Several remarks on numerical solution of the one-dimensional coefficient inverse problem

Abstract

In the work we will introduce some results of numerical experiments, that allowed the author to choose a certain tactic for the numerical solution of onedimensional coefficient inverse problems which are reduced to minimization of the discrepancy functional. Principle distinction from the well-known approach is in the following: we will use the Laplace transform instead of the Fourier transform with respect to the time variable. In the work we will present a new method of derivation of a gradient for the discrepancy functional. The difference of this way is in the following: it is suitable to obtain analytic formulas for the gradient both in the case investigated for one equation and for a system of equations. As was shown in the numerical experiment which results are introduced in the work, represented considerations and recommendations may be useful for numerical solution of the coefficient inverse problem.

Numerical reconstruction of medium parameters of the member of thin anisotropic layers

“…The strategy of minimization of the residual functional [2,3] developed for isotropic media is applied for anisotropic media. We also investigate the case of the transversely isotropic medium with the symmetry axis lying in the plane Oxy.…”

“…For numerical solution of an inverse problem by minimization of a residual functional it is required to solve a direct problem many times. The properties of the residual functional (2.7) are analogous to the investigated functionals, therefore the strategy of minimization developed in the papers [2,3] can be successfully applied to search of the global minimum of the residual functional (2.7) without any chances to be entrapped by local minimuma. In the papers [2,3] properties of the functionals which are similar to the functional (2.7) were investigated for an isotropic medium and the strategy of minimization was developed.…”

“…As noticed above, the strategy of minimization is described in details in [2,3], therefore we do not describe how it is used here. We present only sets of the parameters of the inverse problems, used for the reconstruction.…”

“…Properties of residual functionals and methods of their minimization are investigated in [2,3]. Properties of residual functionals and methods of their minimization are investigated in [2,3].…”

Reconstruction of pressure and shear velocities and boundaries of thin layers in a thinly stratified stack

“…The inverse problem in formulation (5)-(9) and other similar formulations and the properties of the residual functional (10) were studied in much detail in [16][17][18][19], but it was assumed there that the coordinates of the gap points of the medium are known. In particular, it was demonstrated that the inverse problem (5)-(9) can be split into two consecutively solved inverse problems, i.e., it is possible to reconstruct first the velocity of the pressure waves v p , and then, with this velocity being known, to reconstruct the velocity of the shear waves v s .…”