Probabilistic time integration for semi-explicit PDAEs

TL;DR

This paper explores probabilistic time integration methods for semi-explicit PDE-DAEs, demonstrating their effectiveness in capturing solution sensitivities and establishing their theoretical validity for constrained systems.

Contribution

It introduces randomized probabilistic integrators for semi-explicit PDE-DAEs, analyzing their consistency, convergence, and practical calibration.

Findings

Randomized integrators are consistent and convergent.

Probabilistic methods effectively capture solution sensitivities.

Theoretical analysis supports their application to constrained systems.

Abstract

This paper deals with the application of probabilistic time integration methods to semi-explicit partial differential-algebraic equations of parabolic type and its semi-discrete counterparts, namely semi-explicit differential-algebraic equations of index 2. The proposed methods iteratively construct a probability distribution over the solution of deterministic problems, enhancing the information obtained from the numerical simulation. Within this paper, we examine the efficacy of the randomized versions of the implicit Euler method, the midpoint scheme, and exponential integrators of first and second order. By demonstrating the consistency and convergence properties of these solvers, we illustrate their utility in capturing the sensitivity of the solution to numerical errors. Our analysis establishes the theoretical validity of randomized time integration for constrained systems and offers insights into the calibration of probabilistic integrators for practical applications.