A Modified Regularization Method for Inverse Problems of Nonhomogeneous Differential Operator Equation

This paper investigates an abstract nonhomogeneous backward Cauchy problem governed by an unbounded linear operator in a Hilbert space H. The coefficient operator in the equation is assumed to be unbounded, self-adjoint, positive, and to possess a discrete spectrum, with data prescribed at the final time t=T. It is well known that such problems are severely ill-posed. To regularize the problem, we employ a modified approach in which we perturb both the equation and the final condition, rather than treating only one of them. Specifically, the key idea of our work is to simultaneously apply two regularization methods: the quasireversibility method and the quasiboundary value method, to obtain an approximate nonlocal problem depending on two small parameters. We establish stability estimates for the solution of the regularized problem and show that the modified problem is stable, with its solution approximating the exact solution of the original problem. Furthermore, a numerical experiment involving the one-dimensional heat equation is conducted to confirm the practical effectiveness of the proposed method and to illustrate its potential for addressing this type of inverse problem.