Research

The Uncertainty Quantication and its reduction in the multi-scale context are extremely challenging while of paramount importance in applications. Asymptotic analysis and numerical techniques pertaining thereto aim to avoid the formidable cost induced by oscillations while capturing the slow dynamics (or more generally the averaged behaviour) and allows to keep under control uncertainties in the model. The main scope of this proposal is thus (i) the development of robust (UA) numerical schemes for PDEs where largely different scales co-exist and (ii) their extension to the situation where uncertainties exist. The expected outcome is twofold: on the one hand, the design of combined UA-UQ schemes, robust with respect to both stiness of the problem and uncertainties in the model, and on the other hand, a signicant gain of computational cost in applications to plasmas (Vlasov equations) and graphene (quantum models).

Context

A precise numerical simulation of non-linear partial differential equations (PDEs), as those originating from plasma physics or quantum mechanics, is undoubtedly the key to a deeper understanding of many physical phenomena. However, producing accurate solutions of kinetic or Schrödinger equations (modelling gas or quantum dynamics) may be rendered extremely challenging by the presence of multiple scales and experimental uncertainties and thus requires the synergy of expertises, in asymptotic and numerical analysis, and UQ methods.
Conventional methods fail to capture multi-scale phenomena owing to numerical instabilities. The smaller ε is, the higher the computational cost (indeed growing like 1/ε) gets. A possible remedy consists in deriving the asymptotic model in the limit when ε tends to zero, which is vastly easier to solve numerically. The Asymptotic Preserving (AP) paradigm then consists in designing schemes that only use the original equations and yet can capture this limit model inherently and with no additional computational cost. Although very successfull, this approach still entails limitations. AP schemes have been initially developed for dissipative systems and turn out to be inaccurate for intermediate values of ε. As an answer, UA schemes have emerged that are able to handle the whole range of ε-values with the same accuracy and for the same cost. As there is no free lunch, a pre-requisite to the development of UA schemes is to obtain sufficiently many terms of the asymptotic model (in ε, ε², …) and/or to identify a change of variable that removes the potential stiffness of the problem. This methodology has proved to be promising and is the one we wish to extend considerably further in this proposal by considering both time and space oscillations.
UQ has drawn a lot of attention recently and many numerical approaches have been devel-oped in order to control the propagation of data pollution. However, existing methods do not cope with highly-oscillating models. It is our belief that averaging techniques could be used to remedy the dramatic propagation of uncertainties. Accordingly, we aim at obtaining numerical UQ schemes whose accuracy does not depend on ε.

Objectives (for the three years)

Our ambition is to lay the foundations for robust numerical methods: both previously mentioned approaches will thus be pursued in the current project with the aim of designing UA-UQ discretisations in presence of space and time oscillations and uncertainties. To sum up, our ambition is here

• to develop high-order asymptotic models (or sometimes more simply appropriate reformu-lations) for PDEs with high-oscillations, generalizing and cross-fertilizing ideas developed by the two groups;
• to develop new UA schemes based on the acquired knowledge of the asymptotic models; the final objective is to speed up the computations but also to design more versatile methods;
• to redesign UQ methods so as to incorporate the information from asymptotic models and combine them with UA techniques.

The resulting methods will be implemented and applied to kinetic and quantum equations in highly-oscillatory regimes. The project will be articulated around three consecutive stages (of one year each):
Y1 derivation of high-order asymptotic models for kinetic and quantum equations oscillating in both time and space;
Y2 construction of new numerical schemes and UQ techniques in the oscillatory context,
according to the UA paradigm;
Y3 implementation of the new methods as free software. The third stage is concerned with the practical implementations and should be regarded as the main final outcome, together with publications in top-grade journals (JCP, FOCM, SIAM…).

Scientific progress (year 2018)

This first year has been devoted to the development of new asymptotic expansions in situations where they were considered as out of reach or had just not been studied. More specifically, we have addressed the situation of highly-oscillatory problems in which the frequency of oscillations vanishes at specific instants : at these instants, the solution ceases to oscillate. Surprisingly, intuition is somehow misleading, as in contrast to what might be expected at first glance, this makes the problem more difficult and the derivation of asymptotic expansions rather tricky : in the analysis of highly-oscillatory evolution problems, it is indeed commonly assumed that a single frequency is present and that it is either constant or, at least, bounded from below by a strictly positive constant uniformly in time. Allowing for the possibility that the frequency actually depends on time and vanishes at some instants introduces additional difficulties from both  the asymptotic analysis and numerical simulation points of view. Our paper [3] is a first step towards  the resolution of these difficulties. In particular, we show that it is still possible in this situation to  infer the asymptotic behaviour of the solution at the price of more intricate computations and we derive a second order uniformly accurate numerical method.

In parallel, we have introduced a new methodology to design uniformly accurate methods for oscillatory evolution equations where the frequency is constant. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly-oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime. The main advantage of this approach, as compared to two-scale expansions for instance, is (i) that it is implements a specific treatment of the oscillations but keeps the underlying time-step method explicit and,  (ii) it does not increase the dimension of the problem under consideration. The corresponding paper is accepted for publication in FOCM [2].

A first application of this result has been considered for a kinetic graphene model in one space dimension [4] : we have proposed a micro-macro decomposition based numerical approach, which reduces the computational dimension of the nonlinear geometric optics method based numerical method for highly oscillatory transport equation developed. The method solves the highly oscillatory model in the original coordinate, yet can capture numerically the oscillatory space-time quantum solution pointwisely even without numerically resolving the frequency. We prove that the underlying micro-macro equations have smooth (up to certain order of derivatives) solutions with respect to the frequency, and then prove the uniform accuracy of the numerical discretization for a scalar model equation exhibiting the same oscillatory behavior. Numerical experiments verify the theory. During our visit at the University of Wisconsin, we have then envisaged the extension to the 3D-case. However, additional difficulties have arisen and we have oriented our collaboration towards a more tractable case, namely equations arising from the modelisation of surface hoping [6]. This problem is pertaining to the more general class of problems known as avoided crossing (sometimes called intended crossing or anticrossing, i.e. the phenomenon where two eigenvalues of an Hermitian matrix representing a quantum observable and depending on N continuous real parameters cannot become equal except on a manifold of N-2 dimensions). Our study, though preliminary, is of a fundamental interest in quantum physics or chemistry. It is also related to the Landau-Zener approximation which models the transition dynamics of a 2-level quantum mechanical system, and which we intend to analyse in the near future.  Finally, we have developed a generalized polynomial chaos (gPC) based stochastic Galerkin (SG) methods for a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain uncertainties, particularly in coefficients that correspond to the potentials of the molecular system. We first focused on a highly oscillatory scalar model with random uncertainty. Our method is built upon the nonlin-ear geometrical optics (NGO) based method for numerical approximations of deterministic equations, which can obtain accurate pointwise solution even without numerically resolving spatially and temporally the oscillations. With the random uncertainty, we have shown that such a method has oscillatory higher order derivatives in the random space, thus requires a frequency dependent discretization in the random space. We have then modified this method by introducing a new “time” variable based on the phase, which is shown to be non-oscillatory in the random space, based on which we developed a gPC-SG method that can capture oscillations with the frequency-independent time step, mesh size as well as the degree of polynomial chaos. A similar approach is then extended to a semiclassical  model system with a similar numerical conclusion. Various numerical examples attest that these methods indeed capture accurately the solution statistics pointwisely even though none of the numerical parameters resolve the high frequencies of the solution [5].

Scientific progress (year 2019)