Sep 29, 2008
- Oct 1, 2008
Organisers
| Name |
Institution |
| Buckwar, Evelyn |
Heriot-Watt University |
| Gyongy, Istvan |
University of Edinburgh |
| Lord, Gabriel |
Heriot-Watt University |
This is a two and a half day meeting aimed at introducing SPDEs, their analysis, applications and numerics. It is aimed at people with no previous background in SPDEs or stochastics.
Some financial support is available for interested researchers at all levels.
Confirmed Speakers
Mike Christie (Institute of Petroleum Engineering, Heriot-Watt University)
Arnaud Debussche (ENS Cachan Bretagne)
Istvan Gyongy (Mathematics, University of Edinburgh)
Gabriel Lord (Mathematics, Heriot-Watt University)
Catherine Powell (University of Manchester)
Tony Shardlow (Mathematics, University of Manchester)
Andrew Stuart (Mathematics, University of Warwick)
Supporting Institution
Maxwell Institute for Mathematical Sciences (Centre for Analysis and Nonlinear PDE)
Arrangements
Participation
You may register your interest in attending this meeting by completing the application form, which will be available at the top of this page (please keep checking to see if it has gone live). Please note that the application form will close on 31 July 2008. You will receive an e-mail in early August 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 Morag Burton.
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 August. 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 29 and Tuesday 30 September.
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 29 September.
The workshop dinner will take place on the evening of Tuesday 30 September. The workshop grant will cover the cost of this meal.
Poster Session
A poster session will take place after the close of lectures on Monday 29 September. 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 29 September.
Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover tea/coffee throughout the workshop, lunch on the first and second day, the wine reception and the Workshop Dinner. It will also be possible for participants to claim 15.00 GBP towards other local expenses.
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
Provisional Timetable
Monday 29 September
09.00 - 09.50 | Registration
|
09.50 - 10.00 | Welcome
|
10.00 - 11.00 | Tony Shardlow (University of Manchester)
A tutorial on stochastic PDEs (Part 1) |
11.00 - 11.30 | Tea/Coffee
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11.30 - 12.30 | Tony Shardlow (University of Manchester)
A tutorial on stochastic PDEs (Part 2) |
12.30 - 14.00 | Buffet lunch
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14.00 - 15.00 | Istvan Gyongy (University of Edinburgh)
Stochastic PDEs and filtering (Part 1) |
15.00 - 15.30 | Tea/Coffee
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15.30 - 16.30 | Gabriel Lord (Heriot-Watt University)
Stochastic PDEs and numerics (Part 1) Download PDF of slides
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16.30 - 17.30 | Mike Christie (Heriot-Watt University)
Solution error modelling and inverse problems |
17.30 - 19.00 | Wine reception and poster session |
Tuesday 30 September
09.00 - 10.00 | Istvan Gyongy (University of Edinburgh)
Stochastic PDEs and filtering (Part 2) |
10.00 - 11.00 | Gabriel Lord (Heriot-Watt University)
Stochastic PDEs and numerics (Part 2) |
11.00 - 11.30 | Tea/Coffee
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11.30 - 12.30 | Catherine Powell (University of Manchester)
Numerical methods for solving elliptic PDEs with random data Download PDF of slides |
12.30 - 14.00 | Buffet lunch
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14.00 - 15.00 | Arnaud Debussche (ENS Cachan Bretagne)
Weak order for the Euler scheme for a SPDE Download PDF of slides |
15.00 - 15.30 | Tea/Coffee
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15.30 - 16.30 | Istvan Gyongy (University of Edinburgh)
Stochastic PDEs and filtering (Part 3) |
19.00 - | Workshop Dinner at Nargile |
Wednesday 1 October
09.00 - 10.00 | Andrew Stuart (University of Warwick)
Noisy gradient flows and conditioned diffusions Download PDF of paper
|
10.00 - 11.00 | Arnaud Debussche (ENS Cachan Bretagne)
Random modulation of solitons for the Stochastic KdV equations Download PDF of slides |
11.00 - 11.30 | Tea/Coffee
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11.30 - 12.30 | Istvan Gyongy (University of Edinburgh)
Stochastic PDEs and filtering (Part 4) |
12.30 - 14.00 | Close |
Presentations:
| Presentation Details |
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| Christie, Mike |
| Solution error modelling and inverse problems |
View Abstract
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Inverse problems feature in many areas of science and engineering. In practical applications involving industrial CFD codes, the codes can be time consuming to run at high resolution. This may lead to simulations being run at reduced resolution leading to a compromise between the number of runs carried out and the accuracy of each simulation.
