- 31/01/2017Stochastic parameterizations of deterministic dynamical systems: Theory,...
**[43:26]**1646VN4-28b1-2016-2017-02-21Stochastic 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. - 31/01/2017Ergodic Stochastic Differential Equations and Sampling: A numerical analysis...
**[44:28]**1229VN4-28b1-2016-2017-02-21-AErgodic 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 [4]. Finally if there is time we will discuss, how things ideas can be combined with the idea of Multilevel Monte Carlo [5] to produce unbiased estimates of ergodic averages without the need the of an accept-reject correction [6] and optimal computational cost. - 31/01/2017Weak convergence for semi-linear SPDEs.
**[43:09]**1215VN4-28b1-2016-2017-02-21-BJoint 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. - 31/01/2017On stochastic numerical methods for the approximative pricing of financial...
**[41:17]**1336VN4-28b1-2016-2017-02-21-CIn 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 - 01/02/2017Mean-square stability analysis of SPDE approximations.
**[38:52]**727VN4-28b1-2016-2017-02-21-DMean-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. - 01/02/2017Adaptive timestepping for S(P)DEs to control growth.
**[44:43]**1133VN4-28b1-2016-2017-02-21-EWe 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. - 01/02/2017Noise-induced transitions and mean field limits for multiscale diffusions.
**[44:20]**1119VN4-28b1-2016-2017-02-21-FIn 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. - 01/02/2017Accelerated dynamics and transition state theory.
**[45:59]**1045VN4-28b1-2016-2017-02-21-GI 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. - 01/02/2017Long-time homogenization of the wave equation.
**[42:13]**1223VN4-28b1-2016-2017-02-21-HIn 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).

The participation of PhD students, post-docs and advanced students is encouraged and the talks will be mostly accessible also to non-specialists.

Organizers: Martin J. Gander (University of Geneva), Georg Gottwald (University of Sydney), Gilles Vilmart (University of Geneva).

http://www.unige.ch/math/multiscale17/