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

TL;DR

This paper develops a gradient-based method for estimating uncertain diffusion coefficients in elliptic PDEs from noisy data, combining spectral approximation and stochastic optimization.

Contribution

It introduces a novel approach that combines spectral approximation, convergence analysis, and a stochastic augmented Lagrangian method for parameter estimation in elliptic PDEs.

Findings

Proves existence of minimizers and convergence of finite noise solutions.

Develops a stochastic augmented Lagrangian algorithm for numerical estimation.

Demonstrates effectiveness through three numerical examples.

Abstract

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.