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Nonlinear Conjugate Gradient Methods for PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics

Sebastian Blauth

TL;DR

This work develops nonlinear conjugate gradient (NCG) methods on the Riemannian shape space endowed with Steklov-Poincaré-type metrics for PDE-constrained shape optimization. It embeds five NCG variants—$d_k = -g_k + eta_k d_{k-1}$ with $eta_k \\in \{FR,PR,HS,DY,HZ\}$—into a unified gradient-based framework alongside gradient descent and L-BFGS, and introduces a volume-based discretization that extends gradient information to the entire domain via a gradient deformation $oldsymbol{G}$. The authors demonstrate, through four benchmark problems (Poisson, electrical impedance tomography, Stokes, and Navier–Stokes pipe), that NCG methods consistently outperform gradient descent and achieve performance comparable to low-memory L-BFGS, while requiring significantly less memory. The results highlight the practicality and scalability of NCG on PDE-constrained shape optimization, offering a memory-efficient alternative suitable for large-scale industrial problems.

Abstract

Shape optimization based on shape calculus has received a lot of attention in recent years, particularly regarding the development, analysis, and modification of efficient optimization algorithms. In this paper we propose and investigate nonlinear conjugate gradient methods based on Steklov-Poincaré-type metrics for the solution of shape optimization problems constrained by partial differential equations. We embed these methods into a general algorithmic framework for gradient-based shape optimization methods and discuss the numerical discretization of the algorithms. We numerically compare the proposed nonlinear conjugate gradient methods to the already established gradient descent and limited memory BFGS methods for shape optimization on several benchmark problems. The results show that the proposed nonlinear conjugate gradient methods perform well in practice and that they are an efficient and attractive addition to already established gradient-based shape optimization algorithms.

Nonlinear Conjugate Gradient Methods for PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics

TL;DR

This work develops nonlinear conjugate gradient (NCG) methods on the Riemannian shape space endowed with Steklov-Poincaré-type metrics for PDE-constrained shape optimization. It embeds five NCG variants— with —into a unified gradient-based framework alongside gradient descent and L-BFGS, and introduces a volume-based discretization that extends gradient information to the entire domain via a gradient deformation . The authors demonstrate, through four benchmark problems (Poisson, electrical impedance tomography, Stokes, and Navier–Stokes pipe), that NCG methods consistently outperform gradient descent and achieve performance comparable to low-memory L-BFGS, while requiring significantly less memory. The results highlight the practicality and scalability of NCG on PDE-constrained shape optimization, offering a memory-efficient alternative suitable for large-scale industrial problems.

Abstract

Shape optimization based on shape calculus has received a lot of attention in recent years, particularly regarding the development, analysis, and modification of efficient optimization algorithms. In this paper we propose and investigate nonlinear conjugate gradient methods based on Steklov-Poincaré-type metrics for the solution of shape optimization problems constrained by partial differential equations. We embed these methods into a general algorithmic framework for gradient-based shape optimization methods and discuss the numerical discretization of the algorithms. We numerically compare the proposed nonlinear conjugate gradient methods to the already established gradient descent and limited memory BFGS methods for shape optimization on several benchmark problems. The results show that the proposed nonlinear conjugate gradient methods perform well in practice and that they are an efficient and attractive addition to already established gradient-based shape optimization algorithms.

Paper Structure

This paper contains 17 sections, 3 theorems, 70 equations, 13 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. NCG Methods for Finite Dimensional Optimization Problems
  4. Shape Calculus
  5. Riemannian Shape Optimization and Steklov-Poinaré-Type Metrics
  6. Algorithmic Solution of Shape Optimization Problems
  7. General Descent Algorithm for Shape Optimization
  8. NCG Methods for Shape Optimization
  9. Volume-Based Description of the Descent Algorithm
  10. Numerical Comparison of Gradient-Based Algorithms for Shape Optimization
  11. Discretization and Setup
  12. A Shape Optimization Problem Constrained by a Poisson Equation
  13. Shape Identification in Electrical Impedance Tomography
  14. Optimization of an Obstacle in Stokes Flow
  15. Shape Optimization of a Pipe
  16. ...and 2 more sections

Key Result

Proposition 2.2

The reduced cost functional $J$ corresponding to problem eq:poisson_sop is shape differentiable and has the following shape derivative where $I$ is the identity matrix in $\mathbb{R}^d$, $u$ solves the state equation eq:weak_poisson, and $p$ solves the adjoint equation

Figures (13)

  • Figure 1: State variable $u$ for the Poisson problem \ref{['eq:problem_poisson']} on the initial and optimized geometries, obtained by the L-BFGS 5 method.
  • Figure 2: History of the optimization algorithms for the Poisson problem \ref{['eq:problem_poisson']}.
  • Figure 3: Optimized Shapes (blue) compared to the solution of the L-BFGS 5 method (orange) for the Poisson problem \ref{['eq:problem_poisson']}.
  • Figure 4: History of the optimization algorithms for the EIT problem \ref{['eq:problem_eit']}.
  • Figure 5: Optimized Shapes for the EIT problem \ref{['eq:problem_eit']}, $\Omega^\mathrm{in}$ (blue), $\Omega^\mathrm{out}$ (light gray), together with initial (dark gray) and reference (orange) shape of $\Omega^\text{int}$.
  • ...and 8 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3: Structure Theorem
  • Proposition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 3.1