Pierre Sochala

Pierre Sochala, Alexandre Ern, Serge Piperno

Robust and accurate schemes are designed to simulate the coupling between subsurface and overland flows. The coupling conditions at the interface enforce the continuity of both the normal flux and the pressure. Richards' equation governing the subsurface flow is discretized using a Backward Differentiation Formula and a symmetric interior penalty Discontinuous Galerkin method. The kinematic wave equation governing the overland flow is discretized using a Godunov scheme. Both schemes individually are mass conservative and can be used within single-step or multi-step coupling algorithms that ensure overall mass conservation owing to a specific design of the interface fluxes in the multi-step case. Numerical results are presented to illustrate the performances of the proposed algorithms.

Nadège Polette, Olivier Le Maître, Pierre Sochala et al.

This paper addresses the challenge of dimension reduction (DR) in Bayesian inference of high-resolution two-or three-dimensional fields, where a priori parametrizations require a large number of terms. The underlying idea is common to state-of-the-art methods in which the parameter space is decomposed into two subspaces, one informed by the likelihood and one constrained by the prior. DR techniques generally use gradient information from the log-likelihood to derive the corresponding subspaces. However, the gradient may be unavailable or expensive to compute accurately, for instance in the case of simulation-based inference. Inspired by approaches based on likelihood-informed subspaces, we develop a new DR method tailored for settings where gradient computation is not feasible. More specifically, we propose a gradient-free indicator for determining whether a direction is informed by the data. This indicator is derived from the posterior-to-prior covariance ratio introduced in Spantini et al. (2015). We show that, in the linear Gaussian case, this indicator combined with an approximate likelihood leads to a better posterior approximation. The method is then extended to nonlinear cases, and strategies to approximate the posterior covariance are detailed. We demonstrate the effectiveness of this DR through two high-dimensional inference problems arising from groundwater and atmospheric applications.

Michele Botti, Daniele Di Pietro, Pierre Sochala

In this work we propose and analyze a novel Hybrid High-Order discretization of a class of (linear and) nonlinear elasticity models in the small deformation regime which are of common use in solid mechanics. The proposed method is valid in two and three space dimensions, it supports general meshes including polyhedral elements and nonmatching interfaces, enables arbitrary approximation order, and the resolution cost can be reduced by statically condensing a large subset of the unknowns for linearized versions of the problem. Additionally, the method satisfies a local principle of virtual work inside each mesh element, with interface tractions that obey the law of action and reaction. A complete analysis covering very general stress-strain laws is carried out, and optimal error estimates are proved. Extensive numerical validation on model test problems is also provided on two types of nonlinear models.