Research
Descriptions

My research focuses on the following themes.
1. Domain
Decomposition Methods – Parallel Computing:
1.1
The
theory of domain decomposition methods: I have
developed a new machinery to study the convergence problem of both
classical and optimized domain decomposition methods.
1.2
Applications
of domain decomposition methods: I
have used classical and optimized domain decomposition methods to parallelize
the numerical resolution of many problems in various research areas.
Stochastic differential
equations and financial maths
Kinetic equations
Computational fluid
dynamics and primitive equation
Control theory
2. Kinetic Theory: I have built a nonlinear approximation theory, based on an
Adaptive Spectral Method, for Boltzmann equation. My theory is the first bridge
between Kinetic Theory and Nonlinear Wavelet Approximation Theory. With Professor
Miguel Escobedo, I have obtained the first explicit rate of the convergence to
equilibrium of the quantum Boltzmann equation in quantum physics. Moreover, using ideas from
control theory I have designed a
scheme that preserves the long time behavior of the solution of the KolmogorovFokkerPlanck
equation. I have constructed a new method, based on techniques from control theory, to study the asymptotic
behavior of kinetic equations, and to solve a conjecture on the
GoldsteinTaylor model.
Nonlinear approximation
theory for Boltzmann equation
Quantum Boltzmann
equation
Weak coercivity
inequality (inspired by Control Theory)
Structure Preserving
Scheme for the KolmogorovFokkerPlanck Equation (inspired by Control Theory)
3. Dispersive Equation: I am also interested Schrodinger equation and have started
to work on the Hardy uncertainty principle with Professor Luis Vega. I have also
started to learn about Quantum Mechanics in the book of Alberto Galindo and
Pedro Pascual following the advice of Professor Avy Soffer.