## Data Assimilation and Reduced Modeling for High Dimensional Problems

##### CIRM, Luminy, France

##### July 19-August 27, 2021

#### Computation of the auto-diffusion coefficient of a cross-diffusion system with tensor methods

**Supervisors:**V. Ehrlacher (Ecole des Ponts ParisTech and Inria)

**Students:**Open to 1 student

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Project Description:
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At the microscopic level, diffusion phenomena in the buld of a solid composed of different chemical species can be
modeled using lattice-based stochastic hopping models. Hydrodynamic limits (limits as the number of particles tend to
infinity) of such models read as nonlinear diffusion systems, called cross-diffusion systems, where the diffusion
coefficients depend on the local concentrations of each chemical species. Computing these diffusion coefficients
requires to compute, for a given level of concentration, the value of the so-called auto-diffusion coefficient of the
system. This auto-diffusion coefficient can be written equivalently either as the long-time limit of the expectation of
a quantity of interest depending on Markov choin model, or as the solution of a very high-dimensional deterministic
model. The aim of this project is to compare standard methods used for the computation of this coefficient, which use
the first formulation, with tensor methods to solve the high-dimensional optimisation problem.