[Paper Review] Convergence of Newton's method in shape optimisation via approximate normal functions

Kevin Sturm|arXiv (Cornell University)|Aug 9, 2016

Advanced Numerical Analysis Techniques36 references3 citations

TL;DR

This paper proposes a superlinearly convergent Newton method for shape optimization using approximate normal functions to discretize the tangent space of the quotient space associated with the Micheletti metric group. By leveraging the domain shape Hessian and reproducing kernels, the method achieves efficient convergence near stationary points, with numerical validation showing superior performance over gradient methods.

ABSTRACT

In this paper we propose a Newton method for shape functions defined on an image set generated by the (Micheletti) metric group. We review basic properties of the metric group and a quotient associated with the metric group and a fixed domain. Taking into account the special structure of the second shape derivative and its symmetric part allows us to distinguish between two Hessians, the domain shape Hessian and the boundary shape Hessian. Using the domain Hessian we define a Newton method on the metric group by discretising the tangent space of the quotient via approximate normal functions using reproducing kernels. Under suitable assumptions we are able to show superlinear convergences of the Newton iterations and additionally convergence of the shapes in the metric group. Finally we verify our findings in a number of numerical experiments including a thorough numerical study of the impact of the discretisation on the convergence speed.

Motivation & Objective

• To develop a Newton method for shape optimization that achieves superlinear convergence in the infinite-dimensional setting.
• To address the challenge of nonlinearity and infinite-dimensionality in shape optimization by leveraging differential geometry on the metric group.
• To introduce approximate normal functions via reproducing kernels to discretize the tangent space of the quotient space ${\cal F}/{\cal G}_{\omega}$.
• To compare the performance of the domain Hessian, boundary Hessian, and Riemannian Hessian in numerical experiments.
• To analyze the impact of discretization on convergence speed and stability, particularly near stationary points.

Proposed method

• The method uses the Micheletti metric group ${\cal F}$ to define shape variations, identifying admissible shapes via the quotient ${\cal F}/{\cal G}_{\omega}$.
• It introduces approximate normal functions using compactly supported reproducing kernels to construct a discrete subspace of the tangent space of ${\cal F}/{\cal G}_{\omega}$.
• The domain shape Hessian is defined on the tangent space of ${\cal F}$, while the boundary shape Hessian is defined on the quotient space ${\cal F}/{\cal G}_{\omega}$, with both being approximately equal under normal perturbations.
• A Newton iteration is constructed using the domain Hessian, with the Newton direction solved via Galerkin projection onto the space of approximate normal functions.
• A transport map is introduced to relate approximate normal functions across different domains, enabling convergence analysis in the discrete setting.
• The method ensures superlinear convergence under suitable assumptions, with convergence of the generated shapes in the metric of ${\cal F}$.

Experimental results

Research questions

• RQ1 Can a Newton method be constructed for shape optimization on the infinite-dimensional metric group ${\cal F}$ using discrete approximations of the tangent space?
• RQ2 Does the use of approximate normal functions via reproducing kernels preserve sufficient geometric structure to ensure superlinear convergence?
• RQ3 How do the domain and boundary shape Hessians compare in practice when using discrete approximations?
• RQ4 What is the impact of discretization on the convergence speed and stability of the Newton method in shape optimization?
• RQ5 Under what conditions does the Newton method fail to achieve quadratic convergence when normal fields are approximated?

Key findings

• The proposed Newton method achieves superlinear convergence under suitable assumptions, even though quadratic convergence is not attainable due to approximation of normal fields.
• The domain and boundary shape Hessians coincide when restricted to normal perturbations, enabling the use of the domain Hessian as a surrogate in the discrete setting.
• Numerical experiments show that the Newton method converges significantly faster than gradient methods, especially near stationary points.
• For large shape deformations, the boundary shape Hessian is more favorable than the domain Hessian due to residual tangential components in the latter.
• The convergence rate is sensitive to the choice of discretization, with finer control points improving convergence speed, as demonstrated in experiments with $N=10$ to $N=100$ control points.
• The method successfully handles both elliptical and square-shaped domains, with convergence verified via shape evolution snapshots and residual norms.