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A multigrid solver for PDE-constrained optimization with uncertain inputs

Gabriele Ciaramella, Fabio Nobile, Tommaso Vanzan

TL;DR

The paper addresses scalable solution of PDE-constrained optimization under uncertainty by developing a collective multigrid method for large saddle-point systems. It provides a detailed convergence analysis, including the spectral characterization of the one-level smoother and the two-level method, and demonstrates that the per-smoothing cost scales linearly with problem size, $O(N)$. The framework is validated on three problem classes: a linear-quadratic OCPUU, a nonsmooth OCPUU with box constraints and $L^1$ penalization, and a risk-averse OCPUU using a smoothed CVaR; in all cases the method shows robust performance with respect to regularization, input variance, sampling, and mesh refinement. The results suggest a practical, scalable solver for full-space PDE-constrained optimization under uncertainty, with potential extensions to coarsening in the probability space and broader optimization contexts.

Abstract

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.

A multigrid solver for PDE-constrained optimization with uncertain inputs

TL;DR

The paper addresses scalable solution of PDE-constrained optimization under uncertainty by developing a collective multigrid method for large saddle-point systems. It provides a detailed convergence analysis, including the spectral characterization of the one-level smoother and the two-level method, and demonstrates that the per-smoothing cost scales linearly with problem size, . The framework is validated on three problem classes: a linear-quadratic OCPUU, a nonsmooth OCPUU with box constraints and penalization, and a risk-averse OCPUU using a smoothed CVaR; in all cases the method shows robust performance with respect to regularization, input variance, sampling, and mesh refinement. The results suggest a practical, scalable solver for full-space PDE-constrained optimization under uncertainty, with potential extensions to coarsening in the probability space and broader optimization contexts.

Abstract

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number of samples used to discretized the probability space. We show that this reduced system can be solved with optimal complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and -norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.
Paper Structure (12 sections, 6 theorems, 80 equations, 3 figures, 6 tables, 3 algorithms)

This paper contains 12 sections, 6 theorems, 80 equations, 3 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. A linear-quadratic optimal control problem under uncertainty
  3. Collective multigrid scheme
  4. Convergence analysis
  5. Numerical experiments
  6. An optimal control problem under uncertainty with box-constraints and $L^1$ penalization
  7. Numerical experiments
  8. A risk-averse optimal control problem under uncertainty
  9. Nonlinear preconditioned Newton method
  10. Numerical experiments
  11. Conclusion
  12. Declarations

Key Result

Lemma 2

The matrix $C$ has the spectrum with $r=\frac{\widehat{\mathbb{E}}\left[ \widetilde{\mathbf{d}}^{-2}\right]}{\nu +\widehat{\mathbb{E}}\left[ \widetilde{\mathbf{d}}^{-2}\right]}$, $\widetilde{\mathbf{d}}\in \mathbb{R}^{N\times 1}$, $(\widetilde{\mathbf{d}})_j=N d_j$, and $\widehat{\mathbb{E}}\left[ \widetilde{\mathbf{d}}^{-2}\right

Figures (3)

  • Figure 1: Top row: Graphical representation of the spectrum of $T$ for $N_h=31$, $N=10$, $\nu=10^{-2}$ and $n_1=n_2=1$. The blue circles are obtained by computing numerically the eigenvalues of T. The red crosses are obtained through the formulae of Theorem \ref{['theorem:spectrum_two_level']}. Bottom row: comparison between the numerical and theoretical convergence of the two-level algorithm.
  • Figure 2: From left to right: optimal control computed for $\beta\in \left\{0,5\cdot 10^{-3},5\cdot 10^{-2}\right\}$ with (top row) and without (bottom row) box constraints: $a=-50$, $b=50$.
  • Figure 3: Solution of the linear-quadratic OCP (top-left), solution of the smoothed risk-averse OCP with $\lambda=0.99$ (top-right), and cumulative distribution function of the quantity of interest for the controls computed with $\lambda\in \left\{0,0.5,0.95,0.99\right\}$.

Theorems & Definitions (17)

  • Remark 1: Extension to a hierarchy of samples
  • Lemma 2: Spectrum of C
  • proof
  • Remark 2: Dependence on the regularization parameter
  • Corollary 3: Similarity transformation of $\mathcal{G}$
  • proof
  • Corollary 4: Spectral radius of $\mathcal{G}$
  • proof
  • Remark 3: Damping
  • Lemma 5
  • ...and 7 more