My research focuses on the following themes.
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
Computational fluid dynamics and primitive equation
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 Kolmogorov-Fokker-Planck 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 Goldstein-Taylor model.
Nonlinear approximation theory for Boltzmann equation
Quantum Boltzmann equation
Weak coercivity inequality (inspired by Control Theory)
Structure Preserving Scheme for the Kolmogorov-Fokker-Planck 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.