We describe an approach to solving PDEs in which the solution is found by minimizing a squared RKHS norm subject to a penalty, or constraint, defined via the PDE. This links to the subject of Gaussian processes subject to pointwise nonlinear constraints. We introduce a framework for the methodology, establish convergence of the resulting numerical method, and describe numerical results. Forward problems for a nonlinear elliptic PDE, Burgers' equation and a regularized Eikonal equation are presented; and the method is extended to inverse problems and applied to the problem of permeability identification in the Darcy model of flow in a porous medium.
This is in collaboration with Yifan Chen, Bamdad Hosseini and Houman Owhadi.
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