Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method

Abstract

We propose a global convergent numerical method to reconstruct the initial condition of a nonlinear parabolic equation from the measurement of both Dirichlet and Neumann data on the boundary of a bounded domain. The first step in our method is to derive, from the nonlinear governing parabolic equation, a nonlinear systems of elliptic partial differential equations (PDEs) whose solution yields directly the solution of the inverse source problem. We then establish a contraction mapping-like iterative scheme to solve this system. The convergence of this iterative scheme is rigorously proved by employing a Carleman estimate and the argument in the proof of the traditional contraction mapping principle. This convergence is fast in both theoretical and numerical senses. Moreover, our method, unlike the methods based on optimization, does not require a good initial guess of the true solution. Numerical examples are presented to verify these results.

The Carleman-Newton method to globally reconstruct a source term for nonlinear parabolic equation

“…One can follow the arguments in [23,33] to prove the convergence of the constructed sequence {U n } n≥0 to the true solution to (3.13). For more details, see the following papers [22,23,33,31]. This method was used to solve a similar inverse problem to Problem 1.1.…”

“…Remark 4.2. The unique minimizer in (4.11) can be proved by using the same arguments in [22,Theorem 4.1]. For brevity, we do not repeat the proof here.…”

“…The fast convergence will be shown in both analytic and numerical senses. Some papers closely related to the current one are [4,23,22,29]. On the other hand, the coefficient inverse problem for parabolic equations is also very interesting and studied intensively.…”

“…We name it Carleman-Newton method because it is suggested by Carleman estimates and the well-know Newton method in solving nonlinear equations. We draw the reader's attention to the Carleman contraction principle, see [2,4,23,22] to solve related problems to Problem 1.1. Like the proposed Carleman-Newton method, the Carleman contraction principle is very powerful since it can quickly solve nonlinear inverse problems without requesting a good initial guess.…”

“…Therefore, this gradient dependent case is worth studying. Rather than using the Carleman contraction principle as in [2,4,23,22], we develop the Carleman-Newton method to solve the inverse problem under consideration. The Carleman-Newton method in this paper is stronger than the one introduced in [26].…”