Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stochastic Galerkin Method and Hierarchical Preconditioning for PDE-constrained Optimization

Zhendong Li, Akwum Onwunta, Bedřich Sousedík

TL;DR

The paper addresses the challenge of solving large, ill-conditioned KKT systems arising from PDE-constrained optimization with uncertain coefficients by developing hierarchical preconditioners based on truncated stochastic expansions. It combines finite element discretization, stochastic Galerkin (generalized polynomial chaos), and all-at-once time discretization, and proves spectral equivalence between the practical preconditioners and the ideal Schur complement. For steady-state problems, it introduces Schur-complement based and block-diagonal hGS preconditioners, and extends the approach to time-dependent problems via parallel-in-time enabled all-at-once preconditioning (PINT). Numerical experiments demonstrate significant acceleration over mean-based and full-expansion methods, with robustness to parameter variations and scalability to very large problems. This yields robust, scalable solvers for stochastic PDE-constrained optimization and paves the way for applications to more complex PDEs and inequality-constrained settings.

Abstract

We develop efficient hierarchical preconditioners for optimal control problems governed by partial differential equations with uncertain coefficients. Adopting a discretize-then-optimize framework that integrates finite element discretization, stochastic Galerkin approximation, and advanced time-discretization schemes, the approach addresses the challenge of large-scale, ill-conditioned linear systems arising in uncertainty quantification. By exploiting the sparsity inherent in generalized polynomial chaos expansions, we derive hierarchical preconditioners based on truncated stochastic expansion that strike an effective balance between computational cost and preconditioning quality. Numerical experiments demonstrate that the proposed preconditioners significantly accelerate the convergence of iterative solvers compared to existing methods, providing robust and efficient solvers for both steady-state and time-dependent optimal control applications under uncertainty.

Stochastic Galerkin Method and Hierarchical Preconditioning for PDE-constrained Optimization

TL;DR

The paper addresses the challenge of solving large, ill-conditioned KKT systems arising from PDE-constrained optimization with uncertain coefficients by developing hierarchical preconditioners based on truncated stochastic expansions. It combines finite element discretization, stochastic Galerkin (generalized polynomial chaos), and all-at-once time discretization, and proves spectral equivalence between the practical preconditioners and the ideal Schur complement. For steady-state problems, it introduces Schur-complement based and block-diagonal hGS preconditioners, and extends the approach to time-dependent problems via parallel-in-time enabled all-at-once preconditioning (PINT). Numerical experiments demonstrate significant acceleration over mean-based and full-expansion methods, with robustness to parameter variations and scalability to very large problems. This yields robust, scalable solvers for stochastic PDE-constrained optimization and paves the way for applications to more complex PDEs and inequality-constrained settings.

Abstract

We develop efficient hierarchical preconditioners for optimal control problems governed by partial differential equations with uncertain coefficients. Adopting a discretize-then-optimize framework that integrates finite element discretization, stochastic Galerkin approximation, and advanced time-discretization schemes, the approach addresses the challenge of large-scale, ill-conditioned linear systems arising in uncertainty quantification. By exploiting the sparsity inherent in generalized polynomial chaos expansions, we derive hierarchical preconditioners based on truncated stochastic expansion that strike an effective balance between computational cost and preconditioning quality. Numerical experiments demonstrate that the proposed preconditioners significantly accelerate the convergence of iterative solvers compared to existing methods, providing robust and efficient solvers for both steady-state and time-dependent optimal control applications under uncertainty.
Paper Structure (13 sections, 8 theorems, 111 equations, 9 tables, 6 algorithms)

This paper contains 13 sections, 8 theorems, 111 equations, 9 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Steady-state problem
  3. Problem formulation
  4. Schur complement based preconditioner
  5. Block-diagonal hierarchical Gauss-Seidel preconditioner
  6. Time-dependent problem
  7. Problem formulation
  8. PINT-based block-diagonal hierarchical Gauss-Seidel preconditioner
  9. Spectral analysis (time-dependent case)
  10. Numerical experiments
  11. Steady-state case
  12. Time-dependent case
  13. Conclusions

Key Result

Lemma 4.2

\newlabellemma:perturbation Let $A$ and $A_r$ be symmetric positive definite matrices satisfying $(1-\varepsilon_1)A \preceq A_r \preceq (1+\varepsilon_2)A$ in the Loewner order for some $0 < \varepsilon_1,\varepsilon_2 < 1$. Let $B$ be a symmetric positive semidefinite matrix. Then, the eigenvalue

Theorems & Definitions (15)

  • Definition 4.1: Spectral Equivalence
  • Lemma 4.2
  • proof
  • Lemma 4.3: Eigenvalues under Congruence Transformation
  • proof
  • Theorem 4.4: Theorems 4, 6 in Benner-2016-BDP
  • Lemma 4.5
  • proof
  • Lemma 4.6
  • proof
  • ...and 5 more