论文标题
Carleman-Newton方法是全球重建非线性抛物线方程的源术语
The Carleman-Newton method to globally reconstruct a source term for nonlinear parabolic equation
论文作者
论文摘要
我们建议将Carleman估计值和Newton方法结合起来，以从横向边界数据中解决非线性抛物线方程的逆源问题。这个逆源问题的稳定性是有条件的对数。因此，由于常规最小二乘优化而引起的数值结果可能不可靠。为了增强稳定性，我们通过截断代表处理方程解决方案的傅立叶系列的高频项来近似此问题。通过这种情况，我们得出了一个非线性椭圆PDE的系统，该系统由抛物线抛物线方程的求解求解的傅立叶系数组成。我们通过Carleman-Newton方法解决了该系统。 Carleman-Newton方法是一种新开发的算法，用于求解非线性PDE。 Carleman-Newton方法的强度包括（1）不需要良好的初始猜测，并且（2）计算成本并不昂贵。这些功能是严格证明的。拥有该系统的解决方案，我们可以直接将解决方案计算到提出的反问题上。显示了一些数值示例。
We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem 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.
