A stochastic Galerkin method for optimal Dirichlet boundary control problems with uncertain data

2025/06/13 by Max Winkler, Winkler, Max, Hamdullah Yücel +1
Computer Science · Decision Sciences · Engineering · #35R60 #65K15 #65N30 #93E20 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #G.1.8 #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2506.11479

openalex publication_date 2025/06/13 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

The paper deals with a stochastic Galerkin approximation of elliptic Dirichlet boundary control problems with random input data. The expectation of a tracking cost functional with the deterministic constrained control is minimized. Error estimates are derived for the control variable in L²(∂ \mathcal D)-norm and state variable in L²(Ω×\mathcal D)-norm. To solve large linear systems, appropriate preconditioners are proposed for both unconstrained and constrained scenarios. To illustrate the validity and efficiency of the proposed approaches, some numerical experiments are performed.

Citations

• Numerical solution of an optimal control problem with probabilistic and almost sure state constraints
• Error estimates for variational normal derivatives and Dirichlet control\n problems with energy regularization
• Finite element error estimates for normal derivatives on boundary concentrated meshes
• Robust Optimization of PDEs with Random Coefficients Using a Multilevel\n Monte Carlo Method
• Multilevel Monte Carlo analysis for optimal control of elliptic PDEs with random coefficients
• Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems
• The Finite Element Method for Elliptic Problems

Related

• A Multilevel Stochastic Gradient method for PDE-constrained Optimal Control Problems with uncertain parameters
• Meshless discretization of LQ-type stochastic control problems
• A Stochastic Optimal Control Problem for the Heat Equation on the Halfline with Dirichlet Boundary-noise and Boundary-control
• Multilevel Monte Carlo analysis for optimal control of elliptic PDEs with random coefficients
• Preconditioners for robust optimal control problems under uncertainty