Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method

By Andrew V. Knyazev. SIAM Journal on Scientific Computing, 2001

846 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b80a3e1a89714724f5b3aba96a586c1e7d01c3d7

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In Computational Mechanics

In Computational Theory and Mathematics

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Citations in the corpus

Implementing sparse matrix-vector multiplication on throughput-oriented processors (2009, methodology): “Furthermore, in the case of eigen-solvers such as the LOBPCG [13], it is not uncommon to utilize block vectors with ten or more columns.”
Inexact inverse iteration for symmetric matrices (2006, methodology): “Particularly successful for extreme eigenvalues is the LOBPCG method discussed in [11] There are several new features in the present paper.”
Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver (2016, methodology): “‡ Rigorous convergence analysis of CG for eigenvalue computation is very difficult, and virtually all available results merely state that the convergence of CG is at least as fast as that of steepest descent, whereas in reality the convergence of CG is much faster and is more adequately portrayed by”
Rayleigh Quotient Based Optimization Methods For Eigenvalue Problems (2014, methodology): “This idea is basically the same as the one in [18, 23].”
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, methodology): “We will use Locally Optimal Block Preconditioned Conjugate Gradient Algorithm (LOBPCG) by Knyazev [29] for this task.”
Numerical methods for large eigenvalue problems (2002, neutral): “CONTENTS 1 Introduction 519 2 Notation and background 522 3 Single-vector iterations 523 4 Krylov subspace projection methods 525 5 Convergence of polynomial restart methods 545 6 Subspace iteration methods 556 7 The generalized eigenproblem 564 8 Eigenvalue software 571 9 Conclusions and acknowledg”
Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems (2011, methodology): “In the following, we discuss the combination of HTD with LOBPCG [17], an iterative method for computing the smallest eigenvalue(s) of a symmetric matrix A.”
Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies (2015, methodology): “For computing the eigen-pairs, we use the code for the Locally Optimal Block Prec-conditioned Conjugate Gradient (LOBPCG) method available online [37] (this was observed to be faster than MATLAB’s inbuilt sparse eigensolver eigs, which is based on Lanczos iterations).”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, methodology): “• To avoid numerical instabilities due to ill-conditioning of the matrix Uk in the LOBPCG method, it is often [6, 44] recommended to orthogonalize Uk before calculating (λk+1, y).”
Computing Inner Eigenvalues of Matrices in Tensor Train Matrix Format (2013, methodology): “In [7] Knyazev used the optimal vector in the subspace {xi−1,B ei−1, pi−1} as next iterate.”