Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical solution of elliptic distributed optimal control problems with boundary value tracking

Ulrich Langer, Richard Löscher, Olaf Steinbach, Huidong Yang

TL;DR

The paper addresses a boundary value tracking optimal control problem for a linear elliptic PDE with a Neumann boundary value problem, where the control acts as the right-hand side. It develops a state-based variational reformulation with energy regularization in $U=\widetilde{H}^{-1}(\Omega)$ and a conforming tensor-product finite element discretization, enabling rigorous discretization error estimates. A fast solver strategy based on Schur complement reduction yields a boundary system with level-independent convergence; both lumped-mass preconditioning and AMG-preconditioned CG are shown to be effective. Numerical experiments in $d=3$ demonstrate second-order boundary convergence and robust solver performance, validating the theoretical results and the practicality of the approach. Future work will extend the framework to enforce homogeneous Neumann conditions via Lagrange multipliers.

Abstract

We consider some boundary value tracking optimal control problem constrained by a Neumann boundary value problem for some elliptic partial differential equation where the control acts as right-hand side. This optimal control problem can be reformulated asa state-based variational problem that is the starting point for the finite element discretizion. In this paper, we only consider atensor-product finite element discretizion for which optimal discretization error estimates and fast solvers can be derived.Numerical experiments illustrate the theoretical results quantitatively.

Numerical solution of elliptic distributed optimal control problems with boundary value tracking

TL;DR

The paper addresses a boundary value tracking optimal control problem for a linear elliptic PDE with a Neumann boundary value problem, where the control acts as the right-hand side. It develops a state-based variational reformulation with energy regularization in and a conforming tensor-product finite element discretization, enabling rigorous discretization error estimates. A fast solver strategy based on Schur complement reduction yields a boundary system with level-independent convergence; both lumped-mass preconditioning and AMG-preconditioned CG are shown to be effective. Numerical experiments in demonstrate second-order boundary convergence and robust solver performance, validating the theoretical results and the practicality of the approach. Future work will extend the framework to enforce homogeneous Neumann conditions via Lagrange multipliers.

Abstract

We consider some boundary value tracking optimal control problem constrained by a Neumann boundary value problem for some elliptic partial differential equation where the control acts as right-hand side. This optimal control problem can be reformulated asa state-based variational problem that is the starting point for the finite element discretizion. In this paper, we only consider atensor-product finite element discretizion for which optimal discretization error estimates and fast solvers can be derived.Numerical experiments illustrate the theoretical results quantitatively.

Paper Structure

This paper contains 6 sections, 6 theorems, 77 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. State-based variational reformulation
  3. Conforming FE discretization on tensor product meshes
  4. Fast solvers
  5. Numerical results
  6. Conclusions and Outlook

Key Result

Lemma 2.1

Let $y_\varrho \in Y$ be the unique solution of the variational formulation Eqn:GradientEquationVF. For $\overline{y} \in L^2(\Gamma)$ there hold the estimates If $\overline{y} \in H^{1/2}(\Gamma)$ is the Dirichlet trace of $\overline{y}_e \in Y$, then $\| y_\varrho \|_{H^1(\Omega)} \leq \| \overline{y}_e \|_{H^1(\Omega)}$, and If $\overline{y} \in H^1(\Gamma)$ is the Dirichlet trace of $\over

Figures (1)

  • Figure 1: Modified piecewise linear basis functions $\varphi_i(x)$, $i=1,\ldots,n-1$.

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • proof
  • Lemma 6.3
  • proof
  • Lemma 6.4
  • ...and 1 more