Convexity Properties of Inverse Problems with Variational Constraints

When an inverse problem can be formulated so the data are minima of one of the variational problems of mathematical physics, feasibility constraints can be found for the nonlinear inversion problem. These constraints guarantee that optimal solutions of the inverse problem lie in the convex feasible region of the model space. Furthermore, points on the boundary of this convex region can be found in a constructive fashion. Finally, for any convex function over the model space, it is shown that a local minimum of the function is also a global minimum. The proofs in the paper are formulated for deeniteness in terms of rst arrival traveltime inversion, but apply to a wide class of inverse problems including electrical impedance tomography. { 2 { I. Introduction In a series of papers (1-3), the author has developed a stable iterative reconstruction method for rst arrival traveltime inversion. The general theory behind this new approach and its extensions will be described in the present paper. The principle contribution of this work is the observation that, when an inverse problem can be formulated so the data are minima of one of the variational problems of mathematical physics, rigorous physically-based feasibility (or admissibility) constraints can be found for the corresponding nonlinear inversion problems. These constraints guarantee that any optimal solution of the inverse problem found using convex programming techniques lies in a convex feasible region of the model space. Furthermore, points on the boundary of the feasible set can be found in a constructive fashion. Also, for any convex function over the model space, a local minimum of the function is also a global minimum. In light of the structure induced on the model space by the feasibility constraints, we can also obtain a series of results about the structure of the solution set that would not possible to establish otherwise. We have three main goals for the paper: (i) to establish that the idea of using varia-tional/feasibility constraints for inversion is both rigorous and applicable to a wide class of physical problems, (ii) to provide elementary proofs that will be accessible to a broad audience (including physicists, geophysicists, engineers, etc., as well as mathematicians) of the consequences of this idea, and (iii) to present the proofs in an abstract setting so as to be independent of the particular choice of discretization made in practical algorithms for solving the inverse problem. For deeniteness, we use …