  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.
