Space-Dependent Sobolev Gradients as a Regularization for Inverse Radiative Transfer Problems

Abstract

Diffuse optical tomography problems rely on the solution of an optimization problem for which the dimension of the parameter space is usually large. Thus, gradient-type optimizers are likely to be used, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, along with the adjoint-state method to compute the cost function gradient. Usually, the 2 -inner product is chosen within the extraction procedure (i.e., in the definition of the relationship between the cost function gradient and the directional derivative of the cost function) while alternative inner products that act as regularization can be used. This paper presents some results based on space-dependent Sobolev inner products and shows that this method acts as an efficient low-pass filter on the cost function gradient. Numerical results indicate that the use of Sobolev gradients can be particularly attractive in the context of inverse problems, particularly because of the simplicity of this regularization, since a single additional diffusion equation is to be solved, and also because the quality of the solution is smoothly varying with respect to the regularization parameter.

Inner product preconditioned trust-region methods for frequency-domain full waveform inversion

“…Moreover changing the inner product modifies both the gradient and the Hessian and is equivalent to applying a preconditioner [31]. Consequently it also has a major impact on line search based local optimization methods [9,16,23,60]. It is important to highlight here that even though the gradient and the Hessian are modified by the inner product, the misfit to be minimized remains the same.…”