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Philip J. Herbert

Publications and source records attributed to Philip J. Herbert.

17 recordsLinked to original sources
arXiv2026

Shape optimisation for adaptive $r$-refinement: the one-dimensional case with residual based error estimators

We consider $r$-refinement for the finite element discretisation of a Poisson problem. The goal of $r$-refinement is to reposition the nodes of a computational mesh in order to better approximate the finite element error. Since the mesh is being moved, it naturally becomes linked to shape optimisation methods. We propose a standard optimisation algorithm for this $r$-refinement procedure and show that if one seeks to minimise the actual error - which one cannot generally calculate - one has an algorithm which will terminate. The most novel aspect of this work is to apply shape optimisation techniques to the a standard residual error estimator, which is a functional that is differentiable with respect to the mesh, when considered for the Poisson equation in one dimension. To illustrate the approach, a number of numerical experiments are presented, which verify the efficacy of the method.

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arXiv2026

Finite element approximation of the enthalpy formulation for Stefan problems on evolving surfaces

We propose, and analyse, a spatially discrete evolving surface finite element method for the approximation of the enthalpy formulation of the two-phase Stefan problem posed on an evolving surface. Our approach does not rely on mass-lumping and discrete maximum principles. We prove this numerical method is numerically stable, and prove $\mathcal{O}(\sqrt{h})$ error bounds for the temperature in the $L^2_{L^2}$ norm under minimal regularity assumptions by introducing a new projection-type operator. We complement our analysis with discussion on the implementation of this numerical method, where we propose a novel implementation that avoids errors due to numerical quadrature and which has not previously been considered in the literature even in the stationary, flat setting. We also include numerical experiments and experimental order of convergence demonstrations.

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arXiv2026

Asymptotic numerical hypocoercivity of the space-time discontinuous Galerkin method for the Kolmogorov equation

We are concerned with discretisations of the classical Kolmogorov equation by a standard space-time discontinuous Galerkin method. {The} Kolmogorov equation serves as simple, yet rich enough in the present context, model problem for a wide range of kinetic-type equations: although it involves diffusion in one of the two spatial dimensions only, the combined nature of the first order transport/drift term and the degenerate diffusion are sufficient to `propagate dissipation' across the spatial domain in its entirety. This is a manifestation of the celebrated concept of hypocoercivity, a term coined and studied extensively by Villani in \cite{villani}. We show that the {classical} space-time discontinuous Galerkin method {admits} a corresponding hypocoercivity property at the discrete level, asymptotically for large times. To the best of our knowledge, this is the first result of this kind for any standard Galerkin scheme. This property is shown by proving one part of a discrete inf-sup-type stability result for the method in a family of norms dictated by a modified scalar product motivated by the theory in \cite{villani}. This family of norms contains the full gradient of the numerical solution, thereby allowing for a full spectral gap/Poincaré-type inequality at the discrete level, thus, showcasing a subtle, discretisation-parameter-dependent, numerical hypocoercivity property. Further, we show that the space-time discontinuous Galerkin method is inf-sup stable in the family of norms containing the full gradient of the numerical solution, which may be a result of independent interest.

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arXiv2026

Combining diffuse and sharp interface methods in shape optimisation

We develop a concept for the numerical treatment of shape optimization problems based on the combination of phase field and sharp interface methods. On the one hand, phase field methods are very well suited to numerically determine the shape, size and topology of a sought domain, but on the other hand they have problems to sharpen out domains where they e.g. should develop corners. However, this is the strength of a sharp-interface approach developed in our group, which provides shape updates in the Lipschitz topology. This leads to a two-stage process that first determines an optimized shape using the phase field method. The resulting domain is the starting solution for the sharp interface shape optimization method. Both methods are discretized with the finite element method. The starting mesh for the sharp method is constructed from the finite element mesh of the optimal phase field solution using its properly post processed zero-level set. We describe this construction process in detail and investigate the performance of our method on a selection of test problems from the literature and from applications.

