The Quasi-Reversibility Method for Thermoacoustic Tomography in a Heterogeneous Medium
The Quasi-Reversibility Method for Thermoacoustic Tomography in a Heterogeneous Medium
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DOI：
10.1137/06066970x
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发表时间：
2007-11
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通讯作者：
Christian Clason;M. Klibanov
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作者：
Christian Clason;M. Klibanov
在本文中，我们认为热声层析成像的反问题，确定从横向柯西数据的未知初始条件的波动方程与空间变化的系数。这个问题也出现在医学成像和无损检测领域的若干应用中。利用拟可逆性方法，将原来的不适定问题转化为一个四阶偏微分方程的边值问题。我们找到了这个问题的一个弱H^2 $解，并证明了它是一个适定的椭圆问题。误差估计和收敛的近似遵循从可观测性估计的波动方程，这是证明使用Carleman估计。我们推导出一个数值方案的解决方案的拟可逆性问题的$B$样条Galerkin方法，我们给出误差估计。最后，我们提出的数值结果支持这种方法的鲁棒性重建的初始条件，从充分和有限的边界数据。
In this paper we consider thermoacoustic tomography as the inverse problem of determining from lateral Cauchy data the unknown initial conditions in a wave equation with spatially varying coefficients. This problem also occurs in several applications in the area of medical imaging and nondestructive testing. Using the method of quasi-reversibility, the original ill-posed problem is replaced with a boundary value problem for a fourth order partial differential equation. We find a weak $H^2$ solution of this problem and show that it is a well-posed elliptic problem. Error estimates and convergence of the approximation follow from observability estimates for the wave equation, which are proved using a Carleman estimate. We derive a numerical scheme for the solution of the quasi-reversibility problem by a $B$-spline Galerkin method, for which we give error estimates. Finally, we present numerical results supporting the robustness of this method for the reconstruction of initial conditions from full and limited boundary data.