Convergence of a series associated with the convexification method for\n coefficient inverse problems Article Swipe
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· 2020
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.2004.05660
· OA: W4286683695
This paper is concerned with the convergence of a series associated with a\ncertain version of the convexification method. That version has been recently\ndeveloped by the research group of the first author for solving coefficient\ninverse problems. The convexification method aims to construct a globally\nconvex Tikhonov-like functional with a Carleman Weight Function in it. In the\nprevious works the construction of the strictly convex weighted Tikhonov-like\nfunctional assumes a truncated Fourier series (i.e. a finite series instead of\nan infinite one) for a function generated by the total wave field. In this\npaper we prove a convergence property for this truncated Fourier series\napproximation. More precisely, we show that the residual of the approximate PDE\nobtained by using the truncated Fourier series tends to zero in $L^{2}$ as the\ntruncation index in the truncated Fourier series tends to infinity. The proof\nrelies on a convergence result in the $H^{1}$-norm for a sequence of\n$L^{2}$-orthogonal projections on finite-dimensional subspaces spanned by\nelements of a special Fourier basis. However, due to the ill-posed nature of\ncoefficient inverse problems, we cannot prove that the solution of that\napproximate PDE, which results from the minimization of that Tikhonov-like\nfunctional, converges to the correct solution.\n
