SLiSe: Framework to design rational filters for contour-based eigensolvers

We herein present a Julia implementation of SLiSe, a framework for the design of rational filters for contour-based eigensolvers.

The highlights include

• Unlimited degrees of freedom within the design of rational filters,
• Quick convergence with SLiSe filters through minimization of worst-case convergence rate,
• First toolkit to provide support for box-constraints, for fast convergence within contour-based eigensolvers with iterative linear solvers, and
• Fast, ready-to-use rational filters included.

Installation

First, add this repository to Julia via

Pkg.clone("https://github.com/SimLabQuantumMaterials/SLiSeFilters.jl")

Afterwards, add the L-BFGS-B wrapper

Pkg.clone("https://github.com/jey/Lbfgsb.jl")

and run

Pkg.build("Lbfgsb")

to compile the original Fortran library.

Usage

To obtain filters with small worst-case convergence rate, type into Julia:

import SLiSeFilters

# Choose poles per quadrant and gap size
p = 4
G = 0.95

# Starting filter
z, a   = SLiSeFilters.points( "zolo" , p );

# Starting weight function with shift
ws = SLiSeFilters.createWS(  [ sqrt(G),  sqrt(G)^-1, 1.4, 10 ],
                  BitVector( [ 0,    0,   1,  1 ] ) , # free vars within minimization
                             [ 1,  .01,  10, 20 ]   ,
                  BitVector( [ 0,    1,   1,  1 ] ) )   # free vars within minimization

# Compute filter with reduced worst-case convergence rate
z2, a2  = SLiSeFilters.wise( ws, z, a, G )

You find resulting, ready-to-use rational filters in the docs/ folder, as well as Gauss-Legendre filters with reduced worst-case convergence rate.

A Jupyter notebook to demo the underlying SLiSe framework is available in the docs/ folder.

Related packages

A large set of benchmark eigenproblems for the FEAST eigensolver can be obtained from the SpectrumSlicingTestSuite.jl Julia package.

References

• Jan Winkelmann and Edoardo Di Napoli. 2019. Non-linear Least-Squares Optimization of Rational Filters for the Solution of Interior Hermitian Eigenvalue Problems. Frontiers in Applied Mathematics and Statistics, doi: 10.3389/fams.2019.00005.
• Konrad Kollnig, Paolo Bientinesi, and Edoardo Di Napoli. 2021. Rational Spectral Filters with Optimal Convergence Rate. SIAM Journal on Scientific Computing, doi: 10.1137/20M1313933
• Eric Polizzi. 2009. Density-matrix-based algorithm for solving eigenvalue problems. Physical Review B, 79, 11, (Mar. 2009), 115112–115117. doi: 10.1103/PhysRevB.79.115112.
• Richard Byrd, Peihuang Lu, Jorge Nocedal, and Ciyou Zhu. 1995. A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing, 16, 5, (Sept. 1995), 1190–1208. doi: 10.1137/0916069.