Alternating direction multiplier method to estimate an unknown source term in the time-fractional diffusion equation

An estimation for the unknown source term in the time-fractional diffusion equation from measurement data by the alternating direction method of multipliers (ADMM) is considered. The considered model involves a Caputo fractional derivative of order gamma is an element of(0, 1). The inverse source problem is transformed into an optimal control formulation with two distinct cost functions, namely least squares fitting and the L-1-norm. For its resolution, we use the method for all kinds of cost functions. Finally, the efficiency and accuracy of the present method are illustrated by some numerical examples.