we approximate $\pi$ as $\Pr\{\|x\| \leq 1\}\times 2^2$ -```{r, echo=FALSE, dev='jpeg', dev.args=list(quality=100), fig.width=6, fig.height=6, out.width='300px', out.height='300px'} +```{r} +#| label: pi-simulation +#| echo: true set.seed(1231) -p <- matrix(runif(5e3*2, -1, 1), ncol=2) -pcol <- ifelse(sqrt(rowSums(p^2)) <= 1, adjustcolor("blue", .7), adjustcolor("gray", .7)) + +p <- matrix(runif(5e3*2, -1, 1), ncol=2) + +pcol <- ifelse( + sqrt(rowSums(p^2)) <= 1, + adjustcolor("blue", .7), + adjustcolor("gray", .7) + ) + plot(p, col=pcol, pch=18) ``` -# +## The R code to do this @@ -419,7 +836,7 @@ pisim <- function(i, nsim) { # Notice we don't use the -i- } ``` -# +## ```{r parallel-ex2, echo=TRUE, cache=TRUE} library(parallel) @@ -437,40 +854,20 @@ clusterExport(cl, c("nsim", "pisim")) # Benchmarking: parSapply and sapply will run this simulation # a hundred times each, so at the end we have 1e5*100 points # to approximate pi -rbenchmark::benchmark( +microbenchmark::microbenchmark( parallel = parSapply(cl, 1:100, pisim, nsim=nsim), - serial = sapply(1:100, pisim, nsim=nsim), replications = 1 -)[,1:4] + serial = sapply(1:100, pisim, nsim=nsim), + times = 10, + unit = "relative" +) ``` -# Not yet discussed: RcppArmadillo and OpenMP - -* Friendlier than [**RcppParallel**](http://rcppcore.github.io/RcppParallel/)... - at least for 'I-use-Rcpp-but-don't-actually-know-much-about-C++' users (like myself!). - -* Must run only 'Thread-safe' calls, so calling R within parallel blocks can cause - problems (almost all the time). - -* Use `arma` objects, e.g. `arma::mat`, `arma::vec`, etc. Or, if you are used to them - `std::vector` objects as these are thread safe. - -* Pseudo-Random Number Generation is not very straight forward... But C++11 has - a [nice set of functions](http://en.cppreference.com/w/cpp/numeric/random) that can be used together with OpenMP - -* Need to think about how processors work, cache memory, etc. Otherwise you could - get into trouble... if your code is slower when run in parallel, then you probably - are facing [false sharing](https://software.intel.com/en-us/articles/avoiding-and-identifying-false-sharing-among-threads) - -* If R crashes... try running R with a debugger (see - [Section 4.3 in Writing R extensions](https://cran.r-project.org/doc/manuals/r-release/R-exts.html#Checking-memory-access)): - - ```shell - ~$ R --debugger=valgrind - ``` +## Session info -# Session info -```{r} -devtools::session_info() +```{r session, echo=FALSE} +sessionInfo() ``` + +## References diff --git a/07-parallel-computing/style.scss b/07-parallel-computing/style.scss new file mode 100644 index 0000000..d404068 --- /dev/null +++ b/07-parallel-computing/style.scss @@ -0,0 +1,2 @@ +/*-- scss:defaults --*/ +$presentation-font-size-root: 25px;
    +
    + + +
    +

    Before we start

    +

    You can access the content here:

    +
    +
    +
    +
    +

    Fundamentals

    + +
    +
    +

    High-Performance Computing: An overview

    +

    Loosely, from R’s perspective, we can think of HPC in terms of two, maybe three things:

    +
      +
    1. Big data: How to work with data that doesn’t fit your computer

    2. +
    3. Parallel computing: How to take advantage of multiple core systems

    4. +
    5. Compiled code: Write your own low-level code (if R doesn’t has it yet…)

    6. +
    +

    (Checkout CRAN Task View on HPC)

    +
    +
    +

    Some vocabulary for HPC

    +
    +
    +

    High Throughput Computing Cluster (HTC Cluster)

    +
    +
    +

    +
    Open Science Grid Consortium (OSG) https://osg-htc.org
    +
    +
    +
    +

    Supercomputer (HPC Cluster)

    +
    +
    +

    +
    The Exascale-class HPE Cray EX Supercomputer at Oak Ridge National Laboratory (fastest as of June 2024)
    +
    +
    +
    +

    Embarassingly Parallel

    +
    +
    +

    +
    Downloaded from Manning’s “Modern Fortran”
    +
    +
    +
    +

    In terms of scale

    +
    +

    HTC > HPC > Single node > Socket > Core > Thread | SIMD vectorization

    +
    +
    +
    +

    What’s “a core”?

