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Probabilistic time integration for semi-explicit PDAEs

R. Altmann, A. Moradi

TL;DR

The paper develops a probabilistic time-integration framework for semi-explicit PDAEs (parabolic, index-2) by randomizing existing deterministic schemes and injecting noise only in the dynamic kernel part to quantify discretization uncertainty while preserving the constraints. It analyzes mean-square convergence, showing the overall order r equals the minimum of the deterministic order q and the noise order p, and presents four concrete randomized schemes: probabilistic implicit Euler, probabilistic midpoint, probabilistic exponential Euler, and a probabilistic second-order exponential integrator. A numerical example on a constrained semi-linear heat equation validates the theoretical convergence and illustrates how calibration of the noise scale sigma can improve the practical usefulness of the probabilistic solvers. The methods offer a principled way to propagate forward uncertainty through constrained PDE systems and are applicable to Bayesian inverse problems and uncertainty quantification in constrained dynamics.

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.

Probabilistic time integration for semi-explicit PDAEs

TL;DR

The paper develops a probabilistic time-integration framework for semi-explicit PDAEs (parabolic, index-2) by randomizing existing deterministic schemes and injecting noise only in the dynamic kernel part to quantify discretization uncertainty while preserving the constraints. It analyzes mean-square convergence, showing the overall order r equals the minimum of the deterministic order q and the noise order p, and presents four concrete randomized schemes: probabilistic implicit Euler, probabilistic midpoint, probabilistic exponential Euler, and a probabilistic second-order exponential integrator. A numerical example on a constrained semi-linear heat equation validates the theoretical convergence and illustrates how calibration of the noise scale sigma can improve the practical usefulness of the probabilistic solvers. The methods offer a principled way to propagate forward uncertainty through constrained PDE systems and are applicable to Bayesian inverse problems and uncertainty quantification in constrained dynamics.

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.
Paper Structure (14 sections, 6 theorems, 67 equations, 5 figures, 3 algorithms)

This paper contains 14 sections, 6 theorems, 67 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semi-explicit (P)DAEs
  4. Decomposition of the solution and flow map
  5. Deterministic and probabilistic setup
  6. Mean Square Convergence Analysis
  7. Examples of Probabilistic Time Integrators
  8. Probabilistic implicit Euler scheme
  9. Probabilistic midpoint scheme
  10. Probabilistic exponential Euler scheme
  11. Probabilistic exponential integrator of second order
  12. Numerical example
  13. Calibrating Forward Uncertainty Propagation
  14. Conclusion

Key Result

Lemma 3.1

Suppose that for non-negative sequences $\{x^{{n}}\}_{{n}\in[N]_0}$ and $\{y^{{n}}\}_{{n}\in [N]_0}$ and for some constant $L > 0$, we have $x^{{n}+1} \leq y^{{n}}+(1+\tau L)\, x^{{n}}$ for all $n\in[N-1]_0$. Then,

Figures (5)

  • Figure 1.1: The exact trajectory of $V$ of the constrained FitzHugh–Nagumo model from Example \ref{['ex:FitzHugh']} (blue) and $50$ trajectories (red) obtained by the implicit Euler method with Gaussian perturbation on the constraint (left) and on the dynamic part only (right). In both cases the noise scale equals $\sigma = 0.1$.
  • Figure 1.2: Exact value of $V$ (blue) and $50$ trajectories (red) in the same setting as in Figure \ref{['Fig:pert:Fitz:fs']} with different noise scales, namely $\sigma = 0.5$ (left) and $\sigma=1.5$ (right).
  • Figure 4.1: Convergence history for the mean square error for the four probabilistic time integrators introduced in Section \ref{['Sec.PNM']}: implicit Euler method (upper left), midpoint scheme (upper right), exponential Euler (lower left), and the second-order exponential integrator (lower right). Numerical results for different values of $p$ and fixed noise scale $\sigma = 4$.
  • Figure 5.1: The exact trajectory of $V$ (blue) and $100$ realizations (red) computed by the probabilistic implicit Euler method with noise scale $\sigma^*$ (left) and fixed noise scale $\sigma=0.5$ (right).
  • Figure 5.2: A comparison of the observed variations (gray) and their mean (black) in the calibrated probabilistic implicit Euler method using $\sigma^*$ and $\tau=0.04$ for the constrained FitzHugh--Nagumo model from Example \ref{['ex:FitzHugh:revisit']}. The error indicator is marked in red.

Theorems & Definitions (21)

  • Example 1.1
  • Remark 2.3
  • Example 2.5
  • Example 2.6
  • Lemma 3.1: Discrete Gronwall lemma lie22
  • Theorem 3.2: Global mean square convergence
  • proof
  • Remark 3.3
  • Lemma 4.1
  • proof
  • ...and 11 more