The fundamental problem of approximation theory is to resolve a possibly complicated function, called the target function, by simpler, easier to compute functions called approximants. Increasing the resolution of the target function can generally only be achieved by increasing the complexity of the approximants. The understanding of this trade-off between resolution and complexity is the main goal of approximation theory, a classical subject that goes back to the early results on Taylor's and Fourier's expansions of a function.
Modern problems in approximation, driven by applications in biology, medicine, and engineering, are being formulated in very high dimensions, which brings to the fore new phenomena. One aspect of the high-dimensional regime is a focus on sparse signals, motivated by the fact that many real world signals can be well approximated by sparse ones. The goal of compressed sensing is to reconstruct such signals from their incomplete linear information. Another aspect of this regime is the "curse of dimensionality" for standard smoothness classes, which means that the complexity of approximation depends exponentially on dimension. An important step in solving multivariate problems with large dimension has been made in the last 20 years: sparse representations are used as a way to model the corresponding function classes. This approach automatically entails a need for nonlinear approximation, and greedy approximation, in particular.
This program addresses a broad spectrum of approximation problems, from the approximation of functions in norm, to numerical integration, to computing minima, with a focus on sharp error estimates. It will explore the rich connections to the theory of distributions of point-sets in both Euclidean settings and on manifolds and to the computational complexity of continuous problems. It will address the issues of design of algorithms and of numerical experiments. The program will attract researchers in approximation theory, compressed sensing, optimization theory, discrepancy theory, and information based complexity theory.
This page will show upcoming seminars that will be scheduled by organizers, speakers, and participants of the Fall 2014 semester program. Walk-ins are welcomed and encouraged for these seminars. Please check back regularly for updates.
We provide an iCal link for those attendees who wish to sync their own devices and/or calendars to our schedule.
September 8, 2014
9:00 - 5:00
ICESemester program on "High-dimensional Approximation" begins
Visitors are welcome to arrive at ICERM anytime after 9:00AM
To participate in a research cluster please apply through the
semester program visitors
application. Indicate which research cluster you are applying to in the "other comments"
section of the application.
Harmonic analysis provides the mathematical backbone for modern signal and image processing.
It also constitutes an important part of the foundation several scientific and engineering areas,
including communication theory, control science, fluid dynamics, and electromagnetics, that underpin a
much broader set of current applications. Although computer implementation of concepts from harmonic
analysis is prevalent, relatively little attention is given to computational and numerical aspects of
the discipline in its own literature. Further, many of the most capable young mathematicians working
in this area have only modest exposure to the roles of such crucial computational considerations as
finite data effects; e.g., How much error is introduced by truncating this infinite-series representation
of a function in terms of a frame, and where will it be manifested?
On the other hand, new tools and ideas have entered the mainstream of harmonic analysis in recent years
that have not yet become established in areas of applied mathematics where numerical and computational
issues are routinely treated as integral aspects of problem formulation and methodological development.
Among these are tools for non-orthogonal and overcomplete representations in linear spaces and the
exploitation of sparsity and related (e.g., low rank) assumptions in inverse problems of various types.
This research cluster seeks to bridge this perceived gap by (i) fostering understanding and appreciation
of the computational perspective among harmonic analysts and (ii) increasing awareness of emerging
mathematical tools and techniques in applied harmonic analysis among computational mathematicians.