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Approximation of Random Evolution Equations of Parabolic type

Katharina Klioba, Christian Seifert

TL;DR

The paper develops an abstract framework to approximate random parabolic evolution equations driven by finite-dimensional noise, providing convergence rates for a fully discrete scheme in randomness, space, and time. It combines polynomial chaos expansion in randomness with Galerkin spatial discretization and $A$-stable time stepping, and proves a joint convergence rate governed by the randomness regularity ($\ell$), spatial accuracy ($p_x$), and temporal accuracy ($p_t$). A key contribution is a quantified Trotter--Kato argument for form-induced semigroups, enabling control of the randomness discretization error at the level of the semigroup. The theory is illustrated on an anisotropic diffusion problem with random coefficients, showing that, under suitable smoothness, the polynomial chaos approximation attains arbitrary algebraic convergence in randomness, while space-time discretizations maintain deterministic-like rates.

Abstract

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.

Approximation of Random Evolution Equations of Parabolic type

TL;DR

The paper develops an abstract framework to approximate random parabolic evolution equations driven by finite-dimensional noise, providing convergence rates for a fully discrete scheme in randomness, space, and time. It combines polynomial chaos expansion in randomness with Galerkin spatial discretization and -stable time stepping, and proves a joint convergence rate governed by the randomness regularity (), spatial accuracy (), and temporal accuracy (). A key contribution is a quantified Trotter--Kato argument for form-induced semigroups, enabling control of the randomness discretization error at the level of the semigroup. The theory is illustrated on an anisotropic diffusion problem with random coefficients, showing that, under suitable smoothness, the polynomial chaos approximation attains arbitrary algebraic convergence in randomness, while space-time discretizations maintain deterministic-like rates.

Abstract

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.
Paper Structure (22 sections, 33 theorems, 180 equations, 1 figure)

This paper contains 22 sections, 33 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Random PDEs: a literature overview
  3. Main Results
  4. Some words on our method of proof
  5. Outline of the paper
  6. Random Evolution Equations
  7. Approximation Methods
  8. Spatial approximation
  9. Temporal approximation
  10. Randomness approximation
  11. Deterministic Approximation Results
  12. Approximation of Random Evolution Equations
  13. Derivation of the approximate problem
  14. Semi-discretization in space and time
  15. Semi-discretization in randomness
  16. ...and 7 more sections

Key Result

Theorem 1.1

Suppose that the assumptions of Theorem thm:jointrate_symmetric hold. In particular, assume that for some $\ell \in \mathbb{N}_0$, the random family of forms $(a_z)_{z \in \mathbb{R}^N}$ is $\mathbb{P}_Z$-almost surely uniformly bounded and uniformly coercive and satisfies $[z \mapsto a_z(u,v)] \in for $\mathbb{P}_Z$-almost every $z\in\mathbb{R}^N$ and for all $u \in D$, $v \in V$, and multi-indi

Figures (1)

  • Figure 1: The curve $\gamma_r$.

Theorems & Definitions (91)

  • Theorem 1.1
  • Proposition 2.1
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5: based on (6.6) in arendtGalerkin
  • Theorem 3.6
  • Theorem 3.7
  • proof
  • ...and 81 more