A Posteriori Validation of Generalized Polynomial Chaos Expansions*

Generalized polynomial chaos (gPC) expansions are a powerful tool for studying differential equations with random coefficients, allowing, in particular, one to efficiently approximate random invariant sets associated with 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 multiparameter 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-Hohenb erg equation.

Automated summary

Generalized polynomial chaos (gPC) expansions are a powerful tool for efficiently approximating random invariant sets associated with differential equations with random coefficients. This work introduces a new framework for conducting validated continuation in parameter-dependent systems, allowing for rigorous computation of isolated branches of solutions. The proposed methodology is applied to compute random invariant periodic orbits in the Lorenz system and steady states of the Swift-Hohenberg equation.