Gradient-Based Estimation of Uncertain Parameters for Elliptic Partial Differential Equations

This paper addresses the estimation of uncertain distributed diffusion coefficients in elliptic systems based on noisy measurements of the model output. We formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers as well as first order necessary conditions. A spectral approximation of the uncertain observations allows us to estimate the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called finite noise problem in the space of functions with bounded mixed derivatives. We prove convergence of finite noise minimizers to the appropriate infinite dimensional ones, and devise a stochastic augmented Lagrangian method for locating these numerically. Lastly, we illustrate our method with three numerical examples

On a level-set method for ill-posed problems with piecewise non-constant coefficients

We investigate a level-set type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem we propose a Tikhonov-type regularization approach coupled with a level set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability of the regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level set method in some interesting inverse problems arising in elliptic PDE models. Keywords: Level Set Methods, Regularization, Ill-Posed Problems, Piecewise Non-Constant CoefficientsComment: Accepte

Identification of Discontinuous Coefficients in Elliptic Problems Using Total Variation Regularization

. We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its well-established ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for solving the output-least-squares inverse problem. In addition to the basic output-least-squares formulation, we introduce two new techniques to handle large observation errors. First, we use a filtering step to remove as much of the observation error as possible. Second, we introduce two extensions of the output-least-squares model; one model employs observations of the gradient of the state variable while the other uses the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover discontinuous coefficients even under large observation errors. 1. Introduction. Consider the partial differential equation ae \Gammar \Delta (q(x)ru) =..