Simplest Test for the Two-Dimensional Dynamical Inverse Problem (BC-Method)

Belishev, M. I.; Karazeeva, N. A.

doi:10.1007/s10958-019-04567-5

Cited by 4 publications

(3 citation statements)

References 7 publications

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“…To be more precise, for determination of ρ| B T σ one needs to solve a family of the 'shortened' problems (13) with the r.h.s. 1 ξ σ ∈ H ξ σ for all 0 < ξ T and then apply the so-called amplitude formula [4,10]. However, to demonstrate the appearance of Toeplitz matrices, it suffices to consider a single problem (13).…”

Section: Bcpmentioning

confidence: 99%

“…Their further development and possible use in the work with real data is an important perspective goal. Our paper is adjacent to articles [10] and [11], which offer the simple tests for probing algorithms based on the BC-method. It reveals additional opportunities in numerical implementation of the BC-method.…”

Section: Introductionmentioning

confidence: 99%

“…For the sake of determinacy and simplicity, we deal with a concrete inverse problem treated in [4] and [10,11]. Namely, the simplicity is that the geometry of rays in the domain, which the problem is considered in, is Euclidean.…”

Section: Introductionmentioning

confidence: 99%

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Toeplitz matrices in the Boundary Control method

Belishev¹,

Karazeeva²

2021

Preprint

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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 ĈTfor 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 ĈT 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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Section: Bcpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%