Lévy Gradient Descent: Augmented Random Search for Geophysical Inverse Problems

Abstract

A new efficient random search algorithm is introduced for solving inversion problems in geophysical studies. The proposed algorithm is inherently a stochastic optimization method which is built on the concept of gradient descending and Levy flights. Therefore, the algorithm is referred to as the Levy gradient descent (L-GD). In which, the Levy flights is a special class of random walk which consists of many short steps along with a few large steps. Such movements are observed in a varying range of fields, including animals’ foraging patterns, fluid dynamics, transport of light and so on. Meanwhile, the Levy flights typically shows much higher speed in searching for sparsely located targets compared to the well-known Brownian walks, which make them preferable to drive random search algorithms. As shown in the paper, besides optimal solutions of the inverse problems, the L-GD algorithm could also produce estimations on the error distributions of the resultant model parameters. Following a detailed introduction of the methodology, parameter settings of the algorithm are discussed in length through statistical experiments. Subsequently, the proposed algorithm is evaluated using numeric tests and shows attracting properties of the global convergence and significant higher searching efficiency compared to commonly adopted stochastic optimization techniques in geophysical inversions. Moreover, the L-GD algorithm is applied to the inversions of gravity and seismic travel time data and has achieved the same accuracy as gradient-based optimization methods. Meanwhile, though error estimations generated by L-GD algorithm are essentially qualitative, they could still provide valuable information to help evaluating the resultant model parameters, which is of great importance for practical geophysical inversions.