We consider the inverse problem of reconstructing the interior boundary curve of a doubly connected domain from the knowledge of the temperature and the thermal flux on the exterior boundary curve. The use of the Laguerre transform in time leads to a sequence of stationary inverse problems. Then, the application of the modified single-layer ansatz, reduces the problem to a sequence of systems of non-linear boundary integral equations. An iterative algorithm is developed for the numerical solution of the obtained integral equations. We find the Fréchet derivative of the corresponding integral operator and we show the unique solvability of the linearized equation. Full discretization is realized by a trigonometric quadrature method. Due to the inherited ill-possedness of the derived system of linear equations we apply the Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.
Linear Lavrent'ev Integral Equation for the Numerical Solution of a Nonlinear Coefficient Inverse Problem
本文首次对M. M. Lavrent'ev在1964年提出了一个三维波动方程的系数反问题。数据不是超定的。收敛性分析以及数值结果。Lavrent'ev方程的一个有趣的特点是，在不进行任何线性化的情况下，它将一个高度非线性系数的反问题归结为一个第一类线性积分方程。然而，该方程的数值结果，其中使用的系数反问题产生的数据，显示出良好的重建精度。这与1951年导出的经典Gelfand-Levitan方程相似，该方程在一维情况下是有效的。
A domain decomposition preconditioning for the integral equation formulation of the inverse scattering problem
我们提出了区域分解预处理器的解决方案的积分方程制定的正、逆声散射问题的点散射体。我们研究了正问题和反问题，并提出了预处理技术，以加快迭代求解器。对于前向散射问题，我们扩展了{ \em“A restricted additive Schwarz preconditioner for general sparse线性系“，SIAM Journal on Scientific Computing，21（1999），pp. 792--797}，到积分方程。我们结合这个区域分解预条件与低秩校正，这是很容易构建，形成一个新的预条件。对于逆散射问题，我们使用的前向问题的预处理器作为一个积木构建一个预处理器的高斯-牛顿海森。我们提出的数值结果表明，这两个预处理策略的性能。
An integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions
