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arXiv · 2107.03811

Toeplitz matrices in the Boundary Control method

M. I. Belishev·N. A. Karazeeva
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Abstract

Solving inverse problems by dynamical variant of the BC-method is basically reduced to inverting the connecting operator $C^T$ of the dynamical system, for which the problem is stated. Realizing the method numerically, one needs to invert the Gram matrix $\hat C^T=\{(C^Tf_i,f_j)\}_{i,j=1}^N$ for a representative set of controls $f_i$. To raise the accuracy of determination of the solution, one has to increase the size $N$, which, especially in the multidimensional case, leads to a rapid increase in the amount of computations. However, there is a way to reduce it by the proper choice of $f_j$, due to which the matrix $\hat C^T$ gets a specific block-Toeplitz structure. In the paper, we explain, where this property comes from, and outline a way to use it in numerical implementation of the BC-algorithms.

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M. I. Belishev, N. A. Karazeeva. 2021-07-08. Toeplitz matrices in the Boundary Control method. https://arxiv.org/abs/2107.03811

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