A posteriori validation of generalized polynomial chaos expansions

TL;DR

This paper develops a rigorous a posteriori validation method for generalized polynomial chaos expansions, enabling precise error estimates and existence proofs for random invariant sets in complex dynamical systems.

Contribution

It introduces a validated numerics framework for gPC expansions, allowing rigorous computation of random invariant sets and solution branches in parameter-dependent systems.

Findings

Validated error bounds for gPC expansions of invariant sets

Rigorous computation of random periodic orbits in Lorenz system

Validated continuation of steady state branches in PDE models

Abstract

Generalized polynomial chaos expansions are a powerful tool to study differential equations with random coefficients, allowing in particular to efficiently approximate random invariant sets associated to such equations. In this work, we use ideas from validated numerics in order to obtain rigorous a posteriori error estimates together with existence results about gPC expansions of random invariant sets. This approach also provides a new framework for conducting validated continuation, i.e. for rigorously computing isolated branches of solutions in parameter-dependent systems, which generalizes in a straightforward way to multi-parameter continuation. We illustrate the proposed methodology by rigorously computing random invariant periodic orbits in the Lorenz system, as well as branches and 2-dimensional manifolds of steady states of the Swift-Hohenberg equation.