Toeplitz matrices in the Boundary Control method

07/08/2021

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by   M. I. Belishev, et al.

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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 Ĉ^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 Ĉ^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.