In this proceeding, the author presents a very nice review on the use
of tensor numerical methods
for the approximation of the solution of high dimensional
partial differential equations, with a
special focus on tensor formats,
like Tucker, Tensor-Train formats, and especially the quantized
version
of these formats.
In the first section of the paper, the author introduces the
main different
tensor formats that are the most widely used in the literature
for the approximation of functions and
operators.
The author puts an
emphasis on the quantized version of these formats and the approximation
properties they possess, which were proved by the author and coworkers in
several works.
These approximation properties are also illustrated on numerical
examples.
In the second section,
the author presents how these quantized
tensor methods can be used to solve high-dimensional PDEs.
Two main
examples of applications are provided: the first one deals with the approximation
of the solution of the Hartree-Fock equation in electronic structure
calculations, where one of the main numerical issue lies in the approximation
of the electron integrals;
the second one is concerned with the approximation
of time-dependent parabolic equations, and especially here
the chemical
master equation.