Title: Approximating partial differential equations with missing input data
Abstract
We consider the problem of numerically approximating the solution of partial differential equations for which not all the input data are known. A situation of particular interest is when the boundary conditions are unknown, or only partially known. To alleviate this lack of knowledge, we assume to be given linear measurements of the solution. In the context of the Poisson problem with unknown boundary condition, a near optimal recovery algorithm based on the approximation of the Riesz representers of the measurement functionals in some Hilbert spaces is proposed [Binev et al. 2024]. Inherent to this
algorithm is the computation of H^s , s > 1/2, inner products on the boundary of the computational domain. In this work, we borrow techniques used in the analysis of fractional diffusion problems to design and analyze a fully practical near optimal algorithm not relying on the challenging computation of H^s inner-products. We start with the Poisson problem and then extend the idea to the Stokes equations.
