Gradient-based estimation of uncertain parameters for elliptic partial differential equations

This paper discusses the estimation of uncertain distributed diffusion coefficients in elliptic systems based on noisy measurements of the model output. We treat the parameter identification problem as a variational problem over the appropriate stochastic Sobolev spaces and show that minimizers exist and satisfy a saddle point condition. Although a lack of regularity precludes the direct use of gradient-based optimization techniques, a spectral approximation of the observation field allows us to estimate the original problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called finite noise problem, which lends itself readily to more traditional optimization approaches. We prove that the finite noise minimizers converge to the appropriate infinite dimensional ones, and devise and analyze a stochastic augmented Lagrangian method for locating these numerically. We also discuss the numerical discretization of the finite noise problem, using sparse grid hierarchical finite elements, and present three numerical examples to illustrate our method.