Program and videos
The program is the following (last update: 24 January 2017), and includes large breaks to stimulate discussions.
Notice that the posters will remain visible during the whole duration of the workshop.
Place: University of Geneva, Building Science II & III, Quai Ernest-Ansermet 30, 1205 Geneva, Switzerland, auditorium A100 (ground floor).
New! Videos of the presentations are now available on Mediaserver, University of Geneva (see also the links below).
Tuesday 31st January 2017
Wednesday 1st February 2017
Titles and abstracts of the presentations
- Assyr Abdulle (EPF Lausanne)
Note: Unfortunately, the talk of Assyr Abdulle is CANCELED.
Averaging and spectral methods for multiscale stochastic differential equations.
We present recent progress for the numerical solution of stochastic differential equations with multiple scales. We first review averaging-based methods that rely on micro and macro solvers. The efficiency of such methods depends on accurate micro averaging (fast convergence to an invariant distribution). In this direction we briefly mention recent techniques inspired by ideas coming from geometric integration that allow to construct numerical integrators capable of capturing accurately the invariant measure of ergodic SDEs. We finally presents a new method for the solution of multiscale stochastic differential equations. In contrast to averaging-based methods, the new methodology relies on a spectral method to approximate the solution of an appropriate Poisson equation which is used to calculate the coefficients of the macroscopic (homogenized) equation.
 A. Abdulle, W. E, B. Engquist, and E. Vanden-Eijnden, The heterogeneous multiscale method. Acta Numer., 21:1–87, 2012.
 A. Abdulle and G. A. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales. J. Comput. Phys., 231(6):2482–2497, 2012.
 A. Abdulle, G. Vilmart, and K. C. Zygalakis, High order numerical approximation of the invariant measure of ergodic SDEs, SIAM J. Numer. Anal., 52:1600–1622, 2014.
 A. Abdulle, G. A. Pavliotis, and U. Vaes, Spectral methods for multiscale stochastic differential equations, preprint, arXiv:1609.05097, 2016.
- Sonja Cox (University of Amsterdam)
Weak convergence for semi-linear SPDEs.
Joint work with Arnulf Jentzen, Ryan Kurniawan, and Timo Welti.
In numerical analysis for stochastic differential equations, a general rule of thumb is that the optimal weak convergence rate of a numerical scheme is twice the optimal strong convergence rate. However, for SPDEs the optimal weak convergence rate is difficult to establish theoretically. Recently, progress was been made by Jentzen, Kurniawan and Welti for semi-linear SPDEs using the so-called mild Itô formula. We consider this approach for wave equations.
- Antoine Gloria (INRIA Lille and Université Libre de Bruxelles)
Long-time homogenization of the wave equation.
In this talk I'll present recent results on the long-time homogenization of the wave equation in random media. To this aim I'll introduce the notion of Taylor-Bloch waves, at the basis of an approximate spectral theory at low frequencies. For periodic and quasiperiodic coefficients, this allows one to define a family of higher-order homogenized operators which describe the behavior of the solution on arbitrarily large time frames (and encompasses the standard dispersive approximation). I will then turn to the random case, give a short review on quantitative results in the elliptic case, and address the long-time homogenization in this setting. If time allows I'll give the counterpart of these results for the Schrödinger equation with random potential.
This is joint work with Antoine Benoit (ULB).
- Georg Gottwald (University of Sydney)
Stochastic parameterizations of deterministic dynamical systems: Theory, applications and challenges.
There is an increased interest in the stochastic parameterization of deterministic dynamical systems whereby a high-dimensional deterministic dynamical system is reduced to a low-dimensional stochastically driven system.
We discuss standard techniques of stochastic model reduction such as homogenization. Recently rigorous results have been obtained justifying this method. The theory relies on an asymptotic limit of infinite time scale separation which is not always satisfied in real world applications. We present a new method to go beyond this asymptotic limit by employing Edgeworth approximations.
This is joint work with Jeroen Wouters.
- Arnulf Jentzen (ETH Zurich)
On stochastic numerical methods for the approximative pricing of financial derivatives.
In this lecture I intend to review a few selected recent results on numerical approximations for high-dimensional nonlinear parabolic partial differential equations (PDEs), nonlinear stochastic ordinary differential equations (SDEs), and high-dimensional nonlinear forward-backward stochastic ordinary differential equations (FBSDEs). Such equations are key ingredients in a number of pricing models that are day after day used in the financial engineering industry to estimate prices of financial derivatives. The lecture includes content on lower and upper error bounds, on strong and weak convergence rates, on Cox-Ingersoll-Ross (CIR) processes, on the Heston model, as well as on nonlinear pricing models for financial derivatives. We illustrate our results by several numerical simulations and we also calibrate some of the considered derivative pricing models to real exchange market prices of financial derivatives on the stocks in the American Standard & Poor's 500 (S&P 500) stock market index.
