Approximation of random evolution equations of parabolic type

In this paper, we present an abstract framework to obtain convergence rates for the approximation of random evolution equations corresponding to a random family of forms determined by finite-dimensional noise. The full discretization error in space, time, and randomness is considered, where polynomial chaos expansion (PCE) is used for the semi-discretization in randomness. The main result are regularity conditions on the random forms under which convergence of polynomial order in randomness is obtained depending on the smoothness of the coefficients and the Sobolev regularity of the initial value. In space and time, the same convergence rates as in the deterministic setting are achieved. To this end, we derive error estimates for vector-valued PCE as well as a quantified version of the Trotter–Kato theorem for form-induced semigroups. We apply the abstract framework to an anisotropic diffusion model with random diffusion coefficients.

Subjects

Abstract Cauchy problem

approximation

polynomial chaos expansion

convergence rates

strongly continuous semigroups

Parabolic PDEs

random coefficients

47D06

47N40

65J08

35K90

41A25

DDC Class

519: Applied Mathematics, Probabilities

518: Numerical Analysis

515: Analysis

Funding(s)

Projekt DEAL

Lizenz

https://creativecommons.org/licenses/by/4.0/

Publication version

publishedVersion

Name

00028_2026_Article_1158.pdf

Size

795.76 KB

Format

Adobe PDF

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Approximation of random evolution equations of parabolic type