Abstract
Algorithms for solving inverse problems arising in practice may often be viewed as problems in nonlinear programming with the data serving as constraints. Such problems are most easily analyzed when it is possible to segment the solution space into regions that are feasible (satisfying all the known constraints) and infeasible (violating some of the constraints). Then, if the feasible set is compact or (ideally) convex, the solution to the problem will normally lie on the boundary of the feasible set. A nonlinear program may seek the solution by systematically exploring the boundary while satisfying progressively more of the constraints provided by the data. One example of an inverse problem in wave propagation (traveltime tomography) and two examples in vibration (the plucked string and free oscillations of the Earth) are presented to illustrate how the variational structure of these problems may (or may not) be used to create nonlinear programs using implicit variational constraints. A detailed analysis of the string density inversion problem shows that the feasibility set constructed with data consisting of two or more eigenfrequen cies is nonconvex, but the solutions (which are nonunique) nevertheless lie on the feasibility boundary.