ICMS

May 11, 2009 - May 12, 2009

14 India Street, Edinburgh EH3 6EZ

Organisers

Name	Institution
Buckwar, Evelyn	Heriot-Watt University
Lord, Gabriel	Heriot-Watt University

Short report

The numerical solution of stochastic PDEs (SPDEs) is a very active area of research and is of both theoretical and practical interest. Quoting from the feedback summary of the participants: The key future research direction in my field is exactly Numerics for SPDEs. and In connection with the solution of differential equations with random data I see the key future research directions in more effective and accurate methods under simpler and more realistic assumptions on the data. Furthermore the modeling problem seems to me to become more important.
The topics of the talks thus covered a range of aspects of numerical analysis and applications of SPDEs. In particular there were talks treating different types of SPDEs, e.g., Allen-Cahn equation with additive noise, the stochastic wave equation and the stochastic Landau-Lifschitz equation, as well as random PDEs; different numerical schemes and convergence notions, e.g., exponential integrators, Itô-Taylor expansions or splitting schemes for Monte-Carlo methods or pathwise approximation; different types of application area, e.g., groundwater or multiphase flow, ferromagnetism and stochastic diffusive interfaces. The participants commented very favourably on the breadth of topics. The meeting was attended by 30 participants, established researchers from various backgrounds, as well as PhD students. The goals of the meeting were met and, quoting from the feedback form again: By my opinion the academic value was very high, the researchers gave impressive talks about their very recent work.

Download the full report here.

 

Introduction to workshop

In this meeting we aim to bring together researchers focusing on the analysis of stochastic partial differential equations (SPDEs), those concentrating on the applications and researchers designing and analysing numerical schemes for SPDEs. Such a three way interaction is the key to designing efficient and accurate numerical schemes as well as a good understanding of the physical processes and can assist and open new avenues in the mathematical analysis.

Funded by
EPSRC
LMS
EMS

 

Arrangements

Participation
You may register your interest in attending this meeting by completing the application form, which is available at the top of this page. You will receive an e-mail indicating whether you have been allocated a place. Speakers will receive a separate invitation via e-mail.

If you do not receive an e-mail please contact Irene Moore.

Venue
The workshop will take place at the head-quarters of ICMS, 14 India Street, Edinburgh. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, near the city centre.

The ICMS travel pages contain advice on how to travel to Edinburgh. For local information the finding ICMS page shows the location of ICMS and contains useful maps of the city centre.

The seminar room at ICMS has four whiteboards, two overhead projectors, a data projector and laptop.

Wireless access is available throughout the ICMS building. There are also three public PCs which may be used at any time for internet access and to check email.

Accommodation
Participants should make their own accommodation arrangements: it will be possible to reclaim part of these costs through ICMS. Further details will be provided in early April. A list of Edinburgh accommodation of various sorts and prices is available here . Sections 1-3 are particularly relevant.

Invited participants should refer to their personal invitation letter for accommodation arrangements.

Meals and Refreshments
A buffet lunch will be provided on the Monday 11 and Tuesday 12 May.

Morning and afternoon refreshments will be provided throughout the workshop.

There will be an informal wine reception and poster session after the close of lectures on Monday 11 May, followed by a workshop dinner.

Poster Session
A poster session will take place after the close of lectures on Monday 11 May. The number of posters will be limited to 10 and size to A1 Portrait, due to space restrictions. Please indicate on your application form if you are interested in presenting a poster.

Registration
Registration will take place on Monday 11 May at 09.00.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover tea/coffee throughout the workshop, lunch on both days, the wine reception and the workshop dinner. It will also be possible for participants to claim a contribution towards travel, accommodation and local expenses. This contribution will be confirmed soon.

Reimbursement will take place after the workshop. At Registration you will be given an expenses claim form that should be submitted to ICMS, with receipts. Your expenses will be paid directly into your bank account so you will need to provide your account details plus IBAN, SWIFT or Routing numbers as applicable. Please note that we cannot reimburse any item without a receipt.

 

Programme

Monday 11 May

09.00 - 10.00	Registration
10.00 - 10.45	Tony Shardlaw (University of Manchester) Pathwise error bounds for exponential integrator scheme
10.45 - 11.15	Tea/Coffee
11.15 - 12.00	Omar Lakkis (University of Sussex) Computations and Stochastic Diffuse-Interface Models
12.00 - 12.45	Istvan Gyongy (University of Edinburgh) Accelerated finite difference schemes
12.45 - 14.00	Lunch
14.00 - 14.45	Patrick Jenny (ETH Zürich) Stochastic modelling of non-equilibrium multi-phase flow to link pore scale dynamics with Darcy scale behaviour
14.45 - 15.30	Rob Scheichl (University of Bath) Computational challenges in Uncertainty Quantification
15.30 - 16.00	Tea/Coffee
16.00 - 16.45	Wolfgang Nowak (Institut für Wasserbau) Challenges and Demands from the Application Side of Stochastic Modelling
16.45 - 17.30	Erika Hausenblas (University of Salzburg) The Numerical Approximation of the Stochastic Wave equation
17.30 - 18.30	Wine reception

