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A model reduction method based on nonlinear optimization for multiscale stochastic optimal control problems

Jingyi Zhang

TL;DR

This work tackles stochastic PDE‑constrained optimal control with high‑dimensional uncertainty by introducing a nonintrusive, $L^2$‑optimal reduced‑order modeling (DDROM) approach. It relies on a parameter‑separable, affine structure and directly minimizes the $\mathcal{L}^2$ output error, using only observable outputs to build the surrogate and gradient expressions, thereby decoupling from the full PDE matrices. A gradient‑based algorithm, $\mathcal{L}^2$‑Opt‑PSF, updates DDROM matrices to minimize the $\mathcal{L}^2$ error, and GMsFEM provides offline multiscale bases to accelerate computations. Numerical tests on stochastic diffusion and advection–diffusion problems demonstrate accurate, real‑time capable control predictions with complexity independent of the original PDE dimension, effectively bridging theory and engineering practice.

Abstract

This paper presents a nonlinear optimization-based model reduction method for multiscale stochastic optimal control problems governed by stochastic partial differential equations. The proposed approach constructs a non-intrusive, data-driven reduced-order model by employing a parameter-separable structure to handle stochastic dependencies and directly minimizing the L2 norm of the output error via gradient-based optimization. Compared to existing methods, this framework offers three significant advantages: it is entirely data-driven, relying solely on output measurements without requiring access to internal system matrices; it guarantees approximation accuracy for control outputs, aligning directly with the optimization objective; and its computational complexity is independent of the original PDE dimension, ensuring feasibility for real-time control applications. Numerical experiments on stochastic diffusion and advection-diffusion equations demonstrate the method's effectiveness and efficiency, providing a systematic solution for the real-time control of complex uncertain systems and bridging the gap between model reduction theory and practical engineering.

A model reduction method based on nonlinear optimization for multiscale stochastic optimal control problems

TL;DR

This work tackles stochastic PDE‑constrained optimal control with high‑dimensional uncertainty by introducing a nonintrusive, ‑optimal reduced‑order modeling (DDROM) approach. It relies on a parameter‑separable, affine structure and directly minimizes the output error, using only observable outputs to build the surrogate and gradient expressions, thereby decoupling from the full PDE matrices. A gradient‑based algorithm, ‑Opt‑PSF, updates DDROM matrices to minimize the error, and GMsFEM provides offline multiscale bases to accelerate computations. Numerical tests on stochastic diffusion and advection–diffusion problems demonstrate accurate, real‑time capable control predictions with complexity independent of the original PDE dimension, effectively bridging theory and engineering practice.

Abstract

This paper presents a nonlinear optimization-based model reduction method for multiscale stochastic optimal control problems governed by stochastic partial differential equations. The proposed approach constructs a non-intrusive, data-driven reduced-order model by employing a parameter-separable structure to handle stochastic dependencies and directly minimizing the L2 norm of the output error via gradient-based optimization. Compared to existing methods, this framework offers three significant advantages: it is entirely data-driven, relying solely on output measurements without requiring access to internal system matrices; it guarantees approximation accuracy for control outputs, aligning directly with the optimization objective; and its computational complexity is independent of the original PDE dimension, ensuring feasibility for real-time control applications. Numerical experiments on stochastic diffusion and advection-diffusion equations demonstrate the method's effectiveness and efficiency, providing a systematic solution for the real-time control of complex uncertain systems and bridging the gap between model reduction theory and practical engineering.

Paper Structure

This paper contains 14 sections, 3 theorems, 84 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic optimal control problem
  4. Formulation and structure.
  5. $\mathcal{L}_2$-optimal reduced-order modeling
  6. The establishment of the model
  7. Optimization problem with parameter-separable forms
  8. The gradients of the $\mathcal{L}_2$ approximation error
  9. The data-driven, gradient-based algorithm
  10. Generalized multiscale finite element method
  11. Numerical examples
  12. Optimal control for stochastic diffusion equation
  13. Optimal control for stochastic advection-diffusion equation
  14. Conclusion

Key Result

Theorem 2.1

Let $\mathscr{U}_0:=\left\{\underline{u}\in\mathscr{U}:\mathcal{B}(\underline{u},\widetilde{u})=0,\forall\widetilde{u}\in\mathcal{H}_0^1(\Omega)\right\}$ be the kernel space of bilinear form $\mathcal{B}(\cdot,\cdot)$, we can obtain the global existence of a unique solution to the minimization probl

Figures (12)

  • Figure 1: High-contrast coefficient $\kappa_1$
  • Figure 2: Error plots for the quantity of interest $y$ with the finite element solution $y(\mu)$ as the reference, where $\hat{y}(\mu)$ is the interest of the reduced-order model.
  • Figure 3: Relative error plots for the quantity of interest $y$ related only to $U$ or $F$.
  • Figure 4: Error plots for the quantity of interest $y$.
  • Figure 5: Images of solution $U$ at $\overline{\mu}$.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • Definition 3.2
  • Lemma 3.3
  • Theorem 3.4
  • proof