I am a Postdoctoral Research Associate in the Numerical Analysis of Nonlocal PDEs in the group of José A. Carrillo at the University of Oxford. My research focus is on numerical analysis and the development of computational tools for integral equations, PDEs and fractional differential equations with applications in the natural sciences. I am also more generally interested in computational orthogonal polynomials and their applications in the development of multivariate spectral methods.
Orthogonal polynomials can be used to accurately and efficiently approximate functions.
They further enable the construction of spectral methods for solving integral and differential equations as well as quadrature methods.
Equilibrium measure problems arise in the continuous mean-field limit of attractive-repulsive particle systems and have diverse applications ranging from swarming behavior in animals to self-assembly properties of particle systems.
As power law interactions involve Riesz potentials, this special case also relates to fractional differential equations.
By using orthogonal polynomial approximation bases, one can turn very general integral equations into banded sparse linear systems which can be solved with high efficiency and exceptional convergence properties.
We used this to develop solvers for linear and nonlinear integral and integro-differential equations which have applications in the natural sciences and finance.
Strang splitting methods have been successfully used for the numerical treatment of non-relativistic magnetic Schrödinger equations in their appropriate 3D context.
I have worked towards extending these results to also work for relativistic equations which account for quantum mechanical spin.