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Modelisation and numerical simulations

Our research focuses on several important application domains, that we briefly describe in the following.  These applications also allow us to validate the algorithms and the software developed in the previous research topics.

Inverse problems: We focus on methods arising from time reversal techniques and their combination with classical methods in inverse problems.

Numerical methods for wave propagation in multi-scale media: The main concern of this item consists in devising fast and accurate numerical methods for the simulation of electromagnetic waves in media whose geometry and material characteristics are submitted to small scale perturbations in localized regions of the computational domain.

Data analysis in astrophysics: We focus on computationally intensive numerical algorithms arising in the data analysis of current and forthcoming Cosmic Microwave Background (CMB) experiments in astrophysics.  While this application does not involve a PDE, its most complex and time consuming step is solving a generalized least squares problem, which is at the core of our research.

Molecular simulations: Since the beginning of the ERC Synergy Grant EMC2, we started addressing several important limitations of state of the art molecular simulation. In particular, the simulation of very large molecular systems, or smaller systems in which electrons interact strongly with each other, remains out of reach today.

Most of our simulations can be found in the gallery of FreeFEM.

Inverse problems

In this context we focus on the time-reversed absorbing conditions (TRAC) method which combines time reversal techniques and absorbing boundary conditions. Here we present a three-step strategy to solve inverse scattering problems when the time signature of the source is unknown. The proposed strategy combines three recent techniques: (i) wave splitting to retrieve the incident and the scattered wavefields, (ii) time-reversed absorbing conditions for redatuming the data inside the computational domain, (iii) adaptive eigenspace inversion (AEI) to solve the inverse problem. Then, we studied redatuming data which consists in virtually moving the sensors from the original acquisition location to an arbitrary position. This is an essential tool for target oriented inversion. An exact redatuming method which has the peculiarity to be robust with respect to noise was proposed. All simulations were written in FreeFEM.

Wave propagation in multi-scale media

Regarding this subject, we worked on the analysis of periodic homogenization in situation involving an interface between two rapidly oscillating media. By matched asymptotics with homogenization techniques we could derive high order asymptotic expansion and obtain improved transmission conditions for the interface between the two homogenized media. We made use of this approach to devise and test more efficient numerical methods for Helmholtz posed in computational domains with this type of feature.

In the context of seismic imaging using Frequency-domain full-waveform inversion (FWI), new “wide angle” acquisition techniques allow efficient reconstruction of reliable subsurface models with few frequencies. In the frequency domain, FWI relies on successive solutions of large linear systems (several hundreds of millions of unknowns) resulting from the discretization of the Helmholtz equation (acoustic) or elastic wave equations (elastodynamics). In this context, domain decomposition preconditioners are strong candidates for the efficient solution of the forward problems. Two-level DD preconditioners were designed and tested for 3D acoustic simulations on several benchmarks such as the Overthrust model. At frequency f = 20Hz, Finite element discretization with P3 elements on an unstructured tetrahedral mesh adapted to the local wavelength yields a linear system of 2.3 billion unknowns, which can be solved in 37 seconds on 16,960 cores using a two-level DD preconditioner (see here and here) .

Data analysis for astrophysics applications

We focus on Cosmic Microwave Background data analysis. This application involves solving a generalized least squares problem, using Toeplitz algebra, and computing a spherical harmonic transform. In one of the achievements, we consider component separation, which is one of the key stages of any modern cosmic microwave background data analysis pipeline. It involves a series of sequential solutions of linear systems with similar but not identical system matrices, derived for different data models of the same data set. We propose and study efficient solvers adapted to solving time-domain-based component separation systems and their sequences, and which are capable of capitalizing on information derived from the previous solutions.

Molecular simulations

Recently, we address some of the challenges arising in molecular simulations, in particular large molecular systems or strongly correlated systems. In this context the algorithms developed in the previous axis for tensors are tested on molecular systems of interest.

Finally, in the context of the development of efficient algorithms for solving multi-scale problems in plasma physics applications, we propose a new strategy for the parareal algorithm. We solve highly oscillatory Vlasov and Vlasov-Poisson equations with the parareal algorithm, by using for the coarse solver reduced models, obtained from two-scale asymptotic expansions. We demonstrate the accuracy and the efficiency of the strategy in numerical experiments of short time and long time simulations of charged particles submitted to a large magnetic field.