N o d ' ordre : MÉMOIRE Présenté pour obtenir L ' HABILITATION

The rst part of this thesis concerns the formulation of numerical methods that are local in time for the solution of equations with memory. The main idea is that the solution will be updated in the Fourier domain in order to avoid evaluating time convolution integrals that have memory. This work was made possible by the development of a good quadrature[31] of the Fourier integral where a small number of points in the Fourier variable were su cient for a good resolution of the problem in the physical space over a large time interval. First, we developed a numerical method to simulate di usion in unbounded domains with sources and applied it to the modeling of crystal growth using the phase eld model. Then, in order to extend this approach to boundary value problems, we addressed the issue of evaluating the single and double layer potentials on the boundary. Finally, we generalized the idea of replacing time convolution integrals by an e cient quadrature in the transform domain to fractional integrals and derivatives for general fractional orders and obtained a rigorous bound on the quadrature error. Then we applied this approach to a fractional wave equation. The second part of the thesis concerns the speci c application of di usion magnetic resonance imaging (dMRI) in the brain. The e ect on the MRI signal of the water proton magnetization in biological tissue in the presence of magnetic eld gradient pulses can be modeled by a microscopic multiple compartment Bloch-Torrey partial di erential equation (PDE). This PDE can be best understood as imparting a spatially dependent frequency to di using particles in a heterogeneous medium. The dMRI signal is the integral of the solution of this PDE at the echo time. First, we numerically solved this PDE by coupling a standard Cartesian spatial discretization with an adaptive time discretization and studied the di usion characteristics of a tissue model of the brain gray matter made up of cylindrical and spherical cells embedded in the extra-cellular space. Next we formulated a new macroscopic ODE model for the dMRI signal by mathematical homogenization. Then we showed by numerical simulations that this ODE model gives a good approximation of the dMRI signal of the full PDE model at relatively long but still physically realistic di usion times relevant to dMRI in the brain. Finally, I will describe some future research directions in dMRI. The rst is the experimental validation of the PDE model by imaging the ganglia (neuronal network) of the Aplysia (giant sea slug), to be conducted at the MRI center Neurospin. The second is the inclusion of blood ow in the bran micro-vessels in a new PDE model. The third is the formulation of a di erent ODE model valid at shorter di usion times or in the presence of larger cells.