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

M. I. Belishev   ; A. S. Blagoveshchensky   ; N. A. Karazeeva

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 49, Tome 483 (2019), pp. 19-40

Résumé

A dynamical system \begin{align*} &u_{tt}-\Delta u-\nabla \ln \rho \cdot \nabla u = 0 &&\text{in}\quad {\mathbb R^3_+} \times (0,T), \\ &u|_{t=0} = u_t|_{t=0}=0 && \text{in}\quad \overline{\mathbb R^3_+},\\ &u_z|_{z=0}=f && \text{for}\quad 0\leqslant t\leqslant T, \end{align*} is under consideration, where $\rho=\rho(x,y,z)$ is a smooth positive function; $f=f(x,y,t)$ is a boundary control; $u=u^f(x,y,z,t)$ is a solution. With the system one associates a response operator $R: f \mapsto u^f|_{z=0}$. 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, who are interested in numerical realization of methods for solving inverse problems.

Bibliographie

[1] M. I. Belishev, “Boundary control in reconstruction of manifolds and metrics (the BC method)”, Inverse Problems, 13:5 (1997), R1–R45 | DOI | MR | Zbl

[2] M. I. Belishev, “How to see waves under the Earth surface (the BC-method for geophysicists)”, Ill-Posed and Inverse Problems, eds. S. I. Kabanikhin, V. G. Romanov, VSP, 2002, 55–72 | MR

[3] M. I. Belishev, “Dynamical Inverse Problem for the Equation $u_{tt}-\Delta u -\nabla \rho ~\cdot~ \nabla u=0$ (the BC-Method)”, CUBO A Math. J., 10:2 (2008), 17–33 | MR

[4] M. I. Belishev, “Boundary Control Method”, Encyclopedia of Applied and Computational Mathematics, v. 1, 142–146

[5] M. I. Belishev, “Granichnoe upravlenie i tomografiya rimanovykh mnogoobrazii (BC-metod)”, Usp. Mat. Nauk, 72:4 (2017), 3–66 | DOI | MR | Zbl

[6] M. I. Belishev, N. A. Karazeeva, “Prosteishii test dlya dvumernoi dinamicheskoi obratnoi zadachi (BC-metod)”, Zap. nauchn. semin. POMI, 471, 2018, 28–59

[7] M. I. Belishev, V. Yu. Gotlib, “Dynamical variant of the BC-method: theory and numerical testing”, J. Inverse and Ill-Posed Problems, 7:3 (1999), 221–240 | DOI | MR | Zbl

[8] M. I. Belishev, I. B. Ivanov, I. V. Kubyshkin, V. S. Semenov, “Numerical testing in determination of sound speed from a part of boundary by the BC-method”, J. Inverse and Ill-Posed Problems, 24:2 (2016), 159–180 | DOI | MR | Zbl

[9] M. V. De Hoop, P. Kepley, L. Oksanen, “Recovery of a smooth metric via wave field and coordinate transformation reconstruction”, SIAM J. Appl. Math., 78:4 (2018), 1931–1953 | DOI | MR | Zbl

[10] I. B. Ivanov, M. I. Belishev, V. S. Semenov, The reconstruction of sound speed in the Marmousi model by the boundary control method, 24 Sept. 2016, arXiv: 1609.07586v1 [physics.geo-ph]

[11] L. Oksanen, “Solving an inverse obstacle problem for the wave equation by using the boundary control method”, Inverse Problems, 29:3 (2013), 035004 | DOI | MR | Zbl

[12] A. A. Timonov, “On Quantitative Acoustic Imaging Via Boundary Control Method”, Proceedings of the conference “Days of Diffraction” (Steklov's Mathematics Institute of the Russian Academy of Science, St. Petersburg, June 3–7, 2019)