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Deep Uzawa for PDE constrained optimisation

Charalambos G. Makridakis, Aaron Pim, Tristan Pryer

TL;DR

This work presents a numerical solver for optimal control problems constrained by linear and semi-linear second-order elliptic PDEs and includes an extension of Uzawa's algorithm to build approximating sequences for these constrained optimal control problems.

Abstract

In this work, we present a numerical solver for optimal control problems constrained by linear and semi-linear second-order elliptic PDEs. The approach is based on recasting the problem and includes an extension of Uzawa's algorithm to build approximating sequences for these constrained optimal control problems. We prove strong convergence of the iterative scheme in their respective norms, and this convergence is generalised to a class of restricted function spaces. We showcase the algorithm by demonstrating its use numerically with neural network methods that we coin Deep Uzawa Algorithms and show they perform favourably compared with some existing Deep Neural Network approaches.

Deep Uzawa for PDE constrained optimisation

TL;DR

This work presents a numerical solver for optimal control problems constrained by linear and semi-linear second-order elliptic PDEs and includes an extension of Uzawa's algorithm to build approximating sequences for these constrained optimal control problems.

Abstract

In this work, we present a numerical solver for optimal control problems constrained by linear and semi-linear second-order elliptic PDEs. The approach is based on recasting the problem and includes an extension of Uzawa's algorithm to build approximating sequences for these constrained optimal control problems. We prove strong convergence of the iterative scheme in their respective norms, and this convergence is generalised to a class of restricted function spaces. We showcase the algorithm by demonstrating its use numerically with neural network methods that we coin Deep Uzawa Algorithms and show they perform favourably compared with some existing Deep Neural Network approaches.

Paper Structure

This paper contains 23 sections, 8 theorems, 94 equations, 19 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setting and Approach
  3. Model PDE Constrained Optimisation
  4. A strongly coercive Lagrangian
  5. Iterative Approximation
  6. Other Modified Lagrangian Approaches
  7. Convergence Analysis
  8. Non-negativity constraint
  9. Deep Uzawa Algorithm
  10. Neural Network Function Approximation
  11. Training within the Deep Uzawa framework
  12. The Deep Uzawa Iteration
  13. Numerical Experiments: Linear Constraints
  14. Example: Smooth Target
  15. Example: Boundary Layers
  16. ...and 8 more sections

Key Result

Theorem 3.1

Assume that $\Omega\xspace \subset \mathbb{R}^d$ is a convex polygonal domain. Let the sequences $\left\{{u^k}\right\}_{k=0}^\infty \subset \operatorname{H}(\Omega\xspace)$ and $\left\{{f^k}\right\}_{k=0}^\infty \subset \operatorname L\xspace^{2}(\Omega\xspace)$ be generated through the iterative me

Figures (19)

  • Figure 1: Illustration of the architecture within $\mathcal{C}_L$.
  • Figure 2: Example \ref{['sec: 1D Sine\n function target']}. Plots of the state and control (left), and the pointwise error (right), where the exact solutions $u^*, f^* \in \mathcal{V}_N$ are given in equation \ref{['eq: Exact solution 1D\n sine']}, for $\alpha = 10^{-4}$.
  • Figure 3: Example \ref{['sec: 1D Sine function target']}. Plots of the $\operatorname L\xspace^{2}$-error of approximation of state and control as a function of $N_{Uz}$ for various regularisations $\alpha \in [10^{-10}, 1]$.
  • Figure 4: Example \ref{['sec: 1D Sine function target']}. Plots of the $\operatorname L\xspace^{2}$-error of approximation of state and control for $\alpha = 10^{-4}$ and $N_{\text{SGD}} \in [1, 100]$.
  • Figure 5: Example \ref{['sec: 1D Sine function target']}. lots of the $\operatorname L\xspace^{2}$-error of augmented Lagrangian approximation of state and control $u_{\beta,\theta}, f_{\beta,\theta} \in \mathcal{V}_N$, equation \ref{['eq: Augemented Lagrangian Poisson']} for $\beta \in [10^{-10}, 1]$ and $\alpha = 10^{-4}$. The corresponding $\operatorname L\xspace^{2}$-error between the Deep Uzawa state and control is plotted in red.
  • ...and 14 more figures

Theorems & Definitions (16)

  • Remark 2.1: Comparison to the augmented Lagrangian method
  • Remark 2.2: Augmented Lagrangian and an Uzawa iteration
  • Theorem 3.1
  • Remark 3.2: Generalised elliptic operators
  • Theorem 3.3
  • Definition 4.1: Deep-$\mathcal{V}_N$ minimiser
  • Lemma 7.1
  • proof
  • Lemma 7.2
  • proof
  • ...and 6 more