Tony Lelievre

Frederic Legoll, Tony Lelievre, Giovanni Samaey

We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale-in-time systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate slow macroscopic model), which is iteratively corrected using a fine-scale propagator (accurately simulating the full microscopic dynamics). This correction is done in parallel over many subintervals, thereby reducing the wall-clock time needed to obtain the solution, compared to the integration of the full microscopic model. We provide a numerical analysis of the algorithm for a prototypical example of a micro-macro model, namely singularly perturbed ordinary differential equations. We show that the computed solution converges to the full microscopic solution (when the parareal iterations proceed) only if special care is taken during the coupling of the microscopic and macroscopic levels of description. The convergence rate depends on the modeling error of the approximate macroscopic model. We illustrate these results with numerical experiments.

Eric Cances, Virginie Ehrlacher, Tony Lelievre

In this article, we present a family of numerical approaches to solve high-dimensional linear non-symmetric problems. The principle of these methods is to approximate a function which depends on a large number of variates by a sum of tensor product functions, each term of which is iteratively computed via a greedy algorithm. There exists a good theoretical framework for these methods in the case of (linear and nonlinear) symmetric elliptic problems. However, the convergence results are not valid any more as soon as the problems considered are not symmetric. We present here a review of the main algorithms proposed in the literature to circumvent this difficulty, together with some new approaches. The theoretical convergence results and the practical implementation of these algorithms are discussed. Their behaviors are illustrated through some numerical examples.

Sebastien Boyaval, Tony Lelievre

In this work, we develop a reduced-basis approach for the efficient computation of parametrized expected values, for a large number of parameter values, using the control variate method to reduce the variance. Two algorithms are proposed to compute online, through a cheap reduced-basis approximation, the control variates for the computation of a large number of expectations of a functional of a parametrized Ito stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offline, following a so-called greedy procedure, which minimizes the variance among a trial sample of the output parametrized expectations. Numerical results in situations relevant to practical applications (calibration of volatility in option pricing, and parameter-driven evolution of a vector field following a Langevin equation from kinetic theory) illustrate the efficiency of the method.

Tony Lelievre, Mohamed El Makrini, Benjamin Jourdain

The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a {\bf fixed} number of random walkers evolving according to a Stochastic Differential Equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove the convergence of the method for a fixed number of reconfigurations when the number of walkers tends to $+\infty$ while the timestep tends to 0. We confirm our theoretical rates of convergence by numerical experiments. Various resampling algorithms are investigated, both theoretically and numerically

Dan Hu, Tony Lelievre

This short note presents the derivation of a new {\it a priori} estimate for the Oldroyd-B model. Such an estimate may provide useful information when investigating the long-time behaviour of macro-macro models, and the stability of numerical schemes. We show how this estimate can be used as a guideline to derive new estimates for other macroscopic models, like the FENE-P model.