Nonlocal deterministic and stochastic differential equations: analysis and numerics

Research Project, 2019 – 2021

An important problem in spatial statistics is to generate random fields with prescribed covariance properties. A direct approach starting from a given covariance function can be computationally expensive: sampling a random field at n spatial locations generally has a cost of O(n3), since it requires a matrix factorization of an n x n covariance matrix.A relevant class of random fields is the Gaussian Matérn fields, whose covariance functions can be described using three parameters. One advantage of Matérn fields is that they can be generated without explicitly using the covariance function, thereby reducing the computational complexity. Instead, a partial differential equation with spatial white noise as the source term is solved in the spatial domain. The order of the equation is related to the Matérn parameters and it is generally fractional. Thus, a fractional order stochastic partial differential equation must be solved numerically. Since a fractional order differential equation is nonlocal its numerical solution poses special challenges. The purpose of the project is to to device efficient and accurate numerical algorithms for approximating the solution of such equations. The methods will be supported by rigorous regularity and error analysis. They will be implemented using modern finite element software.The results will have significant impact on sciences using spatial statistical tools in modelling, for example, ecology, meteorology, and hydrology.