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Minimal residual rational Krylov subspace method for sequences of shifted linear systems

Hussam Al Daas, Davide Palitta

TL;DR

This work tackles the challenge of solving sequences of shifted linear systems $(A+s_i I_n)x^{(i)}=b^{(i)}$, including difficult cases with nonsymmetric $A$ and complex shifts lacking conjugate pairs. It introduces a minimal residual rational Krylov subspace method (MR-RKSM) that projects onto a rational Krylov space $\mathbf{K}_{m+1}(A,b,\bm{\xi}_m)$ and computes the solution via $X_m=V_{m+1}\underline{K}_mY_m$, where $Y_m$ minimizes the residual, avoiding residual collinearity requirements. A tailored greedy pole-selection strategy further enhances convergence by selecting new poles from the set of shifts based on current residuals, and a fully iterative variant handles inner solves when direct solves are impractical. The method extends to block right-hand sides and demonstrates strong numerical performance against state-of-the-art approaches, especially for challenging complex shifts, while maintaining low memory usage through low-rank representations. The results advocate a matrix-equation formulation as a robust framework for these problems and point to promising directions for further theoretical and practical refinements.

Abstract

The solution of sequences of shifted linear systems is a classic problem in numerical linear algebra, and a variety of efficient methods have been proposed over the years. Nevertheless, there still exist challenging scenarios witnessing a lack of performing solvers. For instance, state-of-the-art procedures struggle to handle nonsymmetric problems where the shifts are complex numbers that do not come as conjugate pairs. We design a novel projection strategy based on the rational Krylov subspace equipped with a minimal residual condition. We also devise a novel pole selection procedure, tailored to our problem, providing poles for the rational Krylov basis construction that yield faster convergence than those computed by available general-purpose schemes. A panel of diverse numerical experiments shows that our novel approach performs better than state-of-the-art techniques, especially on the very challenging problems mentioned above.

Minimal residual rational Krylov subspace method for sequences of shifted linear systems

TL;DR

This work tackles the challenge of solving sequences of shifted linear systems , including difficult cases with nonsymmetric and complex shifts lacking conjugate pairs. It introduces a minimal residual rational Krylov subspace method (MR-RKSM) that projects onto a rational Krylov space and computes the solution via , where minimizes the residual, avoiding residual collinearity requirements. A tailored greedy pole-selection strategy further enhances convergence by selecting new poles from the set of shifts based on current residuals, and a fully iterative variant handles inner solves when direct solves are impractical. The method extends to block right-hand sides and demonstrates strong numerical performance against state-of-the-art approaches, especially for challenging complex shifts, while maintaining low memory usage through low-rank representations. The results advocate a matrix-equation formulation as a robust framework for these problems and point to promising directions for further theoretical and practical refinements.

Abstract

The solution of sequences of shifted linear systems is a classic problem in numerical linear algebra, and a variety of efficient methods have been proposed over the years. Nevertheless, there still exist challenging scenarios witnessing a lack of performing solvers. For instance, state-of-the-art procedures struggle to handle nonsymmetric problems where the shifts are complex numbers that do not come as conjugate pairs. We design a novel projection strategy based on the rational Krylov subspace equipped with a minimal residual condition. We also devise a novel pole selection procedure, tailored to our problem, providing poles for the rational Krylov basis construction that yield faster convergence than those computed by available general-purpose schemes. A panel of diverse numerical experiments shows that our novel approach performs better than state-of-the-art techniques, especially on the very challenging problems mentioned above.

Paper Structure

This paper contains 8 sections, 2 theorems, 40 equations, 2 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. When to expect a low-rank solution
  3. Minimal residual rational Krylov approach
  4. Pole selection
  5. A fully iterative variant
  6. The case of multiple right-hand sides
  7. Numerical experiments
  8. Conclusion

Key Result

Proposition 2.1

Let $A=Q\Lambda Q^{-1}$, $\Lambda=\text{diag}(\lambda_1,\ldots,\lambda_n)$, and assume that there exists a centroid $\mathbf{s}$ such that $\max_{i=1,\ldots,\ell}|s_i-\mathbf{s}|\leq \epsilon$ and that $\min_{j=1,\ldots,n}|\lambda_j+\mathbf{s}|/\epsilon\gg 1$. Then where $c(\epsilon,\mathbf{s},\Lambda)=\epsilon/\min_{j=1,\ldots,n}|\lambda_j+\mathbf{s}|\ll 1$and $\kappa_F(Q)=\|Q\|_F\|Q^{-1}\|_F$.

Figures (2)

  • Figure 1: Example \ref{['ex1']} (2D) - shifts: complex - no conjugate pairs. Relative residual norm achieved by MR-RKSM (blue, solid line), G-EKSM (red, dashed line), and FOM(100) (black, dotted line). For FOM(100) we report the relative residual norm of the first 100 iterations, namely during the first cycle. A similar behavior is also observed in the following restarting cycles.
  • Figure 2: Example \ref{['Ex2']}, $\ell=1024$. Heatmap where we report $\|r_m^{(j)}\|$ in logarithmic scale for all iterations $m$ ($x$-axis) and shift indeces $j$ ($y$-axis). Left: MR-RKSM with poles as in sec. \ref{['Poles selection']}. Right: MR-RKSM with ADM poles Poles1.

Theorems & Definitions (14)

  • Proposition 2.1
  • Proof 1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • Proof 2
  • ...and 4 more