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Numerical techniques for geodesic approximation in Riemannian shape optimization

Estefania Loayza-Romero, Kathrin Welker

TL;DR

The paper addresses PDE-constrained shape optimization on Riemannian shape manifolds by employing outer Sobolev-type metrics on the diffeomorphism group and exploring updates via geodesics rather than solely retractions. It analyzes the computational trade-offs of using explicit geodesic equations and surveys numerical strategies—direct Hamiltonian integration, symplectic model order reduction, SINDy for Hamiltonian dynamics, and discrete geodesic calculus—to reduce the burden of geodesic computations. Benchmark results show outer metrics can achieve comparable or better objective values with substantially fewer iterations than a complete SP-metric while preserving mesh quality, highlighting practical efficiency gains. The work outlines how discrete geodesic calculus and parallel transport enable higher-order methods, including quasi-Newton approaches, within Riemannian shape spaces, advancing scalable, accurate PDE-constrained shape optimization.

Abstract

Shape optimization is commonly applied in engineering to optimize shapes with respect to an objective functional relying on PDE solutions. In this paper, we view shape optimization as optimization on Riemannian shape manifolds. We consider so-called outer metrics on the diffeomorphism group to solve PDE-constrained shape optimization problems efficiently. Commonly, the numerical solution of such problems relies on the Riemannian version of the steepest descent method. One key difference between this version and the standard method is that iterates are updated via geodesics or retractions. Due to the lack of explicit expressions for geodesics, for most of the previously proposed metrics, very limited progress has been made in this direction. Leveraging the existence of explicit expressions for the geodesic equations associated to the outer metrics on the diffeomorphism group, we aim to study the viability of using such equations in the context of PDE-constrained shape optimization. However, solving geodesic equations is computationally challenging and often restrictive. Therefore, this paper discusses potential numerical approaches to simplify the numerical burden of using geodesics, making the proposed method computationally competitive with previously established methods.

Numerical techniques for geodesic approximation in Riemannian shape optimization

TL;DR

The paper addresses PDE-constrained shape optimization on Riemannian shape manifolds by employing outer Sobolev-type metrics on the diffeomorphism group and exploring updates via geodesics rather than solely retractions. It analyzes the computational trade-offs of using explicit geodesic equations and surveys numerical strategies—direct Hamiltonian integration, symplectic model order reduction, SINDy for Hamiltonian dynamics, and discrete geodesic calculus—to reduce the burden of geodesic computations. Benchmark results show outer metrics can achieve comparable or better objective values with substantially fewer iterations than a complete SP-metric while preserving mesh quality, highlighting practical efficiency gains. The work outlines how discrete geodesic calculus and parallel transport enable higher-order methods, including quasi-Newton approaches, within Riemannian shape spaces, advancing scalable, accurate PDE-constrained shape optimization.

Abstract

Shape optimization is commonly applied in engineering to optimize shapes with respect to an objective functional relying on PDE solutions. In this paper, we view shape optimization as optimization on Riemannian shape manifolds. We consider so-called outer metrics on the diffeomorphism group to solve PDE-constrained shape optimization problems efficiently. Commonly, the numerical solution of such problems relies on the Riemannian version of the steepest descent method. One key difference between this version and the standard method is that iterates are updated via geodesics or retractions. Due to the lack of explicit expressions for geodesics, for most of the previously proposed metrics, very limited progress has been made in this direction. Leveraging the existence of explicit expressions for the geodesic equations associated to the outer metrics on the diffeomorphism group, we aim to study the viability of using such equations in the context of PDE-constrained shape optimization. However, solving geodesic equations is computationally challenging and often restrictive. Therefore, this paper discusses potential numerical approaches to simplify the numerical burden of using geodesics, making the proposed method computationally competitive with previously established methods.

Paper Structure

This paper contains 10 sections, 6 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Brief introduction into optimization on shape manifolds
  3. Riemannian shape manifold
  4. Benchmark problem
  5. Efficient approximation of geodesic equations
  6. Direct approach
  7. Symplectic model order reduction
  8. Hamiltonian sparse identification of nonlinear dynamics
  9. Discrete geodesic calculus
  10. Conclusion

Figures (2)

  • Figure 1: Final shapes (upper row) and associated state variables (lower row) for the experiment described in Section \ref{['sec:benchmarkProblem']}.
  • Figure 2: Objective function, $L^2$-norm of shape gradients and mesh quality behavior along the iterations for the different variants of the method for the experiment described in Section \ref{['sec:benchmarkProblem']}.