Monotonicity, bounds and extrapolation of Block-Gauss and Gauss-Radau quadrature for computing $B^T ϕ(A) B$
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arXiv
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2024
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| author | Zimmerling, Jörn Druskin, Vladimir Simoncini, Valeria |
| author_facet | Zimmerling, Jörn Druskin, Vladimir Simoncini, Valeria |
| contents | 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 $B$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_21505 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monotonicity, bounds and extrapolation of Block-Gauss and Gauss-Radau quadrature for computing $B^T ϕ(A) B$ Zimmerling, Jörn Druskin, Vladimir Simoncini, Valeria Numerical Analysis 65F10, 65N22, 65F50, 65F60 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 $B$. |
| title | Monotonicity, bounds and extrapolation of Block-Gauss and Gauss-Radau quadrature for computing $B^T ϕ(A) B$ |
| topic | Numerical Analysis 65F10, 65N22, 65F50, 65F60 |
| url | https://arxiv.org/abs/2407.21505 |