Direct methods of solving multidimensional inverse hyperbolic problems.
The authors present the so-called ``direct methods'' for solving two inverse hyperbolic problems. It is known in the literature there are two groups of solution methods for inverse problems: the first one is based on Volterra operator equations, iterative or optimization procedures which require solutions of direct and adjoint problems at every step, whereas the second group consisting of the Gel'fand-Levitan-Krein method, the method of boundary control, the finite-difference scheme inversion and linearization methods does not require the multiple direct problem solution and allows one to find the solution in a specific point of the medium. The authors refer to the methods of the second group as the ``direct methods''. In this book the authors apply these methods to the following two inverse problems. Problem 1: Let \(\psi(x,y)\), \(\varphi(x,y)\) and \(f(y,t)\) \((x > 0\), \(y \in \mathbb{R}^n\), \(t> 0)\) be given. It is required to find functions \(u(x,y,t)\) and \(q(x,y)\) from the equations \[ \begin{aligned} \frac{\partial ^2 u}{\partial t^2}&=\Delta_{x,y}u - q(x,y)u, \quad x>0, y \in \mathbb{R}^n, t> 0,\\ u(x,y,0)&=\psi(x,y), \quad \frac{\partial u}{\partial t}(x,y,0) =\varphi(x,y),\quad x>0, y \in \mathbb{R}^n,\\ u(0,y,t)&=f(y,t), \quad y \in \mathbb{R}^n, t > 0. \end{aligned} \] Here, \(\Delta_{x,y}u := \partial ^2 u/\partial x^2 + \partial ^2 u/\partial y_1^2 + \dots + \partial ^2 u/\partial y_n^2.\) Problem 2: Let \(H(y,t)\), \(f(y,t)\), \(y \in \mathbb{R}^n\), \(t > 0\) be given. It is required to determine \(\rho(x,y)\) or \(c(x,y)\) or \(\rho(x,y)\) and \(c(x,y)\) simultaneously from the equations \[ \begin{aligned} c^{-2}(x,y)\frac{\partial ^2 v}{\partial t^2}&= \Delta_{x,y}v - \nabla_{x,y} \ln \rho(x,y)\nabla_{x,y}v, \quad x>0, y \in \mathbb{R}^n, t> 0,\\ v|_{t<0} &\equiv 0,\\ \frac{\partial v}{\partial x}(+0,y,t) &=H(y,t), \quad y \in \mathbb{R}^n, t> 0,\\ u|_{x=+0} &= f(y,t), \quad y \in \mathbb{R}^n, t> 0. \end{aligned} \] Here, \(\nabla_{x,y} v=(\partial v/\partial x,\partial v/\partial y_1, \dots,\partial v/\partial y_n).\) In Chapter 1 the finite-difference scheme inversion with the theoretical and numerical results are presented. The idea is quite simple: the inverse problem is replaced by its finite-difference version. After solving the system of nonlinear algebraic equations (using simple techniques) the obtained solution is taken as an approximate solution of the original inverse problem. In Chapter 2 the linearized inverse problem for Problem 2 with \(\rho(x,y) =\) const is considered. The authors prove the uniqueness theorem, obtain a stability estimate, construct a regularizing sequence that converges to the exact solution of the linearized inverse problem and design a finite difference algorithm for solving the inverse problem. In Chapter 3 the Gel'dfand-Levintan-Krein methods are applied to Problem 2 with \(c(x,y)=\) const. These methods consist in reducing a nonlinear inverse problem to a one-parametric family of Fredholm integral equations of the first or second kind. The authors review the developments of the research in this field, formulate the Gel'fand-Levitan-Krein equation for the one-dimensional inverse problem and its multidimensional analog. Afterwards, they describe the Gel'fand-Levitan method for the wave equation, its discrete (one- and multi-dimensional) analogs with the necessary and sufficient condition of unique solvability of the discrete problems. Some numerical examples for the methods are presented. In Chapter 4, the authors study boundary control method (BC-method) which developed by Belishev for Problem 2 when \(c(x,y)=1\). Finally, in Chapter 5 the authors use the projection method to study the two-dimensional Problem 1 and Problem 2 when \(c(x,y)=1\). In both cases, the inverse problems is considered in the form of a system of nonlinear Volterra integral equations. They establish the convergence of the projection method and estimate the rate of convergence. The main idea of the authors' projection method is projecting the multidimensional inverse problems into a finite-dimensional subspace generated by some orthogonal systems of functions. The finite system of one-dimensional inverse problems can be numerically solved by the finite-difference scheme inversion method. Some numerical examples are presented to show the applicability of the method.
system of nonlinear Volterra integral equations
scientific article; zbMATH DE number 2121579