L3

The Helmholtz equation models waves propagating with a fixed frequency. Discretising the Helmholtz equation for high frequencies via standard finite-elements results in linear systems that are large, non-Hermitian, and indefinite. Therefore, when solving these linear systems, one uses preconditioned iterative methods. When one considers uncertainty quantification for the Helmholtz equation, one will typically need to solve many (thousands) of linear systems corresponding to different realisations of the coefficients. At face value, this will require the computation of many preconditioners, a potentially expensive task.

Therefore, we investigate how well a preconditioner for one realisation of the Helmholtz equation works as a preconditioner for another realisation. We prove that if the two realisations are 'nearby' (with a precise meaning of 'nearby'), then the preconditioner is robust (that is, preconditioned GMRES converges in a number of iterations that is independent of frequency). We also give some preliminary computational results indicating the speedup one obtains in uncertainty quantification calculations.

The focus of this talk is how to tackle huge linear least square problems via sketching, a dimensionality reduction technique from randomised numerical linear algebra. The technique allows us to project the huge problem to a smaller dimension that captures essential information of the original problem. We can then solve the projected problem directly to obtain a low accuracy solution or using the projected problem to construct a preconditioner for the original problem to obtain a high accuracy solution. I will survey the existing projection techniques and evaluate the performance of sketching for linear least square problems by comparing it to the state-of-the-art traditional solution methods. More than ten-fold speed-up has been observed in some cases.

This talk is concerned with the mathematical and numerical analysis of a steady phase change problem for non-isothermal incompressible viscous flow. The system is formulated in terms of pseudostress, strain rate and velocity for the Navier-Stokes-Brinkman equation, whereas temperature, normal heat flux on the boundary, and an auxiliary unknown are introduced for the energy conservation equation. In addition, and as one of the novelties of our approach, the symmetry of the pseudostress is imposed in an ultra-weak sense, thanks to which the usual introduction of the vorticity as an additional unknown is no longer needed. Then, for the mathematical analysis two variational formulations are proposed, namely mixed-primal and fully-mixed approaches, and the solvability of the resulting coupled formulations is established by combining fixed-point arguments, Sobolev embedding theorems and certain regularity assumptions. We then construct corresponding Galerkin discretizations based on adequate finite element spaces, and derive optimal a priori error estimates. Finally, numerical experiments in 2D and 3D illustrate the interest of this scheme and validate the theory.

The zeta-function of a manifold varies with the parameters and may be evaluated in terms of the periods. For a one parameter family of CY manifolds, the periods satisfy a single 4th order differential equation. Thus there is a straight and, it turns out, readily computable path that leads from a differential operator to a zeta-function. Especially interesting are the specialisations to singular manifolds, for which the zeta-function manifests modular behaviour. We are also able to find, from the zeta function, attractor points. These correspond to special values of the parameter for which there exists a 10D spacetime for which the 6D corresponds to a CY manifold and the 4D spacetime corresponds to an extremal supersymmetric black hole. These attractor CY manifolds are believed to have special number theoretic properties. This is joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.

Backward Stochastic Differential Equations (BSDEs) provide a systematic way to obtain Feynman-Kac formulas for linear as well as nonlinear partial differential equations (PDEs) of parabolic and elliptic type, and the numerical approximation of their solutions thus provide Monte-Carlo methods for PDEs. BSDEs are also used to describe the solution of path-dependent stochastic control problems, and they further arise in many areas of mathematical finance. 

In this talk, I will discuss the numerical approximation of BSDEs when the nonlinear driver is not Lipschitz, but instead has polynomial growth and satisfies a monotonicity condition. The time-discretization is a crucial step, as it determines whether the full numerical scheme is stable or not. Unlike for Lipschitz driver, while the implicit Bouchard-Touzi-Zhang scheme is stable, the explicit one is not and explodes in general. I will then present a number of remedies that allow to recover a stable scheme, while benefiting from the reduced computational cost of an explicit scheme. I will also discuss the issue of numerical stability and the qualitative correctness which is enjoyed by both the implicit scheme and the modified explicit schemes. Finally, I will discuss the approximation of the expectations involved in the full numerical scheme, and their analysis when using a quasi-Monte Carlo method.

By viewing a stochastic process as a random variable taking values in a path space, the support of its law describes the set of all attainable paths. In this talk, we show that the support of the law of a solution to a path-dependent stochastic differential equation is given by the image of the Cameron-Martin space under the flow of mild solutions to path-dependent ordinary differential equations, constructed by means of the vertical derivative of the diffusion coefficient. This result is based on joint work with Rama Cont and extends the Stroock-Varadhan support theorem for diffusion processes to the path-dependent case.

Abstract: The edge reinforced random walk is a self-interacting process, in which the random walker prefer visited edges with a bias proportional to the number of times the edges were visited. We will gently introduce this model and talk about some of its histories and recent progresses.

We will consider some problems on calculating  the average  angles of random polytopes. Some of them are open.

Consider the asymmetric simple exclusion process with k particles on a linear lattice of N sites. I will present results on the asymptotic of the time needed for the system to reach its equilibrium distribution starting from the worst initial configuration (also called mixing time). Two main regimes appear according to the strength of the asymmetry (in terms of k and N), and in both regimes, the system displays a cutoff phenomenon: the distance to equilibrium falls abruptly from 1 to 0. This is a joint work with Hubert Lacoin (IMPA).

In a joint-work with Andrea Collevecchio and Vladas Sidoravicius,  we study  phase transitions in the recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define the branching-ruin number of a tree, which is  a natural way to measure trees with polynomial growth and therefore provides a polynomial version of the branching number defined by Furstenberg (1970) and studied by R. Lyons (1990). We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk on this tree. We will also mention two other results where the branching-ruin number arises as critical parameter: first, in the context of random walks on heavy-tailed random conductances on trees and, second, in the case of Volkov's M-digging random walk.