The idea behind solution error modelling is to build a statistical model for the errors involved with using a lower resolution code. The model is built from a limited number of runs of both the fine and coarse models. The appropriate error statistics are then interpolated through parameter space to correct the estimates of physical quantities obtained from the lower resolution simulations.
This talk will describe solution error modelling, and illustrate the concepts with applications to 3 problems: a simple viscous fingering problem; the Lorenz equations of atmospheric physics; and an 8 parameter example from the oil industry.
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| Debussche, Arnaud |
| 1. Weak order for the Euler scheme for a SPDE and 2. Random modulation of solitons for the Stochastic KdV equations |
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1.
In this talk, we present recent results on the order of convergence of the Euler scheme for a Stochastic Partial Differential Equation. The strong order of convergence has been studied by many authors. However, very few results are available for the weak order of convergence.
It is well known that the Euler scheme is of strong order 1=2 and weak order 1 in the case of a stochastic differential equation. Two methods are available to prove this result. The first one uses the Kolmogorov equation associated to the stochastic equation and was first used by D. Talay. A second one has been recently discovered by A. Kohatsu-Higa and is based on Malliavin calculus.
In this talk, we generalize such results to the infinite dimensional case. We show how to adapt Talay's method. The main difficulty is due to the presence of unbounded operators in the Kolmogorov equation. A tricky change of unknown allows to treat the case of a linear equation. It also works for an equation whose linear part defines a group, the nonlinear Schrödinger equation for instance. The case of a semilinear equation of parabolic type is more difficult and we use Malliavin calculus, but not in the same way as in Kohatsu-Higa's method. We prove for instance that, in the case of a nonlinear heat equation in dimension one with a space time white noise, the Euler scheme has weak order 1=2, it is well known that the strong order is 1=4.
2. Joint work with Anne de Bouard.
In this work, we consider the Korteweg- de Vries equation perturbed by a random force of white noise type, additive or multiplicative.
In a series of work, in collaboration with Y. Tsutsumi, we have studied existence and uniqueness in the additive case for very irregular noises. These use the functional framework introduced by J. Bourgain. We use similar tools to prove existence and uniqueness for a multiplicative noise. We are not able to consider irregular noises and have to assume that the driving Wiener process has paths in $L^2$ or $H^1$. However, contrary to the additive case, we are able to treat spatially homogeneous noises.
Then, we try to understand the effect of a small noise with amplitude $varepsilon$ on the propagation of a soliton. We prove that, on a time scale proportional to $varepsilon^-2$, a solution initially equal to the soliton but perturbed by a noise of the type above remains close to a soliton with modulated speed and position. The modulated speed and position are semi-martingales and we write the stochastic equations they satisfy. We prove also that a Central Limit Theorem holds so that, on the time scale described above, the solutions can formally be written as the sum of the modulated soliton and a Gaussian remainder term of order $varepsilon$.
In the multiplicative case, we can go further. We prove that the gaussian part converges in distribution to a stationary process. Also, the equations for the modulation parameters allow to give a justification for the phenomenon called ”soliton diffusion” obsverved in numerical simulations: the averaged soliton decays like $t^gamma$. We obtain $gamma=-5/4$.
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| Gyongy, Istvan |
| Stochastic PDEs and filtering |
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View Abstract
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Part 1. Examples of stochastic PDEs
First we consider some examples of stochastic PDEs arising in engineering, physics and biology. We will especially be interested in the stochastic PDEs arising in nonlinear filtering: A noisy "signal and observation" model will be introduced, and the classic problem of computing the best estimate for the signal from the observations will be discussed. In particular, stochastic PDEs for the solution of this problem will be obtained.
Part 2. Methods of solving stochastic PDEs
We revisit the stochastic PDEs presented in Part1 and discuss some methods to prove the existence and uniqueness of their solutions. In particular, we will give an introduction to the method of semigroups and to the L_2 theory of stochastic PDEs.
Part 3. Regularization of ill-posed PDEs by noise
Deterministic PDEs can be ill-posed in the sense that they do not have a solution, or they have more than one solution, or the solutions do not depend continuously on the data. Sometimes, as in the case of the famous Navier-Stokes equation in dimension 3, it is unknown whether the equation is well-posed. Surprisingly, one can change some ill-posed deterministic PDEs into well-posed stochastic PDEs by introducing a small random perturbation. We present some prototypes of this phenomenon.
Part 4. On the connection between deterministic and stochastic PDEs
We revisit the signal and observation model presented in Part1, and are interested now in the "robustness" of the nonlinear filter, i.e., in the "stability" of the estimate with respect to the small changes in the observations. This leads us to investigating the dependence of the solutions of stochastic PDEs on the noise involved in the equations.
