Numerical solution of a time‐like Cauchy problem for the wave equation

Abstract

Let D c R" be a bounded domain with piecewise-smooth boundary, and q ( x , t ) a smooth function on D x [0, T I . Consider the time-like Cauchy problemGiven g , h for which the equation has a solution, we show how to approximate u(x, t ) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results.

Numerical Solution of the Problem of the Computational Time Reversal in the Quadrant

“…We recall that the Lipschitz stability estimate is the best possible. Observe that a similar high accuracy in the presence of a large amount of noise in the data was also demonstrated in the much earlier work [10], where the 2-D wave equation was solved with the lateral Cauchy data at the boundary of a square. Because this square is a bounded domain, then the Lipschitz stability of the problem of [10] takes place [9,7,5], which is similar with the case of the above Inverse Problem 2.…”

“…The first one is of Klibanov and Rakesh [10], in which the method of quasi-reversibility of Lattes and Lions [11] was adapted for the solution of the Cauchy problem (1.1), (1.4) with L = ∆ = ∂ 2 x + ∂ 2 y in the square with the lateral Cauchy data at the boundary of this square (it was shown in the recent book [9] that the quasi-reversibility is a particular case of the Tikhonov regularization method, and convergence rates were established, also, see [6]). A quite good robustness of this method was demonstrated computationally in [10]. This, observation goes along well with computational results of the current publication and might likely be atributed to the existence of a priori Lipschitz stability estimate, which is the best possible one, also see Section 6.…”

Lipschitz stability for hyperbolic inequalities in octants with the lateral Cauchy data and refocising in time reversal

“…This version of the QRM is a special case of the Tikhonov regularizing functional [20]. Convergence of this analog of the QRM was proven in [10], [12] and [14] and numerical experiments were conducted in [12]. These experiments have demonstrated a good stability of this method, at least for those examples which were considered in [12].…”

“…Convergence of this analog of the QRM was proven in [10], [12] and [14] and numerical experiments were conducted in [12]. These experiments have demonstrated a good stability of this method, at least for those examples which were considered in [12]. It was proposed in [13] and [14] to apply the hyperbolic version of the QRM to the computational time reversal in a bounded domain.…”

Biography of Michael V. Klibanov Professor, Ph.D. and Doctor of Science in Physics and Mathematics and an outstanding expert in inverse problems

“…So, since works [12,13], Carleman estimates became one of the important tools in Control Theory, see also [14]. Michael (with co-authors) also published some numerical works, where the theory of [12,13] was applied to numerical studies using the Quasi-Reversibility Method [15][16][17]. It was shown in these publications that the Quasi-Reversibility Method provides a very stable solution for the above-mentioned Cauchy problem.…”