We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of $L^2$. We then solve this system by a predictor–corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows.

我们提出了一种鲁棒的数值方法，通过测量其解的横向柯西信息，可以找到抛物方程的创建或耗竭项的系数。该领域中的大多数论文都使用最佳控制来研究此非线性且严重不适的问题。这种广泛使用的方法的主要缺点是需要一些真正的解决方案的高级知识。在本文中，我们提出了一种新的方法，该方法为解决抛物线方程组的非线性逆问题打开了一扇门，而无需任何初始的真实系数猜测。此要求已通过数字确认。该方法的重点是针对控制方程的特殊基础，针对控制方程的解的傅立叶系数，导出非线性椭圆方程组。${\mathrm{大号}}^2$。然后，我们通过预测器-校正器过程解决该系统，在该过程中，获得第一和第二预测器的计算是有效的。所考虑的反问题的理想解决方案如下。