EPI-SPLINES AND EXPONENTIAL EPI-SPLINES: PLIABLE APPROXIMATION TOOLS

Abstract:

Approximation theory for functions was, at the outset, mostly
concerned with finding best approximating functions that can be (totally)
described by a finite number of parameters. This question took another
dimension when the information about the function was limited, say its
value at some points, but also included some knowledge of its global
properties, typically smoothness level, which should be replicated in the
approximating one. This gave rise to the theory of splines and its
manifold implementations. But in an evolving range of applications, the function is
only defined implicitly by a rather complex system with many side
conditions. In this lecture, we consider such function identification
problems that usually lead to constrained infinite-dimensional
optimization problems and we rely on a new approximating class of
functions, epi-splines, to build approximate solutions. We present a broad
framework for identifying a function that according to some criterion best
represents a given data set and satisfies constraints derived from the
data as well as external information. These function identification
problems lead to constrained infinite-dimensional optimization that
includes as special cases most of the classical fitting, estimation, and regression problems. The
framework allows any constraints, for example related to shape
restrictions, enables studies of information growth
such as from a larger sample, and facilitates the usually unavoidable
approximations needed to make a procedure computationally tractable. The
central components of the framework are epi-splines: the piecewise
polynomial functions that are structurally related to standard splines,
but are more flexible and arise more broadly.

The theoretical foundations of the framework relies heavily on variational
analysis. As much as time allows, a few illustrative examples coming from
curve fitting, uncertainty quantification, variogram construction
(rock components analysis), commodity price and electricity demand
forecasting will be sketched out.

Posted on Oct 21, 2013 in Optimization and Equilibrium, Seminars