Numerical implementation of the BC-method approach for the parabolic inverse problem

Ascertain whether there exist numerical implementations of the Boundary Control method approach of Avdonin–Belishev–Rozhkov (1997) for the dynamic inverse problem of the heat equation v_t(x,t) − v_{xx}(x,t) + q(x)v(x,t) = 0 on (0,L) with Dirichlet boundary control, which uses spectral controllability and a variational principle to recover the spectral data of the Dirichlet operator −d^2/dx^2 + q(x) from the response operator R^T, and, if such implementations exist, identify and document them.

Background

Within the paper’s overview of applications to numerical simulation, the authors recall the Avdonin–Belishev–Rozhkov (1997) method, which exploits spectral controllability and a variational principle to extract spectral data of the operator −d^2/dx^2 + q(x) on (0,L) with Dirichlet boundary conditions from dynamic data of the heat equation. In that approach, the response operator R^T maps the boundary control f(t) to the Neumann-type observation v_x(0,t).

The authors note that, despite theoretical feasibility and the possibility of using arbitrarily small observation times T, practical implementation of this continuous method is challenging. They explicitly state that they are unaware of any numerical implementation attempts of this approach, motivating their replacement of the continuous system by a discrete analogue in the present work.