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 definiteness in terms of first arrival traveltime inversion, but apply to a wide class of inverse problems including electrical impedance tomography.