Workshop on Stochastic Partial Differential Equations

Welcome

This workshop will bring together specialists in the theory and numerical methods for stochastic partial differential equations and their applications, specialists in PDEs and stochastic analysis.

There is no registration fee, but please register here for catering purposes. Additionally to presentations by invited speakers, a limited number of talks can be accepted. Please express your interest in giving a talk by sending an email to Prof. Beniamin Goldys at beniamin.goldys@sydney.edu.au.

List of Speakers

• Zdzislaw Brzezniak (York University)
  
• Madeleine Clare Cartwright (Sydney)
  
• Gaurav Dhariwal (TU Wien)
• Jerome Droniou (Monash)
• Jie Fan (Monash)
• Le Gia (UNSW)
• Beniamin Goldys (Sydney)
• Alex Hening (Tufts University)
• Ngan Le (Monash)
• Gregoire Loeper (Monash)
• Utpal Manna (Indian Institute of Science Education and Research)
• Pierre Portal (ANU)
• Andreas Prohl (Tuebingen University)
• Nimit Rana (York University)
• Thanh Tran (UNSW)

Venue

The workshop will be held at the University of Sydney in Auditorium 2, Administration Building (Building F23). See also the information on how to get there or search on Sydney university's campus map.

Funding Application

PhD students and early career researchers can apply for travel funding held by AMSI on AMSI Research website.

Program

Monday

26/08

Tuesday

27/08

Wednesday

28/08

8:30--9:00

Registration

9:00--9:15

Opening

9:15--10:10

Prohl
Cartwright
Loeper

10:10--10:45

Coffee break
Coffee break
Coffee break

10:45--11:40

Dharival
Le Gia
Brzezniak

11:45--12:40

Hening
Le
Manna

12:40--2:00

Lunch
Lunch
Lunch

2:00--2:55

Portal
Droniou
Goldys

2:55--3:30

Coffee Break
Coffee break
Coffee break

3:30--4:25

Rana
Tran
Fan

18:30--

Dinner

Abstracts of Talks

Stochastic nonlinear Schr\"odinger equation on 3d compact manifolds

Zdzislaw Brzezniak (York University)

Abstract

I will speak about two recent works with Fabian Hornung and Lutz Weis (Karlsruhe). In particular, I will speak about the existence of a global solution to the stochastic nonlinear Schr\"odinger equation (SNSL) on a 3-dimensional manifold with multiplicative Gaussian (and jump) noise.
I will also speak about the uniqueness for such equations in case of Gaussian bilinear noise the proof of which is based on novel Strichartz estimates and Littlewood-Paley and decomposition in time.
I will conclude by mentioning the Large Deviations Principle for SNSL.

A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations

Madeleine Clare Cartwright (Sydney)

Abstract

We propose a formal framework based on collective coordinates to reduce infinite-dimensional stochastic partial differential equations (SPDEs) with symmetry to a set of finite-dimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group. We study SPDEs arising in population dynamics with multiplicative noise and additive symmetry breaking noise. The collective coordinate approach provides a remarkably good quantitative description of the shape of the travelling front as well as its diffusive behaviour, which would otherwise only be available through costly computational experiments. We corroborate our analytical results with numerical simulations of the full SPDE.

Global martingale solution for a stochastic population cross-diffusion system

Gaurav Dhariwal (TU Wien)

Abstract

The existence of global nonnegative martingale solutions to a stochastic cross-diffusion system for an arbitrary but finite number of interacting population species is shown. The random influence of the environment is modelled by a multiplicative noise term. The diffusion matrix is generally neither symmetric nor positive definite, but it possesses a quadratic entropy structure. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzeźniak and co-workers, and Jakubowski's generalization of the Skorokhod theorem. The nonnegativity is proved by an extension of Stampacchia's truncation method due to Chekroun, Park, and Temam.

