Urysohn equations of the form
∫01 k ( s, x ( t ) ) dt = y ( s ) , 0 ≤ s ≤ 1
arise in many areas of application involving the indirect determination of a profile x from measured y. Such problems are typically ill-posed, in the sense that, for a given y, the solution x may not exist, or if it does, may not be unique; and that, even if there is a unique solution x, it may not depend smoothly on y. Results from the literature, reviewed here, show that under appropriate assumptions, the imposition of monotonicity and bound constraints permit the problem to be regularized. The task of solving this nonlinear integral equation can be recast as that of solving an ill-posed linear integral equation followed by a well-posed nonlinear inversion. Again, regularization is possible under monotonicity and bound constraints.
The author introduces straightforward but crude discretizations of both formulations that involve solving least squares problems subject to monotonicity and bound constraints. A formal convergence proof is given, and examples for two model problems are presented. The second formulation proves to be more sensitive to perturbations in y, a crucial issue in this context.
