Section: New Results

Miscellaneous

Participants : Damiano Lombardi, Irene Vignon Clementel.

In [27] an adaptive tensor method is developed to build a parsimonious discretization for the kinetic equations, starting from separated, arbitrary and a priori chosen discretizations for the space and the velocity variables. The method automatically adapts the rank of the decomposition in order to ensure that a criterion on the residual of the equations is satisfied, and the proof of the convergence is provided. The method is tested on the Vlasov-Poisson equation but can be extended to other kinetic equations and to systems in which the domain is the cartesian product of separated domains.

In [42] an a posteriori error estimator for hermitian positive eigenvalue problem is proposed. This estimator, which is based on a residual formulation, is constructed by shifting the operators in such a way that the error between the exact eigenvalues and the approximated ones can be estimated efficiently. It is conditionally certified and sharp.

Diffusion-weighted magnetic resonance imaging (DWI) is a key non-invasive imaging technique for cancer diagnosis and tumor treatment assessment; yet its relation to the underlying tissue structure is not clear. In [36], in order to link low-resolution but non-invasive DWI data with high resolution (invasive) histological information, we developed an image processing and analysis chain, which was used to study the correlation between the DWI diffusion coefficient and tumor cellularity from serial histological slides of a resected non-small cell lung cancer tumor.