Towards a generic numerical analysis framework for fully discrete approximations of non-linear SPDEs

Jerome Droniou (Monash)

Abstract

Usual numerical schemes for stochastic PDEs are based on conforming finite element methods, some of the oldest numerical methods in the world of deterministic PDEs. Although they have desirable features that still make them preferred in some Engineering applications, they also sometimes lack flexibility to provide desirable features, such as: conservativity of physical quantities, applicability to generic polyhedral meshes (as encountered for example in reservoir engineering), parallelisability, capture of sharp features, etc.
Many schemes have been considered for deterministic PDEs to specifically provide some of these features: finite volume methods, mixed finite elements, hybrid schemes, etc. In this talk, we will present some elements of a generic numerical analysis framework for stochastic PDEs, that encompasses many of the these schemes. This framework is based on the Gradient Discretisation Method (GDM) framework for deterministic non-linear elliptic and parabolic PDEs. Convergence analysis for non-linear models is carried out using the discrete functional analysis, translation to the discrete setting of classical functional analysis theorems (Sobolev embeddings, Rellich and Aubin-Simon compactness theorems, etc.). We will illustrate how the "stochastic GDM" can be applied to the stochastic p-Laplace equation and stochastic magnetisation equations.

Multi-type age-structred population model

Jie Fan (Monash)

Abstract

Population process in general setting, where each individual reproduces and dies depending on the state (such as age and type) of the individual as well as the entire population, offers a more realistic framework to population modelling. Formulating the population dynamics as a measure-valued stochastic process allows us to incorporate such dependence. We describe the dynamics of a multi-type age-structured population as a measure-valued process, and obtain its asymptotics. The law of large numbers and central limit theorem can be obtained in the form of PDE and SPDE. Joint work with Kais Hamza, Peter Jagers and Fima Klebaner.

Stochastic Navier-Stokes equations on a thin spherical domain

Le Gia (UNSW)

Abstract

We consider the incompressible Navier-Stokes equations on a thin spherical domain along with free boundary conditions under a random forcing. We show that the martingale solution of these equations converge to the martingale solution of the stochastic Navier-Stokes equation considered on a sphere as the thickness converges to zero.
This is a joint work with Zdzislaw Brzezniak (UK) and Gaurav Dhariwal (Austria).

The existence of stochastic flows in infinite dimensions

Beniamin Goldys (Sydney)

Abstract

It is a classical fact (Bismut, Kunita,...) that ordinary stochastic differential equations with regular coefficients, generate stochastic flows. Later, the existence of stochastic flows was proved for many other finite-dimensional stochastic systems.This property dramatically breaks down in the case of stochastic PDEs. Skorokhod example shows that even a very simple infinite system of independent linear stochastic equations need not generate a stochastic flow. A number of positive results have been obtained since then, but our understanding of stochastic flows in infinite dimensions remains limited. In the first part of the talk we will present new and perhaps surprising examples of the existence and non-existence of stochastic flows in infinite dimensions. In the second part some general results will be presented in the framework of stochastic equations on the algebras of Hilbert space operators. This is a joint work with Szymon Peszat.

Stochastic persistence and extinction

Alex Hening (Tufts University)

Abstract

A key question in population biology is understanding the conditions under which the species from an ecosystem persist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction. The talk is based on joint work with Dang H. Nguyen (University of Alabama).

Weak martingale solutions to the stochastic Landau--Lifshitz--Gilbert equations with multi-dimensional noise via a convergent finite-element scheme

Ngan Le (Monash)

Abstract

In this talk, we propose a convergent linear finite element scheme for the stochastic Landau--Lifshitz--Gilbert equations (SLLGE) in three dimensional domain, with multi-dimensional noise. Firstly, by introducing a suitable transformation, we covert the SLLGE to a highly non-linear time dependent partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent scheme for the numerical solution of the reformulated equation and prove the existence of weak martingale solutions to the SLLGE. This is a joint work with Beniamin Goldys and Joseph Grotowski.

Stochastic PDEs and mixed PDE/Monte-Carlo methods for derivatives pricing

Gregoire Loeper (Monash)

Abstract

In the spirit of [1] we propose a pricing method for derivatives when the underlying diffusion is given by a set of stochastic differential equations, with the objective of reducing the computing time. The numerical method is based on a joint use of Monte-Carlo simulations, PDE or analytical formulas. We show that this method has an natural interpretation in terms of stochastic pde's and from this observation propose a new way of implementing it. We also show how to implement the Least Square Monte-Carlo method proposed in [2] together with the mixed PDE/Monte-Carlo method.