    +
    +
    +

    +
    Source: Original figure from LUMI consortium documentation (LUMI consortium 2023)
    +
    +
    +

    How many cores does your computer has?

    +
    +
    parallel::detectCores()
    +
    +
    [1] 11
    +
    +
    +
    +
    +

    What is parallel computing?

    +
    +
    +
    f <- function(n) n*2
    +
    +f(1:4)
    +
      +
    • Using a single core.
    • +
    • One element at a time.
    • +
    • 3 idle cores.
    • +
    +
    +

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + res <- f(x) + f(x[1]) + f(x[2]) + + + f(x[3]) + + + + f(x[3]) + + + + res + + + + + + + + +

    +
    +
    +
    +

    What is parallel computing?

    +
    +
    +
    f <- function(n) n*2
    +f_pll <- ...magic to parallelize f...
    +f_pll(1:4)
    +
      +
    • Using 4 cores.
    • +
    • 4 elements at a time.
    • +
    • No idle cores.
    • +
    • 4 times faster.
    • +
    +
    +

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + res <- f(x) + f(x[1]) + + + f(x[2]) + + + + f(x[3]) + + + + f(x[3]) + + + + res + + + + + + + + +

    +
    +
    +
    +

    +

    Let’s think before we start…

    + +

    When is it a good idea to go HPC?

    +
    +
    +

    When is it a good idea?

    +
    +
    +
    +
    +

    + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

    +
    Ask yourself these questions before jumping into HPC!
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +

    Pro-tip

    +
    +
    +

    When in doubt, profile your code first! In R, you can use the profvis package. which will give you a visual representation of where your code is spending most of the time.

    +
    +
    +
    +
    +
    +
    +
    +

    When is it a good idea?

    +
    + +
    +
    +
    +
    +

    Things that are easily parallelizable

    +
      +
    • Bootstrapping.
    • +
    • Cross-validation.
    • +
    • Monte Carlo simulations.
    • +
    • Multiple MCMC chains.
    • +
    +
    +
    +

    Things that are not easily parallelizable

    +
      +
    • Regression models.
    • +
    • Within Chain MCMC.
    • +
    • Matrix Algebra (generally)1.
    • +
    • Any sequential algorithm.
    • +
    +
    +
    +
    +
    +

    Good example usage case:

    +
      +
    • Large simulation study running ABMs.
    • +
    • It took about 30 minutes using R + the parallel package… using 50 threads!
    • +
    • It would have taken about 30 mins x 50 = 24 hours using a single thread.
    • +
    +
    +
    +
    +

    Overhead cost

    +
    +
      +
    • Parallelization is not free: Most cost is in sending+receiving data.

    • +
    • In R (and other flavors), you can mitigate by (i) reducing the amount of data communicated, and (ii) reducing the number of times you communicate.

    • +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

    Overhead cost of parallelization: Fitting \(y = \alpha + \beta_k X_k + \varepsilon,\quad k = 1, \dots\) (more about this later)

    +
    +
    +

    Parallel computing in R

    + +Spiderman teaching a toddler to walk
    +
    +

    Parallel computing in R

    +

    While there are several alternatives (just take a look at the High-Performance Computing Task View), we’ll focus on the following R-packages for explicit parallelism

    +

    Some examples:

    +
      +
    • parallel: R package that provides ‘[s]upport for parallel computation, including random-number generation’.
    • +
    +
      +
    • foreach: R package for ‘general iteration over elements’ in parallel fashion.
    • +
    +
      +
    • future: ‘[A] lightweight and unified Future API for sequential and parallel processing of R expression via futures.’
    • +
    +
      +
    • slurmR: R package for working with the Slurm Workload Manager (by yours truly).
    • +
    +

