Seminar: "Matricial Stieltjes Continued Fractions and Block Quadrature: Bounding Errors and Accelerating Krylov Subspace Methods"

Abstract:

In this talk, we revisit the matricial Stieltjes moment problem to explore its connections with block-Krylov subspace methods. We examine how expressing transfer functions and reduced-order models as matricial S-fractions leads directly to rigorous error bounds and convergence acceleration.

By framing the problem through S-fractions, we naturally recover block-Gauss and block-Gauss-Radau quadrature rules. This connection yields a built-in bound for the quadrature error, which serves as an ideal stopping criterion for block-Krylov iterations. I will walk through the formal proof of this bound for elliptical problems and discuss the underlying proof concept, alongside numerical evidence, for the parabolic case.

To push the theory further, we introduce the concept of "matricial S-fraction terminators." By approximating the infinite tail of the continued fraction, these terminators significantly accelerate the convergence of the underlying Krylov methods. Finally, we will step out of the pure theory to see these methods in action. We will briefly demonstrate how this S-fraction framework is applied to industry-scale multiple-input multiple-output (MIMO) PDE systems, specifically in 3D-anisotropic borehole electromagnetic simulations.