# Applied PDE Working Seminar 2014-15

### Meetings are held on Tuesdays at 16.00-17.00 in D1.07.

##### Organiser: Charlie Elliott

These meetings provide an opportunity for individuals to discuss in an informal manner progress on their current work, and describe interesting new problems related to PDEs and their applications and computation.

We have chosen this time to allow us to make a pub visit afterwards if we feel inclined.

Group emails can be sent via applied_maths_pde_workgroup at listserv dot warwick dot ac dot uk.

**Term 2 - 2015**

** 6th January - David O'Connor (Warwick)**

#### A comparison of finite element and Isogeometric analysis

I will motivate this talk by considering the Cahn Hilliard equation and its asymptotic limit. I shall briefly talk about how my current work requires good numerical solvers which are more difficult than first appears. I will then give a review of two techniques available. Firstly using variational inequalities and finite element analysis, secondly, standard Isogeometric analysis. I will attempt to highlight the differences between the two techniques and what work would be required to adapt current schemes to my situation.

### 13th January - Yves van Gennip (Nottingham)

#### PDE techniques for graph problems

Driven by applications in image processing and data analysis, recent years have seen analysts turn their attention to graph based problems. Studies of some of these problems, which can be interpreted as analogues to classical continuum PDE models, show interesting connections between the continuum results and the graph problems.In this talk we will explore some of these graph based PDE type problems and their connections with both continuum results and notions from graph theory. Examples include threshold dynamics, modularity optimization, Ohta-Kawasaki minimization, and their relation to graph cuts, mean curvature flow, and bootstrap percolation.

### 20th January - Jose Carrillo (Imperial)

#### Stability and Pattern formation in Nonlocal Interaction Models

I will review some recent results for first and second order models of swarming in terms of patterns, stationary states, and qualitative properties. I will discuss the stability of these patterns for the continuum and discrete particle cases.

These non-local models appear in collective behavior for animals, control engineering, and molecular structures among others. We first concentrate in the spatial shape of these patterns and the dynamics when inertia terms are neglected. The mathematical question behind consists in finding properties about local minimizers of the total interaction energy. Concerning 2nd order models, we will discuss particular properties of two patterns: flocks and mills. We will discuss the stability of these patterns in the discrete case. In both cases, we will describe the properties obtained for the continuum limits.

### 27th January - Pierre Degond (Imperial)

#### Mathematical models of self-organization: bifurcations and derivation of macroscopic models

We consider self-organizing systems, i.e. systems consisting of a large number of interacting entities which spontaneously coordinate and achieve a collective dynamics. Such systems are ubiquitous in nature (flocks of birds, herds of sheep, crowds, ...). Their mathematical modeling poses a number of fascinating questions such as finding the conditions for the emergence of collective motion. In this talk, we will consider a simplified model first proposed by Vicsek and co-authors and consisting of self-propelled particles interacting through local alignment. We will derive its hydrodynamic limit using the recent concept of generalized collision invariant. We will also rigorously study the multiplicity and stability of its kinetic equilibria. We will illustrate our findings by numerical simulations.

### 3rd February - Faizan Nazar (Warwick)

#### Locality of the TFW equations

In this talk I will discuss the existence and uniqueness of a coupled system of partial differential equations that arises from minimising the Thomas-Fermi-von Weizsäcker energy functional for general infinite nuclear arrangements. This gives rise to stability estimates, which give pointwise control of the electron density in terms of a local nuclear defect. We then discuss the applications of this result, including the neutrality of local defects in TFW theory and the lattice relaxation problem.

**24th February - Bertram Düring **(Sussex)

#### On a structure preserving numerical method for Wasserstein gradient flows

Evolution equations with an underlying gradient flow structure have since long been of special interest in analysis and mathematical physics. In particular, transport equations that allow for a variational formulation with respect to the L2-Wasserstein metric have attracted a lot of attention recently. The gradient flow formulation gives rise to a natural semi-discretisation in time of the evolution by means of the minimising movement scheme, which constitutes a time-discrete minimization problem for the (sum of kinetic and potential) energy. On the other hand, nonlinear diffusion equations of fourth (and higher) order have become increasingly important in pure and applied mathematics. Many of them have been interpreted as gradient flows with respect to some metric structure.

