We propose an efficient inverse solver for identifying the diffusion coefficient based on few random measurements which can be contaminated with noise. We focus mainly on problems involving solutions with steep heat gradients common with sudden changes in the temperature. Such steep gradients can be a major challenge for numerical solutions of the forward problem as they may involve intensive computations especially in the time domain. This intensity can easily render the computations prohibitive for the inverse problems that requires many repetitions of the forward solution. Compared to the literature, we propose to make such computations feasible by developing an iterative approach that is based on the partition of unity finite element method, hence, significantly reducing the computations intensity. The proposed approach inherits the flexibility of the finite element method in dealing with complicated geometries, which otherwise cannot be achieved using analytical solvers. The algorithm is evaluated using several test cases. The results show that the approach is robust and highly efficient even when the input data is contaminated with noise.

我们提出了一种有效的逆求解器，用于基于很少会被噪声污染的随机测量来识别扩散系数。我们主要关注与温度突然变化常见的，具有陡峭热梯度的解决方案有关的问题。对于正向问题的数值解来说，这样的陡峭梯度可能是一个重大挑战，因为它们可能涉及大量的计算，尤其是在时域中。这种强度很容易使反演问题的计算望而却步，因为反问题需要对正解进行多次重复。与文献相比，我们建议通过开发基于单位有限元方法划分的迭代方法来使这种计算可行，从而显着降低计算强度。所提出的方法继承了有限元方法在处理复杂几何形状方面的灵活性，否则无法使用解析求解器实现。使用几个测试用例对算法进行评估。结果表明，即使输入数据被噪声污染，该方法还是可靠且高效的。