论文标题
逆光学断层扫描问题的唯一性，Lipschitz稳定性和重建
Uniqueness, Lipschitz stability and reconstruction for the inverse optical tomography problem
论文作者
论文摘要
在本文中，我们考虑了从Neumann到Dirichlet图中恢复稳态光学断层扫描问题中扩散和吸收系数的反问题。我们首先证明了吸收参数的全球唯一性和Lipschitz稳定性估计，只要扩散是已知的。然后，我们证明了同时恢复扩散和吸收的Lipschitz稳定性。在这两种情况下，参数属于具有先验已知边界的已知有限子空间。证据依赖于单调性结果，结合了局部电位的技术。为了在数值上解决反问题，我们在一类受到两个边界价值问题的可接受参数上提出了一个Kohn-Vogeliustype成本功能。通过Neumann-Todirichlet运算符对最小化问题的重新重新化，使我们能够使用该操作员及其逆的特征性不同，从而获得最佳条件。然后，通过基于准Newton方法的迭代算法进行重建。最后，我们说明了一些数值结果。
In this paper, we consider the inverse problem of recovering a diffusion and absorption coefficients in steady-state optical tomography problem from the Neumann-to-Dirichlet map. We first prove a Global uniqueness and Lipschitz stability estimate for the absorption parameter provided that the diffusion is known. Then, we prove a Lipschitz stability result for simultaneous recovery of diffusion and absorption. In both cases the parameters belong to a known finite subspace with a priori known bounds. The proofs relies on a monotonicity result combined with the techniques of localized potentials. To numerically solve the inverse problem, we propose a Kohn-Vogeliustype cost functional over a class of admissible parameters subject to two boundary value problems. The reformulation of the minimization problem via the Neumann-toDirichlet operator allows us to obtain the optimality conditions by using the Frechet differentiability of this operator and its inverse. The reconstruction is then performed by means of an iterative algorithm based on a quasi-Newton method. Finally, we illustrate some numerical results.
