Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems (Q2037206)

This paper investigates an operator equation where the primary operator is a linear compact map defined on \(L^2\) space and taking values in a general Hilbert space. The solutions satisfy the nonnegative constraints; that is, they are sought in the closed and convex cone \(L^2_+\) of nonnegative elements. The key idea is to define an equation involving an auxiliary operator but without any nonnegative constraints. A perturbed analog of the operator is taken into account. It is shown that the solution set of the auxiliary equation coincides with the original equation under suitable conditions. Since the auxiliary equation has no restrictions, it is used to define two new iterative regularization schemes to incorporate the nonnegative constraints in the original problem. The proposed iterative schemes fall under the broader umbrella of general Landweber-type iterations. The auxiliary operator has the interpretation of being a preconditioner for accelerating the original Landweber iteration. Complete convergence analysis along with the convergence rates are given. The proofs are based on several technical lemmas which are of significance on their own. Application to biosensor tomography is given. Detailed numerical experiments are conducted, which show the feasibility and the efficiency of the developed framework. This well-written paper presents significant algorithmic development, valuable applications, and computations and will be of interest to the large inverse problem community.