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)
• Le Gia (UNSW)
• Beniamin Goldys (Sydney)
• Alex Hening (Tufts University)
• Ngan Le (Monash)
• Gregoire Loeper (Monash)
• Pierre Portal (ANU)
• Andreas Prohl (Tuebingen University)
• Nimit Rana (York University)
• Thanh Tran (UNSW)
• Rongchan Zhu (Beijing Institute of Technology)

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

 Further details will follow soon.

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.

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).

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).

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.