Efficient calculation of Jacobian and adjoint vector products in the wave propagational inverse problem using automatic differentiation
The paper deals with a type of inverse problems, also called identification problems, which consist in finding information on an unknown medium, being known a generated incident disturbance and the corresponding response. The latter has the form of reflected and refracted waves, produced by incident waves which travel in the unknown medium, and are collected by receivers. These nonlinear problems are very large. The number of unknowns and equations could be in the range \(10^3\sim 10^6\), so one cannot compute and store the entire Jacobian. They are posed in terms of optimization problems, solved by means of nonlinear least squares or some other procedure. The purpose of this article is to demonstrate the use of an extended Jacobian framework together with an exploitation of a stencil in Jacobian and adjoint calculations. Automatic differentiation (AD) is applied at the stencil level, and the resulting subprograms fit well within the extended Jacobian framework. The paper contains a description of AD, and an introduction to the inverse problem for acoustic waves. A section is devoted to review the extended Jacobian framework and to show how to exploit the underlying problem structure. Under the title ``Exploiting the stencil structure'', the authors present a section where the basic algorithm of the work is derived. They point out that the \(j\)th component of the finite difference equation \({\mathbf u}^{k+ 1}= F({\mathbf c},{\mathbf u}^k,{\mathbf u}^{k- 1})\), \({\mathbf u}^{-1}={\mathbf u}^0= \mathbf{0}\), reveals that the dependence of \({\mathbf u}^{k+ 1}\) on its vector variables is very sparse. The efficiency of the code is given by the fact that the time-step function is only based on three scalar functions depending on 2, 4 and 5 variables, respectively. The problem of that the derivative and the adjoint codes are slightly more complicated to assemble, is discussed in this section. An extension of the 2-D problem is investigated in the case when the receivers do not lie on the grid points. Numerical tests and 14 figures are presented.
scientific article; zbMATH DE number 1422638