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Preconditioning techniques for generalized Sylvester matrix equations

Yannis Voet

TL;DR

This work considers preconditioning techniques for the iterative solution of generalized Sylvester equations by constructing low Kronecker rank approximations of either the operator itself or its inverse, which can be easily combined with sparse approximate inverse techniques.

Abstract

Sylvester matrix equations are ubiquitous in scientific computing. However, few solution techniques exist for their generalized multiterm version, as they now arise in an increasingly large number of applications. In this work, we consider algebraic parameter-free preconditioning techniques for the iterative solution of generalized multiterm Sylvester equations. They consist in constructing low Kronecker rank approximations of either the operator itself or its inverse. While the former requires solving standard Sylvester equations in each iteration, the latter only requires matrix-matrix multiplications, which are highly optimized on modern computer architectures. Moreover, low Kronecker rank approximate inverses can be easily combined with sparse approximate inverse techniques, thereby enhancing their performance with little or no damage to their effectiveness.

Preconditioning techniques for generalized Sylvester matrix equations

TL;DR

This work considers preconditioning techniques for the iterative solution of generalized Sylvester equations by constructing low Kronecker rank approximations of either the operator itself or its inverse, which can be easily combined with sparse approximate inverse techniques.

Abstract

Sylvester matrix equations are ubiquitous in scientific computing. However, few solution techniques exist for their generalized multiterm version, as they now arise in an increasingly large number of applications. In this work, we consider algebraic parameter-free preconditioning techniques for the iterative solution of generalized multiterm Sylvester equations. They consist in constructing low Kronecker rank approximations of either the operator itself or its inverse. While the former requires solving standard Sylvester equations in each iteration, the latter only requires matrix-matrix multiplications, which are highly optimized on modern computer architectures. Moreover, low Kronecker rank approximate inverses can be easily combined with sparse approximate inverse techniques, thereby enhancing their performance with little or no damage to their effectiveness.
Paper Structure (17 sections, 4 theorems, 66 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 17 sections, 4 theorems, 66 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Krylov subspace methods for matrix equations
  3. Nearest Kronecker product preconditioner
  4. Complexity analysis
  5. Theoretical results
  6. Low Kronecker rank approximate inverse
  7. Kronecker rank $1$ approximate inverse
  8. Complexity analysis
  9. Kronecker rank $q$ approximate inverse
  10. Kronecker rank $q$ sparse approximate inverse
  11. Sparse approximate inverse techniques
  12. Low Kronecker rank sparse approximate inverse
  13. Numerical experiments
  14. RC circuit simulation
  15. Isogeometric analysis
  16. ...and 2 more sections

Key Result

Theorem 3.1

\newlabelth: kr_1_approximation0 Let $M \in \mathbb{R}^{nm \times nm}$ be a block-banded, nonnegative and symmetric positive definite matrix. Then, there exists banded, nonnegative and symmetric positive definite factor matrices $Y$ and $Z$ such that $\phi_M(Y,Z)$ in eq: phi_M_minimization is mini

Figures (3)

  • Figure 1: Convergence history for solving \ref{['eq: circuit_model']} using the (right-preconditioned) GMRES method. The non-preconditioned method converged after $630$ iterations.
  • Figure 2: Convergence history for solving \ref{['eq: mass_operator']} using the (right-preconditioned) Bi-CGSTAB method
  • Figure 3: Convergence history for solving \ref{['eq: conv_diff_eq']} using the (right-preconditioned) GMRES method

Theorems & Definitions (9)

  • Theorem 3.1: van1993approximation
  • Definition 3.2: Sparsity pattern
  • Lemma 3.3
  • Lemma 3.4
  • Proof 1
  • Remark 3.5
  • Remark 4.1
  • Lemma 4.2
  • Proof 2