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... (more)

Information-based complexity (IBC) deals with the computational complexity of continuous problems for which available information is partial, priced and noisy. IBC provides a methodological background for proving the curse of dimensionality as well as provides various ways of vanquishing this curse.

Stochastic computation deals with computational problems that arise in probabilistic models or can be efficiently solved by randomized algorithms. Using IBC background, the complexity of stochastic ordinary (SDE) and partial differential (SPDE) equations have been studied.

Topics covered in the workshop will include: adaptive and nonlinear approximation for SPDEs, infinite-dimensional problems, inverse and ill- posed problems, quasi-Monte Carlo methods, PDEs with random coefficients, sparse/Smolyak grids, stochastic multi-level algorithms, SDEs and SPDEs with nonstandard coefficients, tractability of multivariate problems.

This workshop will bring together researchers from these... (more)

Discrepancy theory deals with the problem of distributing points uniformly over some geometric object and evaluating the inevitably arising errors. The theory was ignited by such famous early results as Herman Weyl's equidistribution theorem and Klaus Roth's theorem on the irregularities of point distributions.

The subject has now grown into a broad field with deep connections tomany areas such as number theory, combinatorics, approximation theory, harmonic analysis, and probability theory, in particular empirical and Gaussian processes. The computational aspects of the subject include searching for well-distributed sets and numerical integration rules. Despite years of research, many fundamental questions, especially in high dimensions, remain wide open, although several important advances have been achieved recently.

The participants of this workshop will share a wide range of views on topics related to discrepancy with an eye towards the recent developments in the subject. The... (more)

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... (more)

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... (more)