Ecient Calculation of Jacobian and Adjoint Vector Products in Wave Propagational Inverse Problem Using Automatic Dierentiation

Wave propagational inverse problems arise in several applications including medical imaging and geophysical exploration. In these problems, one is interested in obtaining the parameters describing the medium from its response to excitations. The problems are characterized by their large size, and by the hyperbolic equation which models the physical phenomena. The inverse problems are often posed as a nonlinear datatting where the unknown parameters are found by minimizing the mis t between the predicted data and the actual data. In order to solve the problem numerically using a gradient-type approach, one must calculate the action of the Jacobian and its adjoint on a given vector. In this paper, we explore the use of automatic di erentiation (AD) to develop codes that perform these calculations. We show that by exploiting structure at 2 scales, we can arrive at a very e cient code whose main components are produced by AD. In the rst scale we exploit the timestepping nature of the hyperbolic solver by using the \Extended Jacobian" framework. In the second ( ner) scale, we exploit the nite di erence stencil in order to make explicit use of the sparsity in the dependence of the output variables to the input variables. The main ideas in this work are illustrated with a simpler, one-dimensional version of the problem. Numerical results are given for both oneand twodimensional problems. We present computational templates that can be used in conjunction with optimization packages to solve the inverse problem.