ABSTRACT We consider the problem of computing the initial condition for a general parabolic equation from the Cauchy lateral data. The stability of this problem is well-known to be logarithmic. In this paper, we introduce an approximate model, as a coupled linear system of elliptic partial differential equations. Solution to this model is the vector of Fourier coefficients of the solutions to the parabolic equation above. This approximate model is solved by the quasi-reversibility method. We will prove the convergence for the quasi-reversibility method as the measurement noise tends to 0. The convergent rate is Lipschitz. We present the implementation of our algorithm in details and verify our method by showing some numerical examples.

中文翻译：

---

通过拟可逆方法从横向柯西数据恢复抛物线方程的初始条件

摘要 我们考虑从柯西横向数据计算一般抛物线方程的初始条件的问题。众所周知，这个问题的稳定性是对数的。在本文中，我们引入了一个近似模型，作为椭圆偏微分方程的耦合线性系统。该模型的解是上述抛物线方程解的傅立叶系数向量。该近似模型通过准可逆性方法求解。我们将证明准可逆方法的收敛性，因为测量噪声趋于 0。收敛速率为 Lipschitz。我们详细介绍了我们算法的实现，并通过展示一些数值例子来验证我们的方法。