A stochastic gradient descent approach with partitioned-truncated singular value decomposition for large-scale inverse problems of magnetic modulus data

Abstract

We propose a stochastic gradient descent approach with partitioned-truncated singular value decomposition (SVD) for large-scale inverse problems of magnetic modulus data. Motivated by a uniqueness theorem in gravity inverse problem and realizing the similarity between gravity and magnetic inverse problems, we propose to solve the level-set function modeling the volume susceptibility distribution from the nonlinear magnetic modulus data. To deal with large-scale data, we employ a mini-batch stochastic gradient descent approach with random reshuffling when solving the optimization problem of the inverse problem. We propose a stepsize rule for the stochastic gradient descent according to the Courant–Friedrichs–Lewy condition of the evolution equation. In addition, we develop a partitioned-truncated SVD algorithm for the linear part of the inverse problem in the context of stochastic gradient descent. Numerical examples illustrate the efficacy of the proposed method, which turns out to have the capability of efficiently processing large-scale measurement data for the magnetic inverse problem. A possible generalization to the inverse problem of deep neural network is discussed at the end.