 

Tuesday 12 May

09.00 - 09.45	Rachel Kuske (University of British Columbia) Competition of different noise sources in delay dynamics
09.45 - 10.30	Arnulf Jentzen (Johann Wolfgang Goethe-University) Higher order numerical approximation of stochastic partial differential equations - a Taylor expansion approach
10.30 - 11.00	Tea/Coffee
11.00 - 11.45	Nicolai Krylov (University of Minnesota) On the It^o and It^o-Wentzell formulas in the theory of SPDEs
11.45 - 12.30	Zdzislaw Brzezniak (University of York) On stochastic PDEs and ferromagnetism
12.45 - 14.00	Lunch

Presentations:

Presentation Details
Brzezniak, Zdzislaw
On stochastic PDEs and ferromagnetism
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Motivated by a recent paper "Magnetic elements at finite temperature and large deviation theory" by R. V. Kohn, M. G. Reznikoff and E. Vanden-Eijnden, J. Nonlinear Sci. 15, 223-253 (2005), we are interested in stochastic parabolic Landau-Lifschitz equations. We investigate existence and uniqueness of solutions, and possibly the Large Deviations principle and stability. This talk is based on a joint work with B. Godys.
Gyongy, Istvan
Accelerated finite difference schemes
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Accelerated finite difference schemes for stochastic PDEs are presented. In particular, sufficient conditions are given under which the convergence of finite difference approximations for stochastic PDEs of parabolic type can be accelerated to any given order of convergence by Richardson's method. The results are applied to numerical solutions of nonlinear filtering problems. The talk is based on joint results with Nicolai Krylov.
Hausenblas, Erika
The numerical approximation of the stochastic wave equation
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We will approximate the Stochastic Wave equation by the leap frog scheme and show that the weak error is two.
Jenny, Patrick
Stochastic modeling of non-equilibrium multi-phase flow to link pore scale dynamics with Darcy scale behaviour
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The prime objective of the proposed probability density function (PDF) modeling framework is to improve our understanding of how the physics and dynamics at the pore scales is linked to the macroscopic equations, which deal with average values. A stochastic particle method (SPM), in which individual (infinitesimal) fluid volumes are modeled, is employed to solve the PDF transport equation. These fluid volumes are represented by computational particles, whose evolution depends on their phase, composition and other properties including memory. Such a Lagrangian approach offers an alternative viewpoint and allows to describe various complex non-equilibrium processes in a more general and natural way than in a classical Eulerian framework. During the presentation the mathematical framework will be derived and its potential will be illustrated with some simple test cases. Moreover, it will be shown how transport equations of joint statistical moments can be derived.
Jentzen, Arnulf
Higher order numerical approximation of stochastic partial differential equations - a Taylor expansion approach
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In this talk, we present Taylor expansions of the solution of a stochastic partial differential equation (SPDE) of evolutionary type and their first applications to numerical analysis. The key instruments for deriving such Taylor expansions are the semigroup approach, i.e. to understand the SPDE as a mild integral equation, and an appropriate recursion technique.
Krylov, Nicolai
On the It^o and It^o-Wentzell formulas in the theory of SPDEs
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I will present several results concerning the It^o and It^o-Wentzell formulas for distribution-valued processes and some related results on the stochastic Fubini theorem and the It^o-Wentzell formula for real-valued processes, which contains It^o's formula as a particular case.
Kuske, Rachel
Competition of different noise sources in delay dynamics
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Transient or unstable behavior is often ignored in considering long time dynamics in the deterministic world. However, stochastic effects can change the picture dramatically, so that the transients can dominate the long range behavior. This talk will contrast the effects of different noise sources in certain systems with delayed feedback, driving both detrimental and beneficial noise-stabilized behaviors composed of transients from the deterministic model. The analysis starts with a canonical model of machine tool dynamics and is extended to consider questions about medical treatments.
Lakkis, Omar
Computing the stochastic Allen-Cahn problem
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Our main goal is the numerical approximation of the Allen-Cahn problem with additive white noise in one-dimensional space. From a compuational view-point, the main difficulty for a rigorous numerical discretization of this SPDE, which is the baby-version of more complicated phase-field model systems, is the presence of the time-space white noise as a forcing term. The discretization has to be conducted in two stages: (1) regularize the white noise and study the regularized problem, (2) approximate the regularized problem and derive a finite element Monte Carlo simulation scheme. We then have to relate our numerical results to theoretical results from the probabilistic analysis of scaling limits conducted mainly by Funaki and Brassesco, De Masi & Presutti in the 1990's [KKL07]. Time allowing I shall review recent advances on the topic where we analyze the behavior of one and multiple interfaces when the interface thickness parameter does not approach the limit. Asymptotic analysis shows that the stochastic solution and its approximation remain "close" to a set of functions where the interface makes sense. This is particularly useful to relate numerical results to theory and it has the potential to define stochastic diffuse interfaces in dimensions higher than 1 [KLR09].
Nowak, Wolfgang
Challenges and demands from the application side of stochastic modelling
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Recent years have shown impressive advances in solution techniques for stochastic partial differential equations (SPDEs). Given a stochastic description of input uncertainty, they evaluate the uncertainty of model output. Most applications of stochastic modeling, however, ask for more than just efficient evaluation of output statistics. In this presentation, four additional challenges are highlighted, which require further advancement of stochastic modeling tools. These are: (1) Stochastic inverse modeling and Bayesian updating, (2) Optimal design of experiments, (3) Extreme value statistics , and (4) the relaxation of crucial assumptions, such as known covariances and (multi-)Gaussianity. Most advanced SPDE solution techniques (such as stochastic Galerkin, stochastic collocation, Karhunen-Loeve moment approaches) are poorly applicable to these challenges, and require further development. The current presentation highlights these limitations and proposes directions for generalization. For illustration purposes, problems from stochastic hydrogeology are used.
Scheichl, Rob
Computational challenges in Uncertainty Quantification
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Uncertainty and its quantification play a major role in decision processes in the modern world. One particular area of huge current interest is the safety assessment of radioactive waste disposal underground. In this talk we will present a common way to model the data uncertainty in this application through stochastic modelling of the rock permeabilities leading to a stochastic PDE for the groundwater flow. The numerical solution of these kinds of stochastic PDEs has become an area of large current activity in the numerical analysis and scientific computing communities. Because of the typically large variances and short correlation lengths in our application, the only feasible solution approach is Monte Carlo type simulation. Therefore, it is of paramount importance to design robust and efficient solvers for PDEs with highly varying heterogeneous coefficients (a typical realisation of the permeability field). Multilevel iterative techniques are indispensable to obtain this goal. However, standard analyses of these techniques only guarantee robustness when the coarse grid(s) resolve the coefficients. Here, we review a sequence of recent papers where simple variants of domain decomposition methods, such as overlapping Schwarz and one-level FETI, are proposed and analysed that are robust to strong coefficient variation (even inside coarse grid elements). In particular, our new theory proves very precisely the previously made empirical observation in the algebraic multigrid literature that the use of energy minimising coarse spaces can lead to robust multilevel methods.
Shardlow, Tony
Pathwise error bounds for exponential integrator scheme
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We introduce an exponential integrator scheme for a reaction diffusion equation with additive space-time forcing. We study the rate of convergence in the pathwise sense and the difficulties of extending the results to locally Lipschitz reaction terms.

