The Carleman-Newton method to globally reconstruct the initial condition for nonlinear parabolic equations
The Carleman-Newton method to globally reconstruct the initial condition for nonlinear parabolic equations
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DOI：
10.1016/j.cam.2024.115827
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发表时间：
2024-02
影响因子：
2.4
通讯作者：
Anuj Abhishek;Thuy T. Le;Loc H. Nguyen;Taufiquar Khan
中科院分区：
文献类型：
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作者：
Anuj Abhishek;Thuy T. Le;Loc H. Nguyen;Taufiquar Khan
本文提出将Carleman估计与Newton方法结合起来，求解非线性抛物型方程的侧向边界反演问题。这个求初始条件的反问题的稳定性是有条件对数的。因此，由于传统的最小二乘优化，数值结果可能不可靠。为了提高稳定性，我们通过截断表示控制方程解的傅立叶级数的高频项来近似这个问题。由此导出了一类非线性椭圆偏微分方程，其解由抛物型控制方程解的傅里叶系数组成。我们用卡莱曼-牛顿法求解这个方程组。Carleman-Newton法是一种求解非线性偏微分方程的新算法。Carleman-Newton方法的优点包括：(1)不需要很好的初始猜测；(2)计算成本不高。这些特征得到了严格的证明。有了这个系统的解，我们就可以直接计算所提出的反问题的解。给出了一些数值算例。
We propose to combine the Carleman estimate and the Newton method to solve an inverse problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse problem for determination of initial condition is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed.