Toeplitz matrices in the Boundary Control method

2021/07/08 by M. I. Belishev, Belishev, M. I., N. A. Karazeeva +1
Computer Science · Mathematics · #35L05 #35Lxx #35R30 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Statistical and numerical algorithms

paper · pdf · doi:10.48550/arxiv.2107.03811

openalex publication_date 2021/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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 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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