Study of multiscale global optimization based on parameter space partition

Abstract

Inverse problems in geophysics are usually described as data misfit minimization problems, which are difficult to solve because of various mathematical features, such as multi-parameters, nonlinearity and ill-posedness. Local optimization based on function gradient can not guarantee to find out globally optimal solutions, unless a starting point is sufficiently close to the solution. Some global optimization methods based on stochastic searching mechanisms converge in the limit to a globally optimal solution with probability 1. However, finding the global optimum of a complex function is still a great challenge and practically impossible for some problems so far. This work develops a multiscale deterministic global optimization method which divides definition space into sub-domains. Each of these sub-domains contains the same local optimal solution. Local optimization methods and attraction field searching algorithms are combined to determine the attraction basin near the local solution at different function smoothness scales. With Multiscale Parameter Space Partition method, all attraction fields are to be determined after finite steps of parameter space partition, which can prevent redundant searching near the known local solutions. Numerical examples demonstrate the efficiency, global searching ability and stability of this method.