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

Abstract:

In this talk we explore quadratures for the block-symmetric form \(B^\top \phi(A, s)B\) where \(A\) is a symmetric nonnegative definite matrix, \(B\) is a tall matrix with \(p\) columns, and \(\phi(\cdot, s)\) is a matrix function with parameter \(s \in \mathbb{R}_+\). These formulations commonly arise in the computation of multiple-input multiple-output transfer functions for diffusion PDEs. We propose an approximation scheme for \(B^\top \phi(A, s)B\) leveraging the block Lanczos algorithm and its representation through Stieltjes matrix continued fractions. Using Stieltjes matrix continued fractions we show that the block-Lanczos algorithm converges monotonically for the resolvent and we extend the notion of Gauss-Radau quadrature to the block case. This leads to a tight error estimate for the block-Lanczos algorithms and allows us to extrapolate by averaging Gauss- and Gauss-Radau estimates. Numerical examples are used to showcase such an approach for 3D diffusive PDEs.