论文标题
通过Carleman-Contraction方法对非线性抛物线方程的初始条件进行全局重建
Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method
论文作者
论文摘要
我们提出了一种全局收敛的数值方法，可以从界限域的边界上的dirichlet和neumann数据的测量中重建非线性抛物线方程的初始条件。我们方法中的第一步是从非线性抛物线方程，这是一种椭圆形偏微分方程（PDE）的非线性系统，其解决方案直接产生了逆源问题的解决方案。然后，我们建立了一个类似收缩映射的迭代方案来解决该系统。这种迭代方案的融合是通过使用卡尔曼估算和传统收缩映射原则证明的论点来严格证明的。这种收敛在理论和数值感官上都是快速的。此外，与基于优化的方法不同，我们的方法不需要对真实解决方案进行良好的初始猜测。提出了数值示例以验证这些结果。
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.