- Annika Lang (Chalmers/University of Gothenburg)
Mean-square stability analysis of SPDE approximations.
Mean-square stability analysis of the zero solution of SDE approximations is well established. In this talk the theory is generalized to martingale-driven SPDE. Since the generalization of the finite-dimensional theory is not suitable, mean-square stability of SPDE is characterized in terms of operators. Applications to Galerkin finite element methods in combination with backward Euler, Crank-Nicolson, and forward Euler approximations of the semigroup and Euler-Maruyama and Milstein schemes for the stochastic integral are presented.
This is joint work with Andreas Petersson and Andreas Thalhammer.
- Tony Lelièvre (Ecole des Ponts ParisTech)
Accelerated dynamics and transition state theory.
I will present multiscale in time algorithms which are used in molecular simulation in order to bridge the gap between the atomistic timescale and the macroscopic timescale. More precisely, I will describe the parallel replica algorithm and its mathematical analysis using the notion of quasi stationary distribution. I will also explain how this notion can be used to justify the construction of jump processes starting from the overdamped Langevin dynamics, using transition state theory and Eyring-Kramers formulas for the rates.
- Gabriel Lord (Heriot Watt University, Edinburgh)
Adaptive timestepping for S(P)DEs to control growth.
We introduce a class of adaptive timestepping strategies for stochastic differential equations such as those arising from the semi-discretization of SPDEs with non-Lipschitz drift coefficients. These strategies work by controlling potential unbounded growth in solutions of a numerical scheme. We prove that the Euler-Maruyama scheme with an adaptive timestepping strategy in this class is strongly convergent and present preliminary results on a semi-implicit scheme and an extension to non-Lipschitz noise terms. We test this alternative to taming on some examples. This is joint work with Conall Kelly.
- Greg Pavliotis (Imperial College, London)
Noise-induced transitions and mean field limits for multiscale diffusions.
In this talk I will present some recent results on the long time behaviour of the overdamped Langevin dynamics for Brownian particles moving in a multiscale, rugged energy landscape. The dynamics of such processes can be quite complicated, in particular in the low temperature regime, since metastable states, corresponding to local minima of the potential, can (co-)exist at all scales. We will show how we can obtain a coarse-grained description for the dynamics at large scales, given by a stochastic differential equation with multiplicative noise, despite the fact that the noise in the original dynamics is additive. We then show that the combined effect of noise and multiscale structure leads to hysteresis effects in the bifurcation diagram for the equilibrium coarse-grained dynamics. In the second part of the talk I will present recent results on the mean field limit of systems of interacting diffusions in a multiscale confining potential. The mean field limit is described by a nonlinear, nonlocal Fokker-Planck equation of McKean-Vlasov type that exhibits phase transitions. The effect of the multiscale structure of the potential on the phase diagram will be discussed in detail.
- Kostas Zygalakis (University of Edinburgh)
Ergodic Stochastic Differential Equations and Sampling: A numerical analysis perspective.
Understanding the long time behaviour of solutions to ergodic stochastic differential equations is an important question with relevance in many field of applied mathematics and statistics. Hence, designing appropriate numerical algorithms that are able to capture such behaviour correctly is extremely important. A recently introduced framework [1,2,3] using backward error analysis allows us to characterise the bias with which one approximates the invariant measure (in the absence of the accept/reject correction). These ideas will be used to design numerical methods exploiting the variance reduction of recently introduced nonreversible Langevin samplers . Finally if there is time we will discuss, how things ideas can be combined with the idea of Multilevel Monte Carlo  to produce unbiased estimates of ergodic averages without the need the of an accept-reject correction  and optimal computational cost.
 K.C. Zygalakis. On the existence and applications of modified equations for stochastic differential equations. SIAM J. Sci. Comput., 33:102-130, 2011.
 A. Abdulle, G. Vilmart, and K. C. Zygalakis. High order numerical approximation of the invariant measure of ergodic sdes. SIAM J. Numer. Anal., 52(4):1600-1622, 2014.
 A. Abdulle, G. Vilmart, and K.C. Zygalakis, Long time accuracy of Lie-Trotter splitting methods for Langevin dynamics. SIAM J. Numer. Anal., 53(1):1-16, 2015.
 A. Duncan, G. A. Pavliotis and T. Lelievre, Variance Reduction using Nonreversible Langevin Samplers, J. Stat. Phys., 163(3):457-491, 2016.
