Relationship between representation formulas for unique regularized solutions of inverse source problems with final overdetermination and singular value decomposition of input-output operators

Abstract

In this paper, the relationship between representation formulas for unique regularized solutions of inverse source, as well as backward problems with final overdetermination for evolution equations and singular value decomposition (SVD) of corresponding input–output operators, is analysed. For each considered inverse problem this representation formula is derived via the solution of appropriate adjoint problems and the parameter of regularization. The Lagrange multiplier method is then used to show that each regularized inverse problem is equivalent to a coupled system of two (direct and adjoint) parabolic (or hyperbolic) problems, and the Lagrange multiplier is the weak solution of the corresponding adjoint problem. In the constant coefficient linear parabolic and hyperbolic equations cases, it is proved that the coupled problem is equivalent to the normal equation corresponding to each inverse/backward problem. As a result, an equivalence of the representation formula for unique regularized solution and the SVD of the input–output operator is obtained. This approach, in particular, allows one to construct regularizing filters for a considered class of inverse problems. Some numerical examples are presented to illustrate the usefulness and effectiveness of the proposed approach.