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Preconditioning correction equations in Jacobi--Davidson type methods for computing partial singular value decompositions of large matrices

Jinzhi Huang, Zhongxiao Jia

TL;DR

This paper analyzes the convergence of MINRES for correction equations in Jacobi–Davidson type SVD computations and introduces an inner preconditioning strategy that leverages current approximate left/right singular vectors to construct a preconditioned, symmetry-preserving correction equation. The resulting inner preconditioned JDSVD (IPJDSVD) method accelerates inner solves without sacrificing outer convergence and is complemented by a thick-restart variant with deflation and purgation to efficiently compute multiple singular triplets. The authors provide convergence results showing faster MINRES convergence when cluster information near the target $\tau$ is exploited, and they develop a practical selection strategy to form the preconditioned equation adaptively. Numerical experiments demonstrate substantial reductions in matrix-vector products and CPU time compared with JDSVD, and reveal IPJDSVD_HYBRID to be competitive with, and often superior to, PRIMME_SVDS, especially for computing the smallest singular triplets.

Abstract

In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD) problems, called JDSVD, a large symmetric and generally indefinite correction equation is solved iteratively at each outer iteration, which constitutes the inner iterations and dominates the overall efficiency of JDSVD. In this paper, a convergence analysis is made on the minimal residual (MINRES) method for the correction equation. Motivated by the results obtained, at each outer iteration a new correction equation is derived that extracts useful information from current subspaces to construct effective preconditioners for the correction equation and is proven to retain the same convergence of outer iterations of JDSVD.The resulting method is called inner preconditioned JDSVD (IPJDSVD) method; it is also a new JDSVD method, and any viable preconditioner for the correction equations in JDSVD is straightforwardly applicable to those in IPJDSVD. Convergence results show that MINRES for the new correction equation can converge much faster when there is a cluster of singular values closest to a given target. A new thick-restart IPJDSVD algorithm with deflation and purgation is proposed that simultaneously accelerates the outer and inner convergence of the standard thick-restart JDSVD and computes several singular triplets. Numerical experiments justify the theory and illustrate the considerable superiority of IPJDSVD to JDSVD, and demonstrate that a similar two-stage IPJDSVD algorithm substantially outperforms the most advanced PRIMME\_SVDS software nowadays for computing the smallest singular triplets.

Preconditioning correction equations in Jacobi--Davidson type methods for computing partial singular value decompositions of large matrices

TL;DR

This paper analyzes the convergence of MINRES for correction equations in Jacobi–Davidson type SVD computations and introduces an inner preconditioning strategy that leverages current approximate left/right singular vectors to construct a preconditioned, symmetry-preserving correction equation. The resulting inner preconditioned JDSVD (IPJDSVD) method accelerates inner solves without sacrificing outer convergence and is complemented by a thick-restart variant with deflation and purgation to efficiently compute multiple singular triplets. The authors provide convergence results showing faster MINRES convergence when cluster information near the target is exploited, and they develop a practical selection strategy to form the preconditioned equation adaptively. Numerical experiments demonstrate substantial reductions in matrix-vector products and CPU time compared with JDSVD, and reveal IPJDSVD_HYBRID to be competitive with, and often superior to, PRIMME_SVDS, especially for computing the smallest singular triplets.

Abstract

In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD) problems, called JDSVD, a large symmetric and generally indefinite correction equation is solved iteratively at each outer iteration, which constitutes the inner iterations and dominates the overall efficiency of JDSVD. In this paper, a convergence analysis is made on the minimal residual (MINRES) method for the correction equation. Motivated by the results obtained, at each outer iteration a new correction equation is derived that extracts useful information from current subspaces to construct effective preconditioners for the correction equation and is proven to retain the same convergence of outer iterations of JDSVD.The resulting method is called inner preconditioned JDSVD (IPJDSVD) method; it is also a new JDSVD method, and any viable preconditioner for the correction equations in JDSVD is straightforwardly applicable to those in IPJDSVD. Convergence results show that MINRES for the new correction equation can converge much faster when there is a cluster of singular values closest to a given target. A new thick-restart IPJDSVD algorithm with deflation and purgation is proposed that simultaneously accelerates the outer and inner convergence of the standard thick-restart JDSVD and computes several singular triplets. Numerical experiments justify the theory and illustrate the considerable superiority of IPJDSVD to JDSVD, and demonstrate that a similar two-stage IPJDSVD algorithm substantially outperforms the most advanced PRIMME\_SVDS software nowadays for computing the smallest singular triplets.
Paper Structure (15 sections, 9 theorems, 116 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 15 sections, 9 theorems, 116 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The basic JDSVD method
  3. Convergence analysis on MINRES for solving \ref{['correction']}
  4. A preconditioned correction equation, MINRES, and practical issues
  5. A preconditioned correction equation
  6. Convergence analysis on MINRES for solving \ref{['precorrection']}
  7. Selection strategy of approximate singular triplets in inner preconditioning
  8. A thick-restart IPJDSVD algorithm with deflation and purgation
  9. A new thick-restart
  10. Deflation and purgation
  11. Thick-restart IPJDSVD algorithm with deflation and purgation
  12. Numerical experiments
  13. A comparison of IPJDSVD and JDSVD
  14. A comparison of IPJDSVD_HYBRID and PRIMME_SVDS
  15. Conclusions

Key Result

Theorem 3.1

\newlabelthm10 Let $\widetilde{B}_1^{\prime}$ be defined as in Mkey. Then the solution $w=\widetilde{W}_1z$ to equation correction, where $z\in\mathbb{R}^{M+N-2}$ is the solution to the linear system If MINRES is applied to newform and correction2 with the initial guesses $\tilde{w}_0$ and $\tilde{z}_0$ related by $\widetilde{w}_0=\widetilde{W}_1\tilde{z}_0$, the $j$th approximate solutions $\ti

Figures (1)

  • Figure 1: Computing $10$ singular triplets of $\mathrm{Alemdar}$ (left) and $\mathrm{bcsstk33}$ (right).

Theorems & Definitions (19)

  • Theorem 3.1
  • Proof 1
  • Theorem 3.2: greenbaum1997iterative
  • Theorem 3.3
  • Proof 2
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.6
  • Proof 3
  • Theorem 3.7
  • ...and 9 more