 M.B. Giles, Mutlilevel Monte Carlo methods, Acta Numerica, 24:259-328, 2015.
 L. Szpruch, S. Vollmer, K. C. Zygalakis and M. B. Giles, Multi Level Monte Carlo methods for a class of ergodic stochastic differential equations. arXiv:1605.01384.
Titles and abstracts of posters
- Asma Barbata (University of Loraine)
Filtering of stochastic nonlinear systems with multiplicative noises. Download poster (PDF).
- Ahmed Hajej (University of Panthéon-Assas)
Stochastic homogenization of a front propagation problem with unbounded velocity.
We study the homogenization of Hamilton-Jacobi equations which arise in front propagation problems in stationary ergodic media. Our results are obtained for fronts moving with possible unbounded velocity. We show, by an example, that the homogenized Hamiltonian, which always exists, may be unbounded. In this context, we show convergence results if we start with a compact initial front. On the other hand, if the media satisfies a finite range of dependence condition, we prove that the effective Hamiltonian is bounded and obtain classical homogenization in this context.
- Tahar Khalifa (Institut supérieur des sciences appliquées et de technologie de Sousse)
Practical output feedback tracking control for a class of stochastic systems. Download poster (PDF).
This study investigates the global adaptive practical tracking for a class of nonlinear stochastic systems with dynamic uncertainties and unmeasured states via dynamic output feedback control. We show that we can extend the work in  to stochastic system and generalize the work in . An output feedback controller is constructed to guarantee that the closed-loop system is globally practically stable in probability and the output can be regulated to the all fixed ball almost surely. Finally an example is provided to demonstrate the effectiveness of the proposed approach.
 A. Ben Abdallah, T. Khlifa, M. Mabrouk, On observer for a class of uncertain non-linear systems. Nonlinear Dynamics, January 2015, Volume 79, Issue 1, pp 359-368.
 Fengzhong Li, Yungang Liu, General Stochastic Convergence Theorem and Stochastic Adaptive Output-Feedback Controller. IEEE Transactions on Automatic Control, 2016, Volume 99.
- Arbaz Khan (Ruprecht-Karls-Universität Heidelberg)
A robust a-posteriori error estimator for Divergence-conforming DG methods for Oseen equation. Download poster (PDF).
It is the aim of this poster to present the a posteriori error analysis for the Oseen equation based on divergence-conforming DG methods. We also show that the ratio of upper bound and lower bound is independent of the Reynold number of the problem. Specific numerical experiments are discussed to validate the theoretical results.
- Atef Lechiheb (Faculté de sciences de Tunis)
On the homogenization problem for stochastic elliptic equations. Download poster (PDF).
In the present work, we are interesting in problem of homogenization theory for one-dimensional elliptic equations with highly oscillatory random coefficients displaying long-range dependence. It is well known that the characterisation of the corrector to homogenization, i.e. the difference between the random solution and the homogenized solution, strongly depends on the correlation property of the underlying random process. Previous studies have shown that, when the random coefficient has short-range correlation, then the rescaled corrector converges to a stochastic integral with respect to standard Brownian motion. In this paper, we study the asymptotic behaviour of the corrector under mild conditions in presence of the long-range correlations and we prove convergence to stochastic integrals with respect to Hermite processes.
- Keith Myerscough (KU Leuven)
Parareal computation of SDEs with time-scale separation. Download poster (PDF).
Goal: Simulate slow-fast SDEs over long time, quickly
Model: Slow-fast system of SDEs, and a macroscopic model taken from the "fast" limit
Method: Parallel-in-time algorithm that iteratively improves the macroscopic result
Result: Reduction in wall clock time
Bonus: Lower variance than full microscopic model
- Przemyslaw Zielinski (KU Leuven)
A micro-macro acceleration method for the Monte Carlo simulation of SDEs. Download poster (PDF).
The purpose of this research is to develop, study and implement a micro-macro method to simulate observables of stiff SDEs. The technique exploits the separation between the fast time scale on which we compute trajectories and the slow evolution of observables, by combining short bursts of paths simulation with extrapolation of a number of macroscopic states forward in time.
In the crucial step of the algorithm, we aim to obtain, after extrapolation, a new ensemble of particles/replicas compatible with given macroscopic states. To address this inference problem, we introduce the matching operator based on the minimisation of suitable distance between probability distributions.
This research focuses on establishing the convergence of the method in the case of time-discretisation only. As a particular example, we analyse the matching based on the logarithmic entropy that also provides a convenient numerical approach.