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| Lord, Gabriel |
| Stochastic PDEs and numerics |
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The course will primarily be focused on understanding a parabolic PDE with noise and its numerical approximation. To interpret the stochastic PDE, we explain the stochastic integral in the Ito sense. There is a brief review of some of the theory for SPDEs before the discretizations are considered. By using a Fourier in space the material is kept straight forward and standard low order schemes are considered for the time discretization. Throughout, the material will be presented in a practical way as to how the equations may be discretized and illustrated through sample computations.
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| Powell, Catherine |
| Numerical methods for solving elliptic PDEs with random data |
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In this talk, we present an introduction to so-called stochastic finite element methods (SFEMs) for solving elliptic PDEs with correlated random inputs. A typical application is groundwater flow modelling where, in Darcy’s equation, it is impossible to specify the permeability coefficients at every spatial location. In stochastic modelling, the idea is to model the uncertain inputs to the governing PDEs as correlated random fields and determine statistical information about the random solution variables.
SFEMs rely on standard finite element schemes for spatial discretisation, but differ in how they handle the discretisation of a probability space. We give an overview of SFEMs based on Monte Carlo, stochastic collocation and stochastic Galerkin methods, outline their convergence properties and then focus on the iterative solution of the resulting linear systems of equations and the associated computational cost. The first two groups of methods are sampling methods and require a large number of deterministic problems to be solved for different realisations of the random inputs. This is relatively straight-forward from a linear algebra point of view. SFEMs based on stochastic Galerkin methods, however, approximate the random components of the solution variables via globally orthogonal polynomials. In some cases, this means that we can solve a single linear system to determine the unknowns in the stochastic expansions of the solution variables. Statistical information about the numerical solution is then directly available. However, the dimension of the system can be orders of magnitude larger than that of the decoupled systems arising in stochastic sampling methods and this is more challenging.
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| Shardlow, Tony |
| A tutorial on stochastic PDEs |
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We begin with a straightforward introduction to the probability theory required for the course. We introduce random fields and focus particularly on Gaussian random fields which occur in many physical applications and are completely described by their mean and spatial correlation structures. We develop the Wiener process as a series expansion in independent Brownian motions.
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| Stuart, Andrew |
| Noisy Gradient Flows and Conditioned Diffusions |
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Conditioned diffusions arise in a variety of application areas such as chemistry, econometrics and signal processing. A conditioned diffusion gives rise to a probability measure on pathspace. I will describe a method for deriving SPDEs for which this measure is invariant. The SPDEs have the form of noisy gradient flows in pathspace.
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Participants
| Name |
Institution |
| Aziz, Mohammed |
University of Oxford |
| Baker, Ruth |
University of Oxford |
| Banas, Lubomir |
Heriot-Watt University |
| Bello, Iyabo |
Heriot-Watt University |
| Buckwar, Evelyn |
Heriot-Watt University |
| Carelli, Erich |
Universitaet Tuebingen |
| Chenchiah, Isaac |
University of Bristol |
| Christie, Mike |
Heriot-Watt University |
| Debussche, Arnaud |
ENS Cachan Bretagne |
| Farkas, Jozsef |
University of Stirling |
| Ferro, Maria |
Università di Napoli 'Federico II' |
| Fischer, Ingo |
Heriot-Watt University |
| Gordon, Andrew |
University Of Manchester |
| Grima, Ramon |
University of Edinburgh |
| Grinfeld, Michael |
University of Strathclyde |
| Gyongy, Istvan |
University of Edinburgh |
| Hamdous, Saliha |
University of Torino |
| Hassan Adamu, Shitu |
Brunel University |
| Heaney, Claire |
Durham University |
| Hussaini, Nafiu |
Brunel University |
| Lloyd, David |
University of Surrey |
| Lord, Gabriel |
Heriot-Watt University |
| Powell, Catherine |
University of Manchester |
| Retkute, Renata |
University of Surrey |
| Riedler, Martin |
Heriot-Watt University |
| Ryser, Marc D. |
McGill University |
| Schmid, Karen |
Heriot-Watt University |
| Shardlow, Tony |
University of Manchester |
| Sherratt, Jonathan |
Heriot-Watt University |
| Sickenberger, Thorsten |
Heriot-Watt University |
| Slastikov, Valeriy |
University of Bristol |
| Smith, David |
University of Reading |
| Stinga, Pablo Raúl |
Universidad Autónoma de Madrid |
| Stuart, Andrew |
University of Warwick |
| Tambue, Antoine |
Heriot-Watt University |
| Thul, Ruediger |
University of Nottingham |
SPDEs: Computations and Applications