Weak solution of a stochastic Landau-Lifshitz-Gilbert equation driven by pure jump noise

Utpal Manna (IISER)

Abstract

In this talk, we consider a stochastic three-dimensional Landau-Lifshitz-Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show existence of weak martingale solutions taking values in a two-dimensional sphere $S^2$ and discuss certain regularity results. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. This is a joint work with Zdzislaw Brzezniak (University of York) and has been published in Commun. Math. Phys. (2019), https://doi.org/10.1007/s00220-019-03359-x.

Maximal regularity of parabolic SPDE with rough coefficients

Pierre Portal (ANU)

Abstract

When considering the SPDE $du(t) = div A(t,.) \nabla u dt + f(t)dW(t)$, what is the minimal regularity requirement on the coefficient matrix $A$ to ensure maximal regularity of the solution (i.e. to make sure $\nabla u$ and $f$ belong to the same space, which is very helpful when solving related non-linear problems)? Traditional space-time regularity assumptions are unrealistic in the stochastic case. In this talk, we see that, without assuming any regularity in time, fairly minimal regularity assumptions in the spatial variables guarantee maximal regularity in spaces such as $L^{p}_{x}L^{2}_{t}$ for $p\geq 2$. For coefficients that are merely bounded and measurable in space and time, we see that some form of maximal regularity can still be obtained (in tent spaces). This is an important step towards solving quasilinear SPDE for a full class of noise terms.
This is joint work with Mark Veraar (Delft).

Numerical approximation of nonlinear SPDEs

Andreas Prohl (U Tuebingen)

Abstract

The stochastic Navier-Stokes equation, or the stochastic version of the harmonic map flow to the 2D sphere are examples for nonlinear SPDEs which only possess weak martingale solutions. I discuss requirements for a discretization of a ’quite general’ nonlinear SPDE to construct a weak martingale solution for vanishing discretization parameters.

The results base upon joint work with M. Ondrejat (Prague) and N. Walkington (Pittsburgh).

Large deviations for stochastic geometric wave equation

Nimit Rana (University of York)

Abstract

We establish here the validity of a large deviation principle (LDP) for the small noise asymptotic of strong solutions to stochastic geometric wave equations with values in a compact Riemannian manifold. As is many recent papers, our proof relies on applying the weak convergence approach of Budhiraja and Dupuis (Probab. Math. Statist., 2000) to an example of SPDEs where solutions are local Sobolev space-valued stochastic processes.

A time-harmonic fluid-solid interaction problem with random interfaces

Thanh Tran (UNSW)

Abstract

A bounded elastic body (obstacle) shares a common boundary with a fluid domain. An incident acoustic wave is sent from the fluid domain to the obstacle. The response of the obstacle and the scattered wave are modelled as a fluid-solid interaction problem, which includes equations of elasticity in the elastic body and the Helmholtz equation in the fluid, together with transmission conditions on the shared boundary which is uncertainly located.
In this talk, we will first present a mixed finite element method that computes the pressure in the fluid and the stress in the elastic body when the interfaces are deterministic. Next we consider the case when the interfaces are uncertainly located. We apply shape calculus to approximate the solution perturbation by the so-called shape derivative and shape Hessian. Correspondingly statistical moments of the solution are approximated by the moments of the shape derivative and shape Hessian which are solutions of related problems on a fixed (deterministic) geometry.
This is a joint project with Salim Meddahi and Antonio Marquez at the University of Oviedo, Spain, and Debopriya Mukherjee at UNSW Sydney.

Organisers

Ben Goldys (The University of Sydney) Daniel Hauer (The University of Sydney)

Ngan Le (Monash University) Thanh Tran (UNSW)

Organisation of this workshop has been possible thanks to financial support from the School of Mathematics and Statistics of the University of Sydney.