# Block preconditioning of stochastic Galerkin problems: New two-sided guaranteed spectral bounds

## TL;DR

This paper develops guaranteed spectral bounds for preconditioned stochastic Galerkin matrices in parametrized diffusion problems, improving understanding of spectral properties and robustness of preconditioning strategies.

## Contribution

It introduces a general approach for two-sided spectral bounds based on operator splitting, applicable to various distributions and less restrictive coefficient conditions.

## Key findings

- Derived new spectral bounds depending on coefficient properties and polynomial type.
- Applicable to multiple classes of block-diagonal preconditioners.
- Bounds are guaranteed and valid for various parameter distributions.

## Abstract

The paper focuses on numerical solution of parametrized diffusion equations with scalar parameter-dependent coefficient function by the stochastic (spectral) Galerkin method. We study preconditioning of the related discretized problems using preconditioners obtained by modifying the stochastic part of the partial differential equation. We present a simple but general approach for obtaining two-sided bounds to the spectrum of the resulting matrices, based on a particular splitting of the discretized operator. Using this tool and considering the stochastic approximation space formed by classical orthogonal polynomials, we obtain new spectral bounds depending solely on the properties of the coefficient function and the type of the approximation polynomials for several classes of block-diagonal preconditioners. These bounds are guaranteed and applicable to various distributions of parameters. Moreover, the conditions on the parameter-dependent coefficient function are only local, and therefore less restrictive than those usually assumed in the literature.