    Implicit parallelism, on the other hand, are out-of-the-box tools that allow the programmer not to worry about parallelization, e.g. such as gpuR for Matrix manipulation using GPU, tensorflow

    +
    +
    + +

    And there’s also a more advanced set of options

    +
      +
    • Rcpp + OpenMP: Rcpp is an R package for integrating R with C++, and OpenMP is a library for high-level parallelism for C/C++ and Fortran.
    • +
    +
      +
    • A ton of other type of resources, notably the tools for working with batch schedulers such as Slurm, HTCondor, etc.
    • +
    +
    +
    +

    The parallel package

    +
    +
    +
      +
    • Explicit parallelism.
    • +
    • Parallel computing as multiple R sessions.
    • +
    • Clusters can be made of both local and remote sessions
    • +
    • Multiple types of cluster: PSOCK, Fork, MPI, etc.
    • +
    +
    +

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + Master session + + + + + + + + child 1 + + + + + + + child 2 + + + + + + child 3 + + + + + + child 4 + + + + Cluster object + + + +

    +
    +
    +
    +

    Parallel workflow

    +

    (Usually) We do the following:

    +
    +
      +
    1. Create a PSOCK/FORK (or other) cluster using makePSOCKCluster/makeForkCluster (or makeCluster)
    2. +
    +
    +
    +
      +
    1. Copy/prepare each R session (if you are using a PSOCK cluster):

      +
        +
      1. Copy objects with clusterExport

      2. +
      3. Pass expressions with clusterEvalQ

      4. +
      5. Set a seed

      6. +
    2. +
    +
    +
    +
      +
    1. Do your call: parApply, parLapply, etc.
    2. +
    +
    +
    +
      +
    1. Stop the cluster with clusterStop
    2. +
    +
    +
    +
    +

    Types of clusters

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    TypeDescriptionProsCons
    PSOCKMultiple machines via socket connectionWorks in all OSsSlowest
    FORKSingle machine via forkingAvoids memory duplicationOnly for Unix-based
    MPI1Multiple machines via Message Passage InterfaceBest alternative for HPC clustersSometimes hard to setup
    +

    Using PSOCK, the slurmR package creates clusters featuring multiple nodes in HPC environments, think hundreds of cores.

    +
    +
    +
    +

    Hands-on

    + +Emergency broadcat: Your R code will get some seriuos speed boost
    +
    +

    Ex 1: Hello world!

    +
    +
    # 1. CREATING A CLUSTER
    +library(parallel)
    +cl <- makePSOCKcluster(4)    
    +
    +# 2. PREPARING THE CLUSTER
    +clusterSetRNGStream(cl, 123) # Equivalent to `set.seed(123)`
    +x  <- 20
    +clusterExport(cl, "x")
    +
    +# 3. DO YOUR CALL
    +clusterEvalQ(cl, {
    +  paste0("Hello from process #", Sys.getpid(), ". x = ", x)
    +})
    +
    +
    [[1]]
    +[1] "Hello from process #24028. x = 20"
    +
    +[[2]]
    +[1] "Hello from process #24029. x = 20"
    +
    +[[3]]
    +[1] "Hello from process #24030. x = 20"
    +
    +[[4]]
    +[1] "Hello from process #24027. x = 20"
    +
    +
    +
    +
    # 4. STOP THE CLUSTER
    +stopCluster(cl)
    +
    +
    +
    +

    Ex 2: Regressions

    +

    Problem: Run multiple regressions on a very wide dataset. We need to fit the following model:

    +

    \[ +y = X_i\beta_i + \varepsilon,\quad \varepsilon\sim N(0, \sigma^2_i),\quad\forall i +\]

    +
    +
    dim(X)
    +
    +
    [1] 500 999
    +
    +
    X[1:6, 1:5]
    +
    +
             x001        x002       x003       x004      x005
    +1  0.61827227  1.72847041 -1.4810695 -0.2471871 1.4776281
    +2  0.96777456 -0.19358426 -0.8176465  0.6356714 0.7292221
    +3 -0.04303734 -0.06692844  0.9048826 -1.9277964 2.2947675
    +4  0.84237608 -1.13685605 -1.8559158  0.4687967 0.9881953
    +5 -1.91921443  1.83865873  0.5937039 -0.1410556 0.6507415
    +6  0.59146153  0.81743419  0.3348553 -1.8771819 0.8181764
    +
    +
    str(y)
    +
    +
     num [1:500] -0.8188 -0.5438 1.0209 0.0467 -0.4501 ...
    +
    +
    +
    +
    +