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arXiv2026

Small deformations of a near cylindrical tube for the Canham-Helfrich Energy with applications to biological membranes

In this article we develop a quadratic energy which approximates the Canham-Helfrich energy for a tube-like surface with clamped boundary and area constraint. The energy is suited to the study of small deformations of biological membranes where the deformations are induced by point forces or point constraints due to the cytoskeleton or a phase dependent spontaneous curvature. Since the deformations we consider are small, we may assume that the surface of interest is a graph over a fixed, undeformed surface. A Lagrangian and the associated Euler-Lagrange equations for the graph are derived. Well-posedness of the Euler-Lagrange equations in suitable spaces is shown. Finally, we provide some illustrative numerical examples.

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arXiv2025

The Stochastic Steepest Descent Method for Robust Optimization in Banach Spaces

Stochastic gradient methods have been a popular and powerful choice of optimization methods, aimed at minimizing functions. Their advantage lies in the fact that that one approximates the gradient as opposed to using the full Jacobian matrix. One research direction, related to this, has been on the application to infinite-dimensional problems, where one may naturally have a Hilbert space framework. However, there has been limited work done on considering this in a more general setup, such as where the natural framework is that of a Banach space. This article aims to address this by the introduction of a novel stochastic method, the stochastic steepest descent method (SSD). The SSD will follow the spirit of stochastic gradient descent, which utilizes Riesz representation to identify gradients and derivatives. Our choice for using such a method is that it naturally allows one to adopt a Banach space setting, for which recent applications have exploited the benefit of this, such as in PDE-constrained shape optimization. We provide a convergence theory related to this under mild assumptions. Furthermore, we demonstrate the performance of this method on a couple of numerical applications, namely a $p$-Laplacian and an optimal control problem. Our assumptions are verified in these applications.

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arXiv2024

A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation

We propose a new stabilised finite element method for the classical Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterised by degenerate diffusion. The stabilisation is constructed so that the resulting method admits a \emph{numerical hypocoercivity} property, analogous to the corresponding property of the PDE problem. More specifically, the stabilisation is constructed so that spectral gap is possible in the resulting ``stronger-than-energy'' stabilisation norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behaviour as the ``time'' variable goes to infinity. We consider both a spatially discrete version of the stabilised finite element method and a fully discrete version, with the time discretisation realised by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical experiments verify the theoretical findings.

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arXiv2023

PDE constrained shape optimisation with first-order and Newton-type methods in the $W^{1,\infty}$ topology

We present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty}$ topology. We propose steepest descent and Newton-like minimisation algorithms for the numerical solution of the respective shape optimisation problems. Our work is built upon previous work of the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), where a $W^{1,\infty}$ framework for star-shaped domains is proposed. To illustrate our approach we present a selection of PDE constrained shape optimisation problems and compare our findings to results from so far classical Hilbert space methods and recent $p$-approximations.

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arXiv2023

Convergence of a steepest descent algorithm in shape optimisation using $W^{1,\infty}$ functions

Built upon previous work of the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), we present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty}$ topology together with a suitable finite element discretisation. For the numerical solution of the respective discrete shape optimisation problems we propose a steepest descent minimisation algorithm with Armijo-Goldstein stepsize rule. We show that the sequence generated by this descent method globally converges, and under mild assumptions also, that every accumulation point of this sequence is a stationary point of the shape functional. Moreover, for the mesh discretisation parameter tending to zero we under mild assumptions prove convergence of the discrete stationary shapes in the Hausdorff complementary metric. To illustrate our approach we present a selection of numerical examples for PDE constrained shape optimisation problems, where we include numerical convergence studies which support our analytical findings.

