We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the "reduced dimensional method". Initially, we use the polynomial-exponential basis to approximate the inverse problem as a system of 1D nonlinear equations. We then employ a Picard iteration based on the quasi-reversibility method and a Carleman weight function. We will rigorously prove that the sequence derived from this iteration converges to the accurate solution for that 1D system without requesting a good initial guess of the true solution. The key tool for the proof is a Carleman estimate. We will also show some numerical examples.

翻译：我们提出了一种全局收敛的计算技术，用于解决利用部分边界数据重构抛物方程中零阶系数的非线性逆问题。该技术称为"降维方法"。首先，我们利用多项式-指数基函数将逆问题近似为一维非线性方程组系统。随后，我们采用基于拟可逆方法和Carleman权函数的Picard迭代。我们将严格证明：该迭代序列收敛到该一维系统的精确解，且无需对真实解提供良好的初始猜测。证明的关键工具是Carleman估计。我们还将展示一些数值算例。