Toeplitz matrices in the Boundary Control method

TL;DR

This paper explores how the boundary control method's numerical implementation can be optimized by exploiting the block-Toeplitz structure of the Gram matrix, reducing computational complexity in inverse problems.

Contribution

It demonstrates the origin of the block-Toeplitz structure in the Gram matrix and proposes a method to utilize this property for more efficient numerical solutions.

Findings

Identification of block-Toeplitz structure in Gram matrix

Reduction of computational complexity in inverse problem solving

Guidelines for choosing controls to induce Toeplitz structure

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.