    Ex 2: Regressions - Serial

    +
    +
    +
    
    +1ans <- apply(
    +  
    +  X      = X,
    +2  MARGIN = 2,
    +3  FUN    = function(x, y) coef(lm(y ~ x)),
    +4  y      = y
    +  )
    +
    +ans[,1:3]
    +##                    x001        x002        x003
    +## (Intercept) -0.03449819 -0.03339681 -0.03728140
    +## x           -0.06082548  0.02748265 -0.01327855
    +
    +
    +
    +
    1
    +
    +Apply calls a function over rows or columns of a matrix. +
    +
    2
    +
    +We are applying over columns of X. +
    +
    3
    +
    +The function fits a linear model (lm) and returns the coefficients (coef). +
    +
    4
    +
    +Since the function also depends on y, we pass it as an argument. +
    +
    +
    +
    +
    +
    +

    Ex 2: Regressions - Parallel

    +
    +
    +
    1cl <- parallel::makePSOCKcluster(4L)
    +2ans <- parallel::parApply(
    +3  cl     = cl,
    +  X      = X,
    +  MARGIN = 2,
    +  FUN    = function(x, y) coef(lm(y ~ x)),
    +  y      = y
    +  )
    +
    +ans[,1:3]
    +##                    x001        x002        x003
    +## (Intercept) -0.03449819 -0.03339681 -0.03728140
    +## x           -0.06082548  0.02748265 -0.01327855
    +
    +
    +
    +
    1
    +
    +Creating a cluster with 4 cores. +
    +
    2
    +
    +Replacing apply with parApply. +
    +
    3
    +
    +Passing the cluster object to parApply. +
    +
    +
    +
    +

    Both results should be the same.

    +
    +
    + +

    Are we going any faster? The microbenchmark package can help us with that:

    +
    +
    library(microbenchmark)
    +microbenchmark(
    +  parallel = parallel::parApply(
    +    cl  = cl,
    +    X   = X, MARGIN = 2,
    +    FUN = function(x, y) coef(lm(y ~ x)),
    +    y   = y
    +    ),
    +  serial = apply(
    +    X   = X, MARGIN = 2,
    +    FUN = function(x, y) coef(lm(y ~ x)),
    +    y   = y
    +    ),
    +    times = 10,
    +    unit = "relative"
    +)
    +
    +
    Unit: relative
    +     expr      min       lq    mean   median       uq      max neval cld
    + parallel 1.000000 1.000000 1.00000 1.000000 1.000000 1.000000    10  a 
    +   serial 2.861088 2.757073 2.75626 2.730507 2.769718 2.693011    10   b
    +
    +
    parallel::stopCluster(cl)
    +
    +
    +
    +

    Ex 3: Bootstrap

    +

    Problem: We want to bootstrap a logistic regression model. We need to fit the following model:

    +

    \[ +P(Y=1) = \text{logit}^{-1}\left(X\beta\right) +\]

    +
    +
    dim(X)
    +
    +
    [1] 100   5
    +
    +
    head(X)
    +
    +
                [,1]       [,2]        [,3]        [,4]        [,5]
    +[1,] -0.13592452  1.1921489 -1.04101654  0.26500638 -0.51561099
    +[2,] -0.04079697 -0.1231379 -0.43190705  1.38694989  0.39568325
    +[3,]  1.01053901 -0.5741648 -0.77781632 -0.29149014 -0.78301461
    +[4,] -0.15826244 -1.4903169  0.37368178 -1.83027672  0.88538861
    +[5,] -2.15663750  2.3638289  0.31256458 -1.62766978 -0.38212891
    +[6,]  0.49864683 -2.9510362  0.07122864 -0.01630346  0.05333596
    +
    +
    y[1:6]
    +
    +
    [1] 1 1 0 0 1 0
    +
    +
    +
    +
    +