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arXiv2023

Shape optimization in $W^{1,\infty}$ with geometric constraints: a study in distributed-memory systems

In this paper we present a shape optimization scheme which utilizes the alternating direction method of multipliers (ADMM) to approximate a direction of steepest descent in $W^{1,\infty}$. The followed strategy is a combination of the approaches presented in Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022) and Müller et al. SIAM SISC 45 (2023). This has appeared previously for relatively simple elliptic PDEs with geometric constraints which were handled using an ad-hoc projection. Here, however, the optimization problem is expanded to include geometric constraints, which are systematically fulfilled. Moreover, this results in a nonlinear system of equations, which is challenging from a computational perspective. Simulations of a fluid dynamics case study are carried out to benchmark the novel method. Results are given to show that, compared to other methods, the proposed methodology allows for larger deformations without affecting the convergence of the used numerical methods. The mesh quality is studied across the surface of the optimized obstacle, and is further compared to previous approaches which used descents in $W^{p,\infty}$. The parallel scalability is tested on a distributed-memory system to illustrate the potential of the proposed techniques in a more complex, industrial setting.

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arXiv2023

An evolving space framework for Oseen equations on a moving domain

This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent Bochner spaces. It is not possible to directly apply established evolving Hilbert space theory due to the incompressibility constraint. After we have established the well-posedness, we derive and analyse a first order time discretisation of the system.

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arXiv2023

Shape Optimisation with $W^{1,\infty}$: A connection between the steepest descent and Optimal Transport

In this work, we discuss the task of finding a direction of optimal descent for problems in Shape Optimisation and its relation to the dual problem in Optimal Transport. This link was first observed in a previous work which sought minimisers of a shape derivative over the space of Lipschitz functions which may be closely related to the $\infty$-Laplacian. We provide some results of shape optimisation using this novel Lipschitz approach, highlighting the difference between the Lipschitz and $W^{1,\infty}$ semi-norms. After this, we provide an overview of the necessary results from Optimal transport in order to make a direct link to the Shape optimisation of star-shaped domains. Demonstrative numerical experiments are provided.

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arXiv2021

A novel $W^{1,\infty}$ approach to shape optimisation with Lipschitz domains

This article introduces a novel method for the implementation of shape optimisation with Lipschitz domains. We propose to use the shape derivative to determine deformation fields which represent steepest descent directions of the shape functional in the $W^{1,\infty}-$ topology. The idea of our approach is demonstrated for shape optimisation of $n$-dimensional star-shaped domains, which we represent as functions defined on the unit $(n-1)$-sphere. In this setting we provide the specific form of the shape derivative and prove the existence of solutions to the underlying shape optimisation problem. Moreover, we show the existence of a direction of steepest descent in the $W^{1,\infty}-$ topology. We also note that shape optimisation in this context is closely related to the $\infty-$Laplacian, and to optimal transport, where we highlight the latter in the numerics section. We present several numerical experiments in two dimensions illustrating that our approach seems to be superior over a widely used Hilbert space method in the considered examples, in particular in developing optimised shapes with corners.

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arXiv2020

A formula for membrane mediated point particle interactions on near spherical biomembranes

We consider a model of a biomembrane with attached proteins. The membrane is represented by a near spherical continuous surface and attached proteins are described as discrete rigid structures which attach to the membrane at a finite number of points. The resulting surface minimises a quadratic elastic energy (obtained by a perturbation of the Canham-Helfrich energy) subject to the point constraints which are imposed by the attachment of the proteins. We calculate the derivative of the energy with respect to protein configurations. The proteins are constrained to move tangentially by translation and by rotation in the axis normal to a reference point. Previous studies have typically restricted themselves to a nearly flat membrane and circular inclusions. A numerically accessible representation of this derivative is derived and employed in some numerical experiments.

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arXiv2020

Second order splitting of a class of fourth order PDEs with point constraints

We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved and flat surfaces but the approach may be applied more generally. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.

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arXiv2019

Small deformations of spherical biomembranes

In this contribution to the proceedings of the 11th Mathematical Society of Japan (MSJ) Seasonal Institute (July 2018) we give an overview of some recent work on a mathematical model for small deformations of a spherical membrane. The idea is to consider perturbations to minimisers of a surface geometric energy. The model is obtained from consideration of second order approximations to a perturbed energy. In particular, the considered problems involve particle constraints and surface phase field energies.

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