Joint CAM/PDE Seminar

Speaker: Joshua Siktar

Affiliation: Texas A&M University

Title: Pointwise error estimates for optimal recovery of solutions to Poisson's Equation

Abstract: We consider an optimal recovery problem for the Poisson problem when the boundary data is unknown but belongs to a model class. A finite number of measurements of the solution to recover are provided to alleviate the lack of information. This presentation focuses on the case where measurements consist of point evaluations on the interior of the computational domain. A key ingredient in the algorithm for numerically computing an optimal recovery is the finite element approximation of the Riesz representers associated with the point evaluation at the measurement locations. Our main result is a pointwise error estimate for the Riesz representers, and we apply this result to obtain sharper estimates measuring the performance of the recovery algorithm in H^1, L-infinity, and for quantities of interest. Numerical results confirm the theoretical error rates in the mesh parameter. Finally, related problems and the discussion of this project within a broader research program will be discussed.