    Ex 3: Bootstrap - Serial

    +
    +
    my_boot <- function(y, X, B=1000) {
    +
    +  # Generating the indices
    +  n <- length(y)
    +  indices <- sample.int(n = n, size = n * B, replace = TRUE) |>
    +    matrix(nrow = n)
    +
    +  
    +
    +
    +  # Fitting the model
    +  apply(indices, 2, function(i) {
    +    glm(y[i] ~ X[i,], family = binomial("logit")) |>
    +      coef()
    +  }) |> t()
    +
    +} 
    +
    +
    +set.seed(3312)
    +ans <- my_boot(y, X, B=50)
    +head(ans)
    +
    +
         (Intercept)     X[i, ]1  X[i, ]2   X[i, ]3  X[i, ]4    X[i, ]5
    +[1,]    2.943576 -0.46986617 2.292807 1.1069735 2.117947  0.7839228
    +[2,]    4.265760 -0.01445575 3.881603 2.5052960 4.300462  0.0542386
    +[3,]    2.702185 -0.40973910 2.315127 1.1693082 3.059388  0.2927383
    +[4,]    4.827939 -1.52854114 2.692226 1.7977035 4.370736  0.7825011
    +[5,]    3.229396 -0.56316370 1.980704 1.4054200 3.949632  0.2806117
    +[6,]    2.933971  0.25911455 2.193838 0.6953409 1.970649 -0.3528708
    +
    +
    +
    +
    +

    Ex 3: Bootstrap - Parallel

    +
    +
    my_boot_pll <- function(y, X, cl, B=1000) {
    +
    +  # Generating the indices
    +  n <- length(y)
    +  indices <- sample.int(n = n, size = n * B, replace = TRUE) |>
    +    matrix(nrow = n)
    +
    +  # Making sure y and X are available in the cluster
    +  parallel::clusterExport(cl, c("y", "X"))
    +
    +  # Fitting the model
    +  parallel::parApply(cl, indices, 2, function(i) {
    +    glm(y[i] ~ X[i,], family = binomial("logit")) |>
    +      coef()
    +  }) |> t()
    +
    +}
    +
    +cl <- parallel::makeForkCluster(4)
    +set.seed(3312)
    +ans_pll <- my_boot_pll(y, X, cl, B=50)
    +head(ans_pll)
    +
    +
         (Intercept)     X[i, ]1  X[i, ]2   X[i, ]3  X[i, ]4    X[i, ]5
    +[1,]    2.943576 -0.46986617 2.292807 1.1069735 2.117947  0.7839228
    +[2,]    4.265760 -0.01445575 3.881603 2.5052960 4.300462  0.0542386
    +[3,]    2.702185 -0.40973910 2.315127 1.1693082 3.059388  0.2927383
    +[4,]    4.827939 -1.52854114 2.692226 1.7977035 4.370736  0.7825011
    +[5,]    3.229396 -0.56316370 1.980704 1.4054200 3.949632  0.2806117
    +[6,]    2.933971  0.25911455 2.193838 0.6953409 1.970649 -0.3528708
    +
    +
    +
    +
    + +

    How much faster?

    +
    +
    microbenchmark::microbenchmark(
    +  parallel = my_boot_pll(y, X, cl, B=1000),
    +  serial   = my_boot(y, X, B=1000),
    +  times    = 1,
    +  unit     = "s"
    +)
    +
    +
    Unit: seconds
    +     expr       min        lq      mean    median        uq       max neval
    + parallel 0.1918902 0.1918902 0.1918902 0.1918902 0.1918902 0.1918902     1
    +   serial 0.5202934 0.5202934 0.5202934 0.5202934 0.5202934 0.5202934     1
    +
    +
    parallel::stopCluster(cl)
    +
    +
    +
    +

    Ex 4: Overhead cost

    +

    Problem: Revisit of the overhead cost of parallelization. We want to fit the following model \[y = X_k\beta_k + \varepsilon,\quad k = 1, \dots\]

    +
    +
    +
                [,1]       [,2]       [,3]       [,4]       [,5]
    +[1,] -1.91835255 -0.2359106 -1.4642601 -0.5320349 -0.4639574
    +[2,] -0.09017806 -0.1022420 -0.6735899  1.6146947 -2.3792154
    +[3,] -1.25551672  1.2079800  0.2159515 -0.1323614  0.9867689
    +[4,]  1.28006769 -0.2806277 -0.2026345 -0.7375033 -0.1067501
    +
    +
    +
    [1]  0.3570720  0.4850507  1.0281664 -0.7044579  1.1378356 -0.9032009
    +
    +
    +
    +
    +
    +
    + +
    +

    Important

    +
    +
    +

    For this exercise only, we are excluding the time required to setup and stop the cluster. Those times are usually negligible for large computations but are also part of the overhead cost.

