Multilevel quadrature formulae for the optimal control of random PDEs

TL;DR

This work tackles PDE-constrained optimal control under uncertainty by introducing a general multilevel quadrature framework that combines solutions of discretized optimal control problems across multiple spatial and probabilistic accuracy levels. The method uses forward/adjoint PDE solves at different levels and postprocesses adjoint information to form a single, efficient approximation $u^{\text{MQ}}(L)$ of the optimal control, with convergence and complexity results for unconstrained linear-quadratic problems. Under mild assumptions on spatial discretization and positive-weight, unbiased quadrature rules, the authors derive an explicit error bound and show that MLMC can achieve a target tolerance $\varepsilon$ with favorable cost scaling, often outperforming standard Monte Carlo approaches. Numerical experiments in 1D and 2D domains confirm improved computational complexity and even illustrate robustness to control constraints, highlighting practical impact for high-dimensional or weakly regular random PDE problems.

Abstract

This manuscript presents a framework for using multilevel quadrature formulae to compute the solution of optimal control problems constrained by random partial differential equations. Our approach consists in solving a sequence of optimal control problems discretized with different levels of accuracy of the physical and probability discretizations. The final approximation of the control is then obtained in a postprocessing step, by suitably combining the adjoint variables computed on the different levels. We present a general convergence and complexity analysis for an unconstrained linear quadratic problem under abstract assumptions on the spatial discretization and on the quadrature formulae. We detail our framework for the specific case of a MultiLevel Monte Carlo (MLMC) quadrature formula, and numerical experiments confirm the better computational complexity of our MLMC approach compared to a standard Monte Carlo sample average approximation, even beyond the theoretical assumptions.

Figures (5)

• Figure 1: Reference solutions $u^\star$ for the two dimensional problem without (left) and with (right) constraints.
• Figure 2: Results of the preliminary analysis to fit the constants $C_1$, $\alpha$, $C_2$, $\beta$ and $C_3$ for the unconstrained case (top row) and constrained case (bottom row) for $d=2$.
• Figure 3: Verification of \ref{['eq:lemma']} without (left) and with (center) box constraints. The right panel shows the growth of the computational time to assemble the finite element matrices and to solve the optimality system as the problem size increases.
• Figure 4: For the unconstrained problem, we report the error decay (left panel), the computational complexity (center panel) and the computational time (right panel). The top row refers to $d=1$, the bottom row to $d=2$.
• Figure 5: For the constrained problem, we report the error decay (left panel), the computational complexity (center panel) and the computational time (right panel). The top row refers to $d=1$, the bottom row to $d=2$.

Theorems & Definitions (19)

• Proposition 3.1
• Proof 1
• Remark 3.2: Proposition \ref{['lemma:error_bound']} for deterministic quadrature formulae
• Remark 3.3: On the use of two different quadrature formulae on each level
• Example 4.3
• Lemma 4.4
• Proof 2
• Lemma 4.6
• Proof 3
• Remark 4.7: Lemma \ref{['lemma:surplus']} for deterministic quadrature formulae