Simultaneous identification of initial value and source strength in a transmission problem for a parabolic equation

We are concerned with an inverse problem of simultaneously recovering the initial value and source strength in a transmission problem for a parabolic equation from extra measurements. First, we establish a conditional stability of the inverse problem by combining the Carleman estimate with the logarithmic convexity theory. Second, we construct a regularized minimizing functional and transform the inverse problem into an optimization problem. The existence and stability of solutions to the optimization problem are analyzed rigorously. Then, we introduce a surrogate functional and propose an iterative thresholding algorithm for solving the optimization problem. The algorithm is cheap and easy to be implemented numerically. Finally, we present several numerical examples for the two-dimensional (2 D ) and three-dimensional (3 D ) cases to show the validity of the proposed algorithm.