论文标题
与系数逆问题相关的系列的收敛性
Convergence of a series associated with the convexification method for coefficient inverse problems
论文作者
论文摘要
本文涉及与某个版本的凸化方法相关的系列的收敛性。该版本最近是由解决系数逆问题的第一作者的研究小组开发的。凸化方法旨在构建具有Carleman重量功能的全球凸出Tikhonov样函数。在先前的作品中，严格凸出的加权Tikhonov样功能的构建假定截短的傅立叶序列（即有限的序列而不是无限序列），用于由总波场产生的函数。在本文中，我们证明了此截断的傅里叶序列近似的收敛属性。更确切地说，我们表明，使用截短的傅立叶系列获得的近似PDE的残差倾向于在$ l^{2} $中为零，作为截断的傅立叶级数中的截断指数倾向于无穷大。证明依赖于融合结果，结果$ h^{1} $ - $ l^{2} $ - 正交投影的序列上的有限二维子空间的序列，该子空间由特殊傅立叶的元素跨越。但是，由于系数逆问题的不足性质，我们无法证明该近似PDE的解决方案是由于该Tikhonov样功能的最小化而导致的，它会收敛到正确的解决方案。
This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman Weight Function in it. In the previous works the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in $L^{2}$ as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the $H^{1}$-norm for a sequence of $L^{2}$-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.
