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New Randomized Global Generalized Minimum Residual (RGl-GMRES) method

Achraf Badahmane, Xian-Ming GU

TL;DR

A new Randomized Global Generalized Minimum Residual (RGlGMRES) algorithm for efficiently computing solutions to large scale linear systems with multiple right hand sides is developed and matrix sketching is introduced.

Abstract

In this paper, we develop a new Randomized Global Generalized Minimum Residual (RGlGMRES) algorithm for efficiently computing solutions to large scale linear systems with multiple right hand sides.The proposed method builds on a recently developed randomized global Gram Schmidt process, in which sketched Frobenius inner products are employed to approximate the exact Frobenius inner products of high-dimensional matrices. We give some new convergence results of the randomized global GMRES method for multiple linear systems. In the case where the coefficient matrix A is diagonalizable, we derive new upper bounds for the randomized Frobenius norm of the residual. In this paper, we study how to introduce matrix sketching in this algorithm. It allows us to reduce the dimension of the problem in one of the main steps of the algorithm. To validate the effectiveness and practicality of this approach, we conduct several numerical experiments, which demonstrate that our RGl-GMRES method is competitive with the GlGMRES method for solving large scale problems with multiple right-hand sides.

New Randomized Global Generalized Minimum Residual (RGl-GMRES) method

TL;DR

A new Randomized Global Generalized Minimum Residual (RGlGMRES) algorithm for efficiently computing solutions to large scale linear systems with multiple right hand sides is developed and matrix sketching is introduced.

Abstract

In this paper, we develop a new Randomized Global Generalized Minimum Residual (RGlGMRES) algorithm for efficiently computing solutions to large scale linear systems with multiple right hand sides.The proposed method builds on a recently developed randomized global Gram Schmidt process, in which sketched Frobenius inner products are employed to approximate the exact Frobenius inner products of high-dimensional matrices. We give some new convergence results of the randomized global GMRES method for multiple linear systems. In the case where the coefficient matrix A is diagonalizable, we derive new upper bounds for the randomized Frobenius norm of the residual. In this paper, we study how to introduce matrix sketching in this algorithm. It allows us to reduce the dimension of the problem in one of the main steps of the algorithm. To validate the effectiveness and practicality of this approach, we conduct several numerical experiments, which demonstrate that our RGl-GMRES method is competitive with the GlGMRES method for solving large scale problems with multiple right-hand sides.
Paper Structure (9 sections, 7 theorems, 60 equations, 1 figure, 6 tables, 3 algorithms)

This paper contains 9 sections, 7 theorems, 60 equations, 1 figure, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Global GMRES (Gl-GMRES) method
  3. Gl-Arnoldi method
  4. Convergence analysis of the Gl-GMRES method
  5. RGl-GMRES method
  6. Introduction to random sketching
  7. Randomized Global Arnoldi (RGl-Arnoldi) method
  8. Numerical Results
  9. Conclusion

Key Result

Proposition 1

Assume that $h_{i+1,i} \neq 0$ for $i = 1,\ldots,k$, and let where $V_i$, for $i=1,\ldots,k$, are the matrices computed by Algorithm 1. Then the following relations hold: where $H_k \in \mathbb{R}^{k \times k}$ is an upper Hessenberg matrix and $\widetilde{V}_k = [V_1, \ldots, V_{k+1}]$. The global GMRES method constructs, at step $k$, the approximation $X_k$ satisfying the following two relati

Figures (1)

  • Figure 1:

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Definition 2.4
  • Proposition 1
  • Theorem 2.1
  • Definition 3.1: Subspace embedding
  • Proposition 2
  • Definition 3.2: Subspace embedding
  • ...and 14 more