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A reconstructed discontinuous approximation for distributed elliptic control problems

Ruo Li, Haoyang Liu, Jun Yin

TL;DR

The paper addresses distributed elliptic optimal control problems and introduces a reconstructed discontinuous Galerkin (RDA) approach to achieve high-order accuracy with one unknown per element. It develops an interior-penalty DG discretization in the reconstructed space, along with a local reconstruction operator and a linearly convergent projected gradient descent solver for the discrete optimality system. The authors establish rigorous a priori and a posteriori error estimates in $L^2$ and energy norms, including sharp bounds under box constraints, and provide numerical experiments validating convergence and estimator efficiency. The work offers a computationally efficient framework for PDE-constrained optimization with strong convex costs and adaptive error control.

Abstract

In this paper, we present and analyze an internal penalty discontinuous Galerkin method for the distributed elliptic optimal control problems. It is based on a reconstructed discontinuous approximation which admits arbitrarily high-order approximation space with only one unknown per element. Applying this method, we develop a proper discretization scheme that approximates the state and adjoint variables in the approximation space. Our main contributions are twofold: (1) the derivation of both a priori and a posteriori error estimates of the $L^2$-norm and the energy norms, and (2) the implementation of an efficiently solvable discrete system, which is solved via a linearly convergent projected gradient descent method. Numerical experiments are provided to verify the convergence order in a priori estimate and the efficiency of a posteriori error estimate.

A reconstructed discontinuous approximation for distributed elliptic control problems

TL;DR

The paper addresses distributed elliptic optimal control problems and introduces a reconstructed discontinuous Galerkin (RDA) approach to achieve high-order accuracy with one unknown per element. It develops an interior-penalty DG discretization in the reconstructed space, along with a local reconstruction operator and a linearly convergent projected gradient descent solver for the discrete optimality system. The authors establish rigorous a priori and a posteriori error estimates in and energy norms, including sharp bounds under box constraints, and provide numerical experiments validating convergence and estimator efficiency. The work offers a computationally efficient framework for PDE-constrained optimization with strong convex costs and adaptive error control.

Abstract

In this paper, we present and analyze an internal penalty discontinuous Galerkin method for the distributed elliptic optimal control problems. It is based on a reconstructed discontinuous approximation which admits arbitrarily high-order approximation space with only one unknown per element. Applying this method, we develop a proper discretization scheme that approximates the state and adjoint variables in the approximation space. Our main contributions are twofold: (1) the derivation of both a priori and a posteriori error estimates of the -norm and the energy norms, and (2) the implementation of an efficiently solvable discrete system, which is solved via a linearly convergent projected gradient descent method. Numerical experiments are provided to verify the convergence order in a priori estimate and the efficiency of a posteriori error estimate.

Paper Structure

This paper contains 18 sections, 9 theorems, 154 equations, 19 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model Problem and the RDA Method
  3. Model Problem
  4. The Reconstruction Operator
  5. The Reconstructed Discontinuous Approximation Method
  6. Discrete System.
  7. Iterative Method
  8. A Priori Estimates
  9. A General Result
  10. Super-convergence Analysis.
  11. A Posteriori Estimates
  12. A General Result
  13. Sharp A Posteriori Estimate.
  14. Numerical Results
  15. Examples for Convergence Order.
  16. ...and 3 more sections

Key Result

Lemma 1

Let $a_{h}$ be defined with sufficiently large $\mu$, then there exist constants $C,c$ such that

Figures (19)

  • Figure 1: Error of Example 1 for $m=1.$ Take $h_u=h$(left) and $h_u=4h^2$(right).
  • Figure 2: Error of Example 1 for $m=2.$ Take $h_u=4h^2$(left) and $h_u=16h^3$(right).
  • Figure 3: Error of Example 1 for $m=3$. Take $h_u=16h^3$.
  • Figure 4: Recovery error of Example 1. Take $h_u=h.$
  • Figure 5: Error of Example 2 for $m=1.$ Take $h_u=h$(left) and $h_u=4h^2$(right).
  • ...and 14 more figures

Theorems & Definitions (19)

  • Remark 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • ...and 9 more