Participants

Name	Institution
Banas, Lubomir	Heriot-Watt University
Bocharov, Boris	University of Edinburgh
Brzezniak, Zdzislaw	University of York
Buckwar, Evelyn	Heriot-Watt University
Ding, Jie	University of Edinburgh
Ernst, Oliver	TU Bergakademie Freiberg
Ferro, Maria	Università di Napoli 'Federico II'
Gordon, Andrew	University Of Manchester
Gyongy, Istvan	University of Edinburgh
Hall, Eric	University of Edinburgh
Hausenblas, Erika	University of Salzburg
Jenny, Patrick	ETH Zurich
Jentzen, Arnulf	Johann Wolfgang Goethe-University
Kruse, Raphael	University of Bielefeld
Krylov, Nicolai	University of Minnesota
Kuske, Rachel	University of British Columbia
Lakkis, Omar	University of Sussex
Lord, Gabriel	Heriot-Watt University
Lythe, Grant	University of Leeds
Mugler, Antje	University of Applied Science Zwickau
Nowak, Wolfgang	Institut für Wasserbau / SimTech
Riedler, Martin	Heriot-Watt University
Scheichl, Rob	University of Bath
Schmid, Karen	Heriot-Watt University
Sebastian, Geiger	Heriot-Watt University
Shardlow, Tony	University of Manchester
Sickenberger, Thorsten	Heriot-Watt University
Stanciulescu, Vasile	University of Leicester
Starkloff, Hans-Jörg	University of Applied Sciences Zwickau
Yi, Liu	AWE plc