When it comes to solving equations of gradient flow type numerically, schemes that respect the equation's special structure are of particular interest. In this talk we present a fully discrete variant of the minimising movement scheme for the numerical solution of the nonlinear fourth-order Derrida-Lebowitz-Speer-Spohn equation in one space dimension, and discuss possible extensions to higher approximation order.

### 3rd March - Anotida Madzvamuse (Sussex)

#### Stability analysis of non-autonomous reaction-diffusion systems on evolving domains: the effects of cross-diffusion

In this talk, I will present stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, I will derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. The analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of these theoretical predictions, I will present evolving evolving surface finite element solutions exhibiting patterns generated by a short-range inhibition, long-range activation reaction-diffusion model with linear cross-diffusion in the presence of domain evolution.

### 10th March - Julian Braun (Augsburg)

#### Connecting atomistic and continuum models of nonlinear elasticity

The connection between atomistic and continuous macroscopic models of elasticity theory for crystalline solids is typically given by the Cauchy-Born rule. It states that the microscopic deformation of atoms should follow the macroscopic deformation gradient. I will discuss conditions under which the Cauchy-Born rule holds true. In particular, I will present a rather general existence and convergence result for local energy minimizers under fixed boundary displacements for atomistic and continuum models that are related by the Cauchy-Born rule.

**Term 1 - 2014**

**23rd September - Klaus Deckelnick (Otto-von-Guericke-Universität Magdeburg) **

**Double obstacle phase field approach for an elliptic inverse problem with discontinuous coefficients**

We consider the inverse problem of recovering interfaces along which the piecewise constant diffusion coefficient in an elliptic PDE has jump discontinuities. We employ a least squares approach together with a perimeter regularization. A suitable relaxation of the perimeter leads to a sequence of Cahn--Hilliard type functionals for which we obtain a --convergence result. Using a finite element discretization of the elliptic PDE and a suitable adjoint problem we derive an iterative method in order to approximate discrete critical points. We prove convergence of the iteration and present results of numerical tests. This is joint work with Charlie Elliott (Warwick) and Vanessa Styles (Sussex).

### NOTE: This seminar will be held in B3.02 instead of the usual room.

### 30th September - David McCormick (Warwick)

#### Non-resistive magnetohydrodynamics in Sobolev and Besov spaces

In this talk I will examine local-in-time existence and uniqueness of solutions to the viscous, non-resistive MHD equations. I will first outline some results in Sobolev spaces, before concentrating on more recent results in Besov spaces. Besov spaces are often the most natural spaces to work in for nonlinear parabolic PDEs, and here they provide an optimal result in the edge case where Sobolev spaces fail. I will introduce Besov spaces from scratch, before going on to outline the key ideas in proving the a priori estimates required to prove existence and, if time permits, uniqueness. This is joint work with Jean-Yves Chemin (Paris 6), James Robinson and Jose Rodrigo.

### NOTE: This seminar will be held in MS.04 instead of the usual room.

### 7th October - Tom Ranner (Leeds)

#### A computational approach to an optimal partition problem on surfaces

We explore an optimal partition problem on surfaces using a computational approach. The problem is to minimise the sum of the first Dirichlet Laplace--Beltrami operator eigenvalues over a given number of partitions of a surface. We consider a method based on eigenfunction segregation and perform calculations using modern high performance computing techniques. We first test the accuracy of the method in the case of three partitions on the sphere then explore the problem for higher numbers of partitions and on other surfaces.

### 14th October - Andrew Duncan (Imperial)

#### Diffusion on Rapidly Varying Surfaces

Lateral diffusion of particles on rough and/or rapidly fluctuating membranes is a common problem in biology. I will describe a multi-scale analysis approach to studying the macroscopic behaviour of passive particles diffusing laterally on such surfaces and will show how this approach allows one to compute effective properties of the macroscopic diffusion as well as provide a means to rigorously justify a number of existing results.

### 21st October - Yulong Lu (Warwick)

#### A Bayesian level set method for geometric inverse problems

The level set method was initially proposed by Osher and Sethian for tracking evolving interfaces in fluid dynamics. In the past two decades, the level set method has been combined with optimization techniques for solving geometric inverse problems where the unknown is a geometric shape. While these level-set-based optimization approaches have been used for various applications, their mathematical theory has not been yet fully developed. For example, the convergence of these algorithms is known only in some particular case.

