An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value problem. Standard ways of solving these non-local problems include direct temporal and spatial discretization as well as the shooting method, which may be computationally expensive in higher dimensions. In the present article, we present approaches based on low-rank approximation via Arnoldi algorithm to bypass the computational limitations of the mentioned classical methods. Regardless of the dimension of the problem, we prove that the Arnoldi approach can be effectively used to turn the inverse problem into a simple initial value problem at the cost of only computing one-dimensional matrix functions while still retaining the same accuracy as the classical approaches. Numerical results in dimensions d=1,2,3 are provided to validate the theoretical findings and to demonstrate the efficiency of the method for growing dimensions.

翻译：考虑一类线性抛物型方程中未知热源的反问题。此类问题通常可转化为带有时间非局部条件的直接问题，而非初始值问题。求解这些非局部问题的标准方法包括时空直接离散化以及打靶法，这些方法在高维情况下计算成本较高。本文提出基于Arnoldi算法的低秩逼近方法，以克服上述经典方法的计算局限性。我们证明了无论问题维度如何，Arnoldi方法都能有效将反问题转化为简单的初始值问题，仅需计算一维矩阵函数即可保持与经典方法相同的精度。最后给出了维度d=1,2,3的数值结果，以验证理论发现并展示该方法在维度增长时的计算效率。