Simplest test for two-dimensional dynamical inverse problem (the BC-method)

M. I. Belishev ; N. A. Karazeeva

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 48, Tome 471 (2018), pp. 38-58

Résumé

A dynamical system \begin{align*} {tt}-\Delta u-\nabla\ln\rho\cdot\nabla u=0\text{in}\quad\mathbb R^2_+\times(0,T)\\ |_{t=0}=u_t|_{t=0}=0\text{in}\quad\mathbb R^2_+\\ |_{y=0}=f\text{for}\quad0\leqslant t\leqslant T, \end{align*} is under consideration, where $\mathbb R^2_+:=\{(x,y)\in\mathbb R^2\mid y>0\}$; $\rho=\rho(x,y)$ is a smooth positive function; $f=f(x,t)$ is a boundary control; $u=u^f(x,y,t)$ is a solution. With the system one associates a response operator $R\colon f\mapsto u^f|_{y=0}$. The inverse problem is to recover the function $\rho$ via the response operator. The short presentation of the local version of the BC-method, which recovers $\rho$ via the data given on a part of the boundary, is provided. If $\rho$ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. The way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, which are interested in numerical realization of methods for solving inverse problems.