Ron DeVore
Estimating Parameters from Measurements for a Family of Elliptic Parametric PDEs
We consider a family of parametric elliptic pdes −div(a∇u)) = f on a domain D with zero boundary conditions and diffusion coefficient a satisfying the ellipticity condition r ≤ a ≤ R on D with r > 0, If a varies continuously over a parameter set, then the solution ua describes a solution manifold. We study the question of whether we can determine a when ua is known and the related question of how well we can approximate a when we have partial information of ua given by data observations. In the case that a is uniquely determined by ua, we study the related question of how smooth is the inverse map ua → a. Concerning the recovery of a from partial measurements of the state, we discuss possible strategies and their cost in terms of the entropy of the solution manifold.