Solving Electromagnetic Inverse Problem Using Adaptive Gradient Descent Algorithm

The nonlinear conjugate gradient (NLCG) algorithm is one of the popular linearized methods used to solve the frequency-domain electromagnetic (EM) geophysical inverse problem. During NLCG iterations, the model gradient guides the searching direction while the line-search algorithm determines the step length of each iteration. Normally, the line search requires solving the corresponding forward problem a few times. Since line search is usually computationally inefficient, we introduce the adaptive gradient descent (AGD) algorithm to accelerate solving the frequency-domain EM inverse problem within the linearized framework. The AGD algorithm is a variant of the classical gradient descent method and has been well-developed and widely used in deep learning. Rather than the time-consuming line search, its core idea is to algebraically manipulate the cumulative gradients and updates of the model from previous iterations to estimate the model parameter variables at the current iteration. For the inversion of magnetotelluric (MT) data, we here designed and implemented a framework using the AGD algorithm combined with the cool-down scheme to tune the regularization parameter. To improve the convergence performance of the AGD algorithm [specifying to Adam and root-mean-square propagation (RMSProp)], we proposed a tolerance strategy which has been tested numerically. To optimize the global learning rate, we carried out some comparative trials in the proposed inversion framework. The inverted results of synthetic and real-world data showed that both the AGD algorithms (Adam and RMSProp) can recover comparable results and save more than a third of CPU time compared with the NLCG algorithm.

Automated summary

The nonlinear conjugate gradient (NLCG) algorithm is a popular linearized method for solving the frequency-domain electromagnetic (EM) geophysical inverse problem. To accelerate the solution process within the linearized framework, the adaptive gradient descent (AGD) algorithm is introduced, which manipulates the cumulative gradients and updates from previous iterations. A framework combining the AGD algorithm with a cool-down scheme is implemented for the inversion of magnetotelluric (MT) data, and a tolerance strategy is proposed to optimize the global learning rate. The inverted results demonstrate that both AGD algorithms (Adam and RMSProp) can achieve comparable results and save more than a third of CPU time compared to the NLCG algorithm.