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Monotonicity, bounds and extrapolation of Block-Gauss and Gauss-Radau quadrature for computing $B^T φ(A) B$

Jörn Zimmerling, Vladimir Druskin, Valeria Simoncini

TL;DR

This work develops a block Lanczos-based framework for computing transfer-type matrix functions $F(s)=B^T \phi(A,s) B$ with SPD $A$, focusing on $\phi(A,s)=(A+sI)^{-1}$ and extending to Stieltjes functions. It introduces a block Gauss quadrature via the block Lanczos process and a corresponding block Gauss-Radau variant that fixes zero as a spectral node, yielding monotone two-sided error bounds. By representing the block quadrature as a matricial Stieltjes continued fraction (S-fraction) and its pencil form, the authors derive practical error bounds and averaging-based acceleration schemes that improve convergence for PDEs on unbounded domains and for large graph-Laplacians. Numerical experiments on 2D diffusion, 3D Maxwell diffusion, and graph-Laplacian transfer functions demonstrate the tightness of bounds and the effectiveness of averaging and, in some cases, random subspace enrichment for faster convergence.

Abstract

In this paper, we explore quadratures for the evaluation of $B^T φ(A) B$ where $A$ is a symmetric positive-definite (s.p.d.) matrix in $\mathbb{R}^{n \times n}$, $B$ is a tall matrix in $\mathbb{R}^{n \times p}$, and $φ(\cdot)$ represents a matrix function that is regular enough in the neighborhood of $A$'s spectrum, e.g., a Stieltjes or exponential function. These formulations, for example, commonly arise in the computation of multiple-input multiple-output (MIMO) transfer functions for diffusion PDEs. We propose an approximation scheme for $B^T φ(A) B$ leveraging the block Lanczos algorithm and its equivalent representation through Stieltjes matrix continued fractions. We extend the notion of Gauss-Radau quadrature to the block case, facilitating the derivation of easily computable error bounds. For problems stemming from the discretization of self-adjoint operators with a continuous spectrum, we obtain sharp estimates grounded in potential theory for Padé approximations and justify averaging algorithms at no added computational cost. The obtained results are illustrated on large-scale examples of 2D diffusion and 3D Maxwell's equations and a graph from the SNAP repository. We also present promising experimental results on convergence acceleration via random enrichment of the initial block $B$.

Monotonicity, bounds and extrapolation of Block-Gauss and Gauss-Radau quadrature for computing $B^T φ(A) B$

TL;DR

This work develops a block Lanczos-based framework for computing transfer-type matrix functions with SPD , focusing on and extending to Stieltjes functions. It introduces a block Gauss quadrature via the block Lanczos process and a corresponding block Gauss-Radau variant that fixes zero as a spectral node, yielding monotone two-sided error bounds. By representing the block quadrature as a matricial Stieltjes continued fraction (S-fraction) and its pencil form, the authors derive practical error bounds and averaging-based acceleration schemes that improve convergence for PDEs on unbounded domains and for large graph-Laplacians. Numerical experiments on 2D diffusion, 3D Maxwell diffusion, and graph-Laplacian transfer functions demonstrate the tightness of bounds and the effectiveness of averaging and, in some cases, random subspace enrichment for faster convergence.

Abstract

In this paper, we explore quadratures for the evaluation of where is a symmetric positive-definite (s.p.d.) matrix in , is a tall matrix in , and represents a matrix function that is regular enough in the neighborhood of 's spectrum, e.g., a Stieltjes or exponential function. These formulations, for example, commonly arise in the computation of multiple-input multiple-output (MIMO) transfer functions for diffusion PDEs. We propose an approximation scheme for leveraging the block Lanczos algorithm and its equivalent representation through Stieltjes matrix continued fractions. We extend the notion of Gauss-Radau quadrature to the block case, facilitating the derivation of easily computable error bounds. For problems stemming from the discretization of self-adjoint operators with a continuous spectrum, we obtain sharp estimates grounded in potential theory for Padé approximations and justify averaging algorithms at no added computational cost. The obtained results are illustrated on large-scale examples of 2D diffusion and 3D Maxwell's equations and a graph from the SNAP repository. We also present promising experimental results on convergence acceleration via random enrichment of the initial block .
Paper Structure (18 sections, 8 theorems, 36 equations, 5 figures, 2 algorithms)

This paper contains 18 sections, 8 theorems, 36 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Notation
  4. Block Gauss Quadratures via the Block Lanczos Algorithm
  5. Gauss quadrature and the matrix S-fraction
  6. Block Gauss-Radau quadrature
  7. Two-sided error bounds and convergence acceleration via averaging of Gauss and Gauss-Radau rules
  8. Monotonically convergent lower and upper bounds
  9. Acceleration via averaged Gauss-Radau and Gauss rules
  10. Numerical Examples
  11. Transfer functions of PDE operators on unbounded domains
  12. 2D Diffusion
  13. 3D Electromagnetic diffusion
  14. Graph Laplacian
  15. Conclusions
  16. ...and 3 more sections

Key Result

lemma thmcounterlemma

Let $T_m$ be as defined in equation (eq:T). Then there exist $\hat{\boldsymbol \kappa}_{i}\in{\mathbb{R}}^{p\times p}$, $\hat{\boldsymbol \kappa}_1=I_p$ and $\boldsymbol{\gamma}_{i}\in{\mathbb{R}}^{p\times p}$ all full rank such that where and $\boldsymbol \alpha_1=(\hat{\boldsymbol \kappa}_{1}^{-1})^T\boldsymbol{\gamma}_1^{-1}\hat{\boldsymbol \kappa}_{1}^{-1}=\boldsymbol{\gamma}_1^{-1}$, $\bold

Figures (5)

  • Figure 1: Grid, head conductivity $\sigma(\mathbf{x})$ and transducer locations of the heat diffusion textcase.
  • Figure 2: Convergence and quadrature rule averaging for the 2D heat diffusion problem.
  • Figure 3: Convergence of the block Lanczos method and averaging for the 3D electromagnetic case.
  • Figure 4: Convergence, error bound and averaged quadrature rules for the normalized graph Laplacian test case
  • Figure 5: Acceleration of convergence by adding random vectors to the starting block vector $B$ for shift $s=10^{-2}$.

Theorems & Definitions (12)

  • remark thmcounterremark
  • lemma thmcounterlemma
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • corollary thmcountercorollary
  • remark thmcounterremark
  • remark thmcounterremark
  • theorem 1
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • ...and 2 more