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 data-fitting where the unknown parameters are found by minimizing the misfit 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 differentiation (AD) to develop codes that perform these calculations. We show that by exploiting structure at 2 scales, we can arrive at a very efficient code whose main components are produced by AD. In the first scale we exploite the time-stepping nature of the hyperbolic solver by using the "Extended Jacobian" framework. In the second (finer) scale, we exploit the finite difference 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 one- and two- dimensional problems. We present computational templates that can be used in conjunction with optimization packages to solve the inverse problem.
Bibliographical noteFunding Information:
1 This research is sponsored in part by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Grants DE-FG02-97ER25013 and DF-FG02-94ER25225, and also by the National Science Foundation Grant DMS 9503114, and by the Air Force Office of Scientific Research Grant F49620-95-I-0305. We acknowledge helpful discussions with William Symes, who has a similar on-going effort on automatic differentiation as ours . Some of the ideas in this work were inspired by his presentation at the Institute for Mathematics and its Applications, Minnesota, in July 1997.
- Automatic differentiation
- Finite difference computation
- Hyperbolic pdes
- Inverse problems seismic inversion
- Jacobian and adjoint products
- Wave propagation