Numerical Solution of the Problem of the Computational Time Reversal in the Quadrant

Abstract

The problem of the computational time reversal is posed as the inverse problem of the determination of an unknown initial condition with a finite support in a hyperboilc equation, given the Cauchy data at the lateral surface. A stability estimate for this ill-posed problem implies refocusing of the time reversed wave field. Two such two-dimensional inverse problems are solved numerically in the case when the domain is a quadrant and the Cauchy data are given at finite parts of coordinate axis. The previously obtained Lipschitz stability estimate (if proven) rigorously explains and numerical results confirm the experimentally observed phenomenon of refocusing of time reversed wave fields.

A new version of the quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem

“…A comprehensive numerical study of the proposed numerical method was conducted. This study has demonstrated robustness of our technique with respect up to 50% random noise in the data, similarly with previous publications [4], [12], [15]. This study has also demonstrated that this method is capable to image sharp peaks, which is important for the application to thermoacoustic tomography, for example.…”

“…The method of multipliers was recently extended to the case of non-zero lower order terms in Theorem 3.5 of the book [6]. The above problems were previously solved numerically in [4], [7], [12] and [15]. The work [7] was the first one, where the problem of thermoacoustic tomography was formulated and solved numerically as Inverse Problem 1, i.e., as the hyperbolic Cauchy problem with the lateral data.…”

“…On the other hand, since cancerous tumors can be modeled as narrow peaks, it is interesting to image those peaks in the application to thermoacoustic tomography considered below. These first two ideas for J ε are taken from [15]. Mainly because of the second difference we cannot apply previously derived convergence results and thus, need to prove convergence of our new version of the QRM.…”

“…Note that while smooth slowly varying functions were reconstructed numerically in [4] and [12], our numerical experiments reconstruct both those functions and δ−like functions. δ−like functions were also reconstructed in [15] for the Inverse Problem 2. However, the numerical technique of [15] is different from one of the current paper.…”

“…δ−like functions were also reconstructed in [15] for the Inverse Problem 2. However, the numerical technique of [15] is different from one of the current paper. The method of [7] and [15] is based on the representation of the function u(x, t) via truncated Fourier series and minimization of the resulting residual least squares functional.…”