Solving Fredholm integrals of the first kind with tensor product structure in 2 and 2.5 dimensions

Abstract

We present an efficient algorithm to solve a class of two- and 2.5-dimensional (2-D and 2.5-D) Fredholm integrals of the first kind with a tensor product structure and nonnegativity constraint on the estimated parameters of interest in an optimization framework. A zeroth-order regularization functional is used to incorporate a priori information about the smoothness of the parameters into the problem formulation. We adapt the Butler-Reeds-Dawson (1981) algorithm to solve this optimization problem in three steps. In the first step, the data are compressed using singular value decomposition (SVD) of the kernels. The tensor-product structure of the kernel is exploited so that the compressed data is typically a thousand fold smaller than the original data. This size reduction is crucial for fast optimization. In the second step, the constrained optimization problem is transformed to an unconstrained optimization problem in the compressed data space. In the third step, a suboptimal value of the smoothing parameter is chosen by the BRD method. Steps 2 and 3 are iterated until convergence of the algorithm. We demonstrate the performance of the algorithm on simulated data.