    +
    +
    +
    +
    +
    +

    Ex 4: Overhead cost - Naive

    +

    Let’s start with the naive approach: fitting the model and returning the full output.

    +
    +
    library(parallel)
    +cost_serial <- system.time(lapply(1:ncol(X), function(i) lm(y ~ X[,i])))
    +
    +# Running the benchmark
    +cl <- makePSOCKcluster(4)
    +clusterExport(cl, c("X", "y"))
    +cost_pll <- system.time(parLapply(cl, 1:ncol(X), function(i) lm(y ~ X[,i])))
    +
    +# Stopping the cluster
    +stopCluster(cl)
    +
    +
    +
    + + + + + + + + + + + + + + + + + +
    Elapsed time (s)
    Serial2.429
    Parallel naive4.489
    +
    +
    +

    The problem: we are returning a lot of information that we may not need:

    +
    +
    # Approximate size of the output of apply/parApply
    +format(ncol(X) * object.size(lm(y ~ X[,1])), units="GB")
    +
    +
    [1] "7.2 Gb"
    +
    +
    +
    +
    +

    Ex 4: Overhead cost - Less receiving

    +

    Instead of capturing the full output, we can just return the coefficients.

    +
    +
    cl <- makePSOCKcluster(4)
    +clusterExport(cl, c("X", "y"))
    +cost_pll_coef <- system.time(
    +  parLapply(cl, 1:ncol(X), function(i) coef(lm(y ~ X[,i])))
    +  )
    +
    +# Stopping the cluster
    +stopCluster(cl)
    +
    +
    +
    + + + + + + + + + + + + + + + + + + + + + +
    Elapsed time (s)
    Serial2.429
    Parallel naive4.489
    Parallel coef0.672
    +
    +
    +

    The coefficients are much smaller, significantly reducing the overhead cost to about 0.8 Mb.

    +
    +
    +

    Ex 4: Overhead cost - Less doing

    +

    Since we only get coefficients, we can use a lighter version of lm called lm.fit.

    +
    +
    cl <- makePSOCKcluster(4)
    +X1 <- cbind(1, X)
    +clusterExport(cl, c("X1", "y"))
    +cost_pll_lite <- system.time({
    +  parLapply(cl, 1:ncol(X), function(i) coef(lm.fit(y = y, x=X1[,c(1, i),drop=FALSE]))
    +  )
    +})
    +
    +# Stopping the cluster
    +stopCluster(cl)
    +
    +
    +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + +
    Elapsed time (s)
    Serial2.429
    Parallel naive4.489
    Parallel coef0.672
    Parallel lite0.252
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +

    Pro-tip

    +
    +
    +

    Using a Fork cluster instead of a PSOCK cluster can further reduce the overhead cost. Both X and y would have been automatically available in the Fork cluster at 0 cost.

    +
    +
    +
    +
    +
    +
    +

    Conclusion

    +
    +
      +
    • Parallel computing is a powerful tool to speed up your R code.

    • +
    • It’s not always the best solution, you have to think first!

    • +
    • In R the parallel package is a good starting point for explicit parallelism.

    • +
    • When parallelizing, think about the overhead cost and how to “do less”:

      +
        +
      1. Reduce the amount of data communicated (send/receive).
      2. +
      3. Pass only what you need (i.e., communicated data x2).
      4. +
      5. Use lighter versions of functions when possible.
      6. +
    • +
    +
    +
    +
    +

    + +

    Thanks!

    +

    + gvegayon
    ggvy.cl
    george.vegayon@utah.edu
    Presentation created with and quarto.org +

    +
    +
    +

    See also

    + +

    For more, checkout the CRAN Task View on HPC

    +
    +
    +

    Bonus track: Simulating \(\pi\)

    +
      +
    • We know that \(\pi = \frac{A}{r^2}\). We approximate it by randomly adding points \(x\) to a square of size 2 centered at the origin.

