Robust Parameter Inversion Using Stochastic Estimates

Abstract

For parameter inversion problems governed by systems of partial differential equations, such as those arising in Diffuse Optical Tomography (DOT), even the cost of repeated objective function evaluation can be overwhelming. Despite the linear (in the state variable) nature of the DOT problem, the nonlinear parameter inversion process is dominated by the computational burden of solving a large linear system for each source and frequency. To compute the Jacobian for use in Newton-type methods, an adjoint solve is required for each detector and frequency. When a three-dimensional tomography problem may have nearly 1,000 sources and detectors, the computational cost of an optimization routine is a large burden. While techniques from model order reduction can partially alleviate the computational cost, obtaining error bounds in parameter space is typically not feasible. In this work, we examine two different remedies based on stochastic estimates of the objective function. In the first manuscript, we focus on maximizing the efficiency of using stochastic estimates by replacing our objective function with a surrogate objective function computed from a reduced order model (ROM). We use as few as a single sample to detect a misfit between the full-order and surrogate objective functions. Once a sufficiently large difference is detected, it is necessary to update the ROM to reduce the error. We propose a new technique for improving the ROM with very few large linear solutions. Using this techniques, we observe a reduction of up to 98% in the number of large linear solutions for a three-dimensional tomography problem. In the second manuscript, we focus on establishing a robust algorithm. We propose a new trust region framework that replaces the objective function evaluations with stochastic estimates of the improvement factor and the misfit between the model and objective function gradients. If these estimates satisfy a fixed multiplicative error bound with a high, but fixed, probability, we show that this framework converges almost surely to a stationary point of the objective function. We derive suitable bounds for the DOT problem and present results illustrating the robust nature of these estimates with only 10 samples per iteration.