Mathematics Colloquia and Seminars

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Finding a function that solves a system of equations, an inclusion, or an optimization problem is intrinsically solving an infinite-dimensional problem. Closed form solutions for a problem of this type are by and large out of the question, one has to resort to finding a finite-dimensional approximating problem that is guaranteed to generate an approximating solution. Epi-splines are determined by a finite number of parameters, which renders them appropriate for the development of such approximations, and are dense, in the desired topology, in the unusually rich classes of extended-valued semicontinuous functions. Being piecewise polynomial functions, epi-splines are structurally related to splines, but the objective being pursued is fundamentally different. In fact, classical polynomial splines can be viewed as a strategic subfamily of epi-splines that are specifically suited to deal with problems where the goal is to construct a smooth curve that interpolates the values of a function and possibly some derivatives at a finite number of points. Epi-splines support the solution of infinite-dimensional problems and are therefore focused on approximation, involving an arbitrary criterion, instead of interpolation. An approximating epi-spline also takes into account any external information that might be available about the function one tries to approximate such as monotonicity, bounds, and maximum curvature. We describe the construction of epi-splines and their solid foundation in variational analysis.