Tuesday September 22th10:00-10:30 Accueil 10:30-11:15 N. Touzi 11:15-12:00 L. Goudenege 14:00-14:45 B. Jourdain 14:45-15:30 G. Stoltz 15:30-16:00 Pause 16:00-16:45 A. Lang 16:45-17h30 W. Stannat ----------------------------------Wednesday September 23th09:30-10:15 A. Abdulle 10:15-11:00 C. Powell 11:00-12:00 Pause et posters 14:00-14:45 M. Bossy 14:45-15:30 C. Prieur

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**Assyr Abdulle**

“Numerical methods for stiff and ergodic stochastic differential equations”

**Mireille Bossy**

“Simulation and rate of convergence of conditional McKean-Vlasov kinetic process”

(joint work with Laurent Violeau)

**Ludovic Goudenege**

“Numerical simulation of rare events in metastable dynamics”

Key words : “SPDEs” and “stochastic numerical simulations”

**Benjamin Jourdain**

“Multitype sticky particles and diagonal hyperbolic systems of PDEs”

(joint work with J. Reygner)

In dimension one, under the sticky dynamics, particles move with constant velocity between collisions and then stick together with conservation of the mass and momentum.

This dynamics can be seen as the small noise limit of systems of diffusions interacting through their rank in the drift. According to Brenier and Grenier 1998, the large scale behaviour of this dynamics is given by the entropy solution to a scalar conservation law.

We introduce a multitype version of the sticky particles dynamics where each particle has a type, only sticks with particles of the same type and undergoes a velocity change when colliding a particle with another type. Under a uniform strict hyperbolicity assumption saying that the ranges of velocities for each type are disjoint intervals, we prove that the large scale behaviour of this dynamics is given by weak solutions to a diagonal hyperbolic system of PDEs. We then derive a Lp stability estimate on the particle system uniform in the number of particles. This allows to construct a nonlinear semigroup solving the system in the sense of Bianchini and Bressan 2005 and stable in Wasserstein distances of all orders.

We analyse the L1 error of this approximation procedure, by splitting it into the discretisation error of the initial data and the non-entropicity error induced by the evolution of the particle system. We prove that the error at time t is bounded from above by a term of order (1+t)/n, where n denotes the number of particles, and give an example showing that this rate is optimal. We last analyse the additional error introduced when replacing the multitype sticky particle dynamics by an iterative scheme based on the typewise sticky particle dynamics, and illustrate the convergence of this scheme by numerical simulations.

**Annika Lang**

“Stochastic and random partial differential equations: a shared simulation problem”

**Catherine Powell**

“Fast and Low Rank Solvers for Stochastic Galerkin Approximations”

We discuss two pieces of recent work (see [1] and [2]) that lead to significant computational savings in the numerical solution of the discrete systems associated with stochastic Galerkin finite element (SGFEM) approximation of PDEs with random coefficients. It is well known that SGFEM discretisations of PDEs with stochastically nonlinear coefficients (eg PC expansions) lead to linear systems with rather horrible block dense matrices. In contrast, SGFEM discretisations of PDEs with stochastically linear coefficients (eg KL expansions) lead to nice block sparse matrices. These are cheaper to manipulate and precondition in the framework of standard Krylov subspace iteration. For many PDE problems, especially those with stochastically nonlinear coefficients, it remains very challenging to design cheap solvers for SGFEM linear systems.

In the first part of the talk, we will discuss mixed formulations of second-order elliptic problems, where the diffusion coefficient is the exponential of a random field, and the priority is to approximate the flux. After reformulating the PDE as a first-order system in which the logarithm of the diffusion coefficient appears on the left-hand side, we obtain a new problem which is stochastically linear, leading to block sparse matrices for which efficient solvers can more easily be designed. We discuss the pros and cons of this reformulation strategy.

In the second part of the talk we replace the usual Kronecker formulation of SGFEM linear systems with equivalent multi-term matrix equations and discuss a new iterative solver that builds up a reduced approximation space on-the-fly from rational Krylov spaces. Numerical results demonstrate that huge computational savings can be made, compared to solving the usual Kronecker formulation with standard Krylov iteration schemes when the number of random variables is large.

References

[1] E. Ullmann and C.E. Powell, Solving Log-transformed Random Diffusion Problems by Stochastic Galerkin Mixed Finite Element Methods. To appear in SIAM/ASA J. Uncertainty Quantification (2015).

[2] C.E. Powell, D. Silvester and V. Simoncini, An Efficient Low Rank Iterative Solver for Stochastic Galerkin Matrix Equations. Submitted, (2015).

Key words: Stochastic Galerkin approximation, PDEs with random coefficients, mixed finite elements, linear algebra, low rank algorithms.

**Clémentine Prieur**

“Goal-oriented error estimation for the reduced basis method, application to sensitivity analysis”

**Wilhelm Stannat**

” Numerical approximation of stochastic nerve axon equations”

Keywords: stochastic nerve axon equations, continuous time-approximation, strong approximation error, lattice approximation

**Gabriel Stoltz**

“Error estimates for transport coefficients in molecular dynamics”

In this talk, I will present error estimates for transport coefficients, which can be obtained either by the linear response of appropriately perturbed stochastic dynamics, or, equivalently, through the time integration of correlation functions. I will consider Langevin dynamics, numerically integrated with splitting schemes; and overdamped Langevin dynamics, possibly corrected by a Metropolis/Hastings procedure in order to remove the bias on the invariant measure.