Multivariate numerical derivative by solving an inverse heat source problem

A method for approximating multivariate numerical derivatives is presented from multidimensional noise data in this paper. Starting from solving a direct heat conduction problem using the multidimensional noise data as an initial condition, we conclude estimations of the partial derivatives by solving an inverse heat source problem with an over-specified condition, which is the difference of the solution to the direct problem and the given noise data. Then, solvability and conditional stability of the proposed method are discussed for multivariate numerical derivatives, and a regularized optimization is adopted for overcoming instability of the inverse heat source problem. For achieving partial derivatives successfully and saving amount of computation, we reduce the multidimensional problem to a one-dimensional case, and give a corresponding algorithm with a posterior strategy for choosing regularization parameters. Finally, numerical examples show that the proposed method is feasible and stable to noise data.