    • +
    • So, we approximate \(\pi\) as \(\Pr\{\|x\| \leq 1\}\times 2^2\)

    • +
    +
    +
    set.seed(1231)
    +
    +p <- matrix(runif(5e3*2, -1, 1), ncol=2)
    +
    +pcol <- ifelse(
    +  sqrt(rowSums(p^2)) <= 1,
    +  adjustcolor("blue", .7),
    +  adjustcolor("gray", .7)
    +  )
    +
    +plot(p, col=pcol, pch=18)
    + +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    +

    +

    The R code to do this

    +
    +
    pisim <- function(i, nsim) {  # Notice we don't use the -i-
    +  # Random points
    +  ans  <- matrix(runif(nsim*2), ncol=2)
    +  
    +  # Distance to the origin
    +  ans  <- sqrt(rowSums(ans^2))
    +  
    +  # Estimated pi
    +  (sum(ans <= 1)*4)/nsim
    +}
    +
    +
    +
    +

    +
    +
    library(parallel)
    +# Setup
    +cl <- makePSOCKcluster(4L)
    +clusterSetRNGStream(cl, 123)
    +
    +# Number of simulations we want each time to run
    +nsim <- 1e5
    +
    +# We need to make -nsim- and -pisim- available to the
    +# cluster
    +clusterExport(cl, c("nsim", "pisim"))
    +
    +# Benchmarking: parSapply and sapply will run this simulation
    +# a hundred times each, so at the end we have 1e5*100 points
    +# to approximate pi
    +microbenchmark::microbenchmark(
    +  parallel = parSapply(cl, 1:100, pisim, nsim=nsim),
    +  serial   = sapply(1:100, pisim, nsim=nsim),
    +  times    = 10,
    +  unit     = "relative"
    +)
    +
    +
    Unit: relative
    +     expr      min       lq     mean   median       uq     max neval cld
    + parallel 1.000000 1.000000 1.000000 1.000000 1.000000 1.00000    10  a 
    +   serial 1.882948 1.899173 1.975634 1.896836 2.155636 2.04383    10   b
    +
    +
    +
    +
    +

    Session info

    +
    +
    +
    R version 4.4.1 (2024-06-14)
    +Platform: aarch64-apple-darwin23.4.0
    +Running under: macOS Sonoma 14.6.1
    +
    +Matrix products: default
    +BLAS:   /opt/homebrew/Cellar/openblas/0.3.27/lib/libopenblasp-r0.3.27.dylib 
    +LAPACK: /opt/homebrew/Cellar/r/4.4.1/lib/R/lib/libRlapack.dylib;  LAPACK version 3.12.0
    +
    +locale:
    +[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
    +
    +time zone: America/Denver
    +tzcode source: internal
    +
    +attached base packages:
    +[1] parallel  stats     graphics  grDevices utils     datasets  methods  
    +[8] base     
    +
    +other attached packages:
    +[1] microbenchmark_1.4.10
    +
    +loaded via a namespace (and not attached):
    + [1] digest_0.6.37     codetools_0.2-20  multcomp_1.4-25   fastmap_1.2.0    
    + [5] Matrix_1.7-0      xfun_0.45         lattice_0.22-6    TH.data_1.1-2    
    + [9] splines_4.4.1     zoo_1.8-12        knitr_1.47        htmltools_0.5.8.1
    +[13] rmarkdown_2.27    mvtnorm_1.2-5     cli_3.6.3         grid_4.4.1       
    +[17] sandwich_3.1-0    compiler_4.4.1    tools_4.4.1       evaluate_1.0.0   
    +[21] survival_3.6-4    yaml_2.3.8        rlang_1.1.4       jsonlite_1.8.9   
    +[25] MASS_7.3-60.2    
    +
    +
    +
    +
    +

    References

    +
    + +
    +
    +
    +LUMI consortium. 2023. “Documentation - Distribution and Binding.” https://docs.lumi-supercomputer.eu/runjobs/scheduled-jobs/distribution-binding/. +
    +
    +
    + +
    +

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