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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.

Simulation of compositional multiphase Darcy flow in heterogeneous porous media with different type of applications: simulation of reservoir models, simulation of basin models, simulation
of geological CO2 underground storage.

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.

Numerical solution of Maxwell’s Equations for a full medical imaging system

This work is carried out within the ANR project MEDIMAX and aims at developing an efficient and robust inversion tool associated with the electromagnetic forward problem. The inversion algorithm relies on a general parallel open source simulation code for the direct problem, based on the high-level integrated development environment FreeFem++ coupled with the HPDDM library for high-performance domain decomposition methods. The code can be used for modeling the scattering of arbitrary electromagnetic waves in highly heterogeneous media, over a wide frequency range.

The medical application considered in the project is the microwave tomographic imaging of cerebrovascular accidents (CVAs) or brain strokes. This work has been carried out in collaboration with EMTensor GmbH, Vienna (Austria), an electromagnetic medical imaging company, and has been awarded the Bull-Joseph Fourier 1st Prize 2015. It demonstrates on synthetic data the feasibility of a new microwave imaging technique allowing for the characterization of CVAs, beginning with the very first instance of patient care in an ambulance and extending to continuous patient monitoring during hospitalization. CVAs are classified in two categories: ischemic strokes (80% of cases), resulting from the occlusion of a cerebral artery, and hemorrhagic strokes (20% of cases), provoked by a bleeding vessel. From a medical point of view, the fast detection and characterization of CVAs are crucial for patient survival.

EMTensor’s experimental system consists of an electromagnetic reverberating chamber surrounded by five layers of 32 antennas each, able to work alternately as transmitters or receivers at a fixed frequency, typically 1 Ghz. The inversion algorithm reconstructs an image of the head from the measurements (the 160×160 scattering matrix), corresponding to a map of the dielectric properties of the different head tissues, which are well differentiated at this frequency range. The inversion is done using a L-BFGS algorithm, which involves repeatedly solving the direct problem modeled by Maxwell’s equations in the chamber, with a complex and highly heterogeneous medium.

Results shown below are obtained using synthetic data corresponding to a very precise numerical brain model from EMTensor with a simulated hemorrhagic stroke. We add 10% Gaussian white noise to the data. We can create parallelism by reconstructing the image in successive slices and solving five smaller inverse problems independently, each problem corresponding to one ring of antennas. This, together with the parallelism coming from the domain decomposition method as well as solving for multiple right-hand sides, allows us to reconstruct an image corresponding to one ring in less than two minutes (94 seconds) using 4096 computing cores.


Imaginary part of the reconstructed permittivity in the whole chamber (right) from synthetic data created from a numerical brain model with stroke (left) with 10% white Gaussian noise.


Left: real (top row) and imaginary (bottom row) part of the reconstructed permittivity during 30 BFGS iterations (right column) using synthetic data created from a numerical brain model with stroke (left column) with 10% white Gaussian noise.

Above: Total time required to solve the inverse problem for the top ring (30 BFGS iterations) and obtain the corresponding reconstructed images in Left.


Click the picture to download the movie, showing the decomposition into 16 subdomains, the jumps in the material coefficients (blue = steel, red = rubber) and the amplitude of the deformation field.