Improving performance of contour integral-based nonlinear eigensolvers with infinite GMRES

TL;DR

The relationship between polynomial eigenvalue problems and their scaled linearizations is analyzed, and a novel weighting strategy is provided which can significantly accelerate the convergence of infinite GMRES in this particular context.

Abstract

In this work, the infinite GMRES algorithm, recently proposed by Correnty et al., is employed in contour integral-based nonlinear eigensolvers, avoiding the computation of costly factorizations at each quadrature node to solve the linear systems efficiently. Several techniques are applied to make the infinite GMRES memory-friendly, computationally efficient, and numerically stable in practice. More specifically, we analyze the relationship between polynomial eigenvalue problems and their scaled linearizations, and provide a novel weighting strategy which can significantly accelerate the convergence of infinite GMRES in this particular context. We also adopt the technique of TOAR to infinite GMRES to reduce the memory footprint. Theoretical analysis and numerical experiments are provided to illustrate the efficiency of the proposed algorithm.

Figures (5)

• Figure 1: We use infGMRES to solve several linear systems of the gun problem. The contour is a circle centered at $66762$ with a radius $45738$. The quadrature nodes $\xi_j$ from $16$ linear systems, $T(\xi_j)x=z$, $j=0$, $\dotsc$, $15$, lie equidistantly on the contour. The only Taylor expansion point is located at the center of the contour, and the infGMRES method with $m=32$ maximum iterations is used in both figures. To illustrate the accuracy, we plot the relative residuals $\lVert T(\xi_j)x_{0,j}-z\rVert_2/\bigl(\lVert T(\xi_j)\rVert_2\lVert x_{0,j}\rVert_2 +\lVert z\rVert_2\bigr)$, where $x_{0,j}$ stands for the approximate solution of the $j$th linear system (same terms as in Algorithm \ref{['alg:infgmres']}). In practice, for approximating eigenpairs, we usually need these linear systems to be solved to an accuracy higher than $10^{-10}$. The original infGMRES method (left) failed at all $16$ points, whereas the infGMRES applied to the variable-substituted system $T(5a\tilde{\xi}+c)$ (right), although still not accurate enough, demonstrates a seemingly improved performance.
• Figure 2: A brief guideline on choosing expansion points: If a single point at the center does not work, choose expansion points equidistantly on the contour or on an inner ellipse, based on the condition number of the problem. Continue doubling the number of expansion points till the accuracy is satisfactory.
• Figure 3: Interest field, expansion points and eigenvalues of different test problems.
• Figure 4: The proportional chart of Beyn's methods with infinite GMRES. The times consumed by different operations are illustrated.
• Figure 5: Solving linear systems of the gun problem by infGMRES without weighting (b, c), with scaling weighting \ref{['eq:scalB2009']} (d, e) and with our weighting \ref{['eq:ourwt']} (f, g). (a) shows a full picture of the distribution of the expansion points and part of the quadrature nodes. The residuals plotted in the figures are the relative residuals of solving linear systems. For a certain quadrature node $\xi_j$, the relative residual is defined as $\lVert T(\xi_j)x_{0,j}-z\rVert_2/\bigl(\lVert T(\xi_j)\rVert_2 \lVert x_{0,j}\rVert_2+\lVert z\rVert_2\bigr)$, where $x_{0,j}$ stands for the approximate solution.

Theorems & Definitions (11)

• Lemma 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Lemma 2
• proof
• Remark 4: How to determine $d_j$'s in practice
• Remark 5
• Remark 6: Accuracy of linear systems and the quadrature rule