We develop a new method--- **Bayesian level set method** for solving geometric inverse problems. The main idea is to represent the geometry by the level set technique and then to solve the inverse problem in the Bayesian framework. In concrete, given the data , with a prior measure prescribed for the level set function , the Bayes' formula gives the posterior measure for from which the geometric information can be extracted. Our work consists of a rigorous analysis for the continuity of the level set map arising from our level set representation, whereby the measurability of the observational map is proved, justifying the Bayes' Theorem in Bayesian inverse problems.

The proposed Bayesian level set method is applied to solving the inverse problem of determining the discontinuity of permeability in the subsurface flow. For this Bayesian inverse problem, we prove the existence and well-posedness of the posterior measure on the space of level set functions. We also apply pCN-MCMC methods to characterize the posterior measure and provide numerical experiments to demonstrate the feasibility of our methodology.

This is joint work with Marco Iglesias (Nottingham) and Andrew Stuart (Warwick).

### 28th October - Andrea Cangiani (Leicester)

#### On Extending Finite Element Methods to Polytopic Meshes

Can we extend the FEM to general meshes while maintaining the ease of implementation and computational cost comparable to that of standard FEM? Within this talk, I will present two approaches that achieve just that (and much more): the Virtual Element Method (VEM) and a discontinuous Galerkin (dG) method.

### 4th November - Graham Hobbs (Warwick)

#### Membrane deformation and induced interactions between pointwise inclusions

I will explore various models for the deformation of biomembranes by protein molecules which are embedded in or attached to the membrane, termed protein inclusions. Mathematically we focus on minimisation problems involving the Helfrich energy functional. Inclusions will be modelled by enforcing pointwise constraints on the membrane displacement or curvature. I will present results detailing the existence and behaviour of minimal energy states for these models and explore the membrane mediated interactions between inclusions.

### 11th November - Matthew Dunlop (Warwick)

#### A multiplicative noise model for Bayesian inversion

In the theory of Bayesian inverse problems, the data is typically assumed to be corrupted by additive noise. We introduce a model for the data that also incorporates multiplicative noise, which is more realistic in some circumstances. We prove existence and

well-posedness of the posterior, as well as posterior consistency in the small noise limit. More specifically, we show convergence of the posterior modes, characterised as minimisers of a sequence of random functionals, to the truth when the magnitude of the noise goes to zero. The large data limit is also discussed. Finally some numerics are

presented, illustrating the improved mixing of MCMC chains simulating the posterior arising from the multiplicative model versus that coming from the purely additive model.

### 18th November - Oliver Dunbar (Warwick)

#### Knot energies in the numerical detection of self-intersecting moving membranes

Cell motility is the ability for cells to move in a coordinated fashion in reaction to a stimulus. One may choose to model this process as a coupled surface advection-diffusion equation to some geometric evolution law. Unfortunately it may be observed that unphysical self-intersections of the membrane can occur, and the idea developed in this talk seek to detect and with the view of prevention of this phenomena. The Mobius energy is well studied in algebra as a tool for observing knotting in curves, and it thus it is sensitive to self-intersection. We present ideas of discretisation of this energy for moving unknown curves with the aims of retaining acceptable order of convergence and efficiency, then incorporate this energy into the cell motility model and analyse its effectiveness.

### 25th November - Andrew Binder (University of Minnesota)

#### Analysis of the Surface Cauchy-Born Method in 1D

Andrew Binder, Mitchell Luskin, Christoph Ortner, and Harold Park

Surface effects can play a significant role in the determination of material properties on smaller scales. Despite the fact that surface effects are most influential in smaller systems, the systems may still be large enough that simulating the system atomistically will be computationally infeasible. The recently developed surface Cauchy-Born method was designed to efficiently model such systems and improve upon the regular Cauchy-Born method by better capturing surface effects. The accuracy of the surface Cauchy-Born method is investigated in the specific case of a linearized 1D system interacting according to a many-body potential as well as through a nonlinear analysis that considers the convergence of the method to the atomistic result for a more general 1D system.