### 20.09.15 - 25.09.15, Seminar 15391

# Algorithms and Complexity for Continuous Problems

### Diese Seminarbeschreibung wurde vor dem Seminar auf unseren Webseiten veröffentlicht und bei der Einladung zum Seminar verwendet.

## Motivation

The goal of the seminar is to bring together researchers from different communities working on computational aspects of continuous problems.

Continuous computational problems arise in diverse areas of science and engineering. Examples include path and multivariate integration, partial differential equations, approximation, optimization, as well as operator equations. Understanding the complexity of such problems and construction of efficient algorithms is both important and challenging.

Among the topics to be included are:

**Compressed Sensing **

Compressed sensing is a rapidly developing modern mathematical technology.
It allows effective reconstruction of sparse or almost sparse signals from a few samples, much less than the Shannon rate requires for the reconstruction of arbitrary signals. First real world applications have been developed or are on the horizon. These include MRI, radar and sparse signal approximation in image processing.

**Learning Theory**

During the last decade, the field of learning theory has witnessed an enormous advance and growth. This progress was both triggered and made possible by successfully merging two quite different communities, namely the machine learning community, which traditionally resides in computer science, on the one hand and the mathematically oriented community of non-parametric statistics on the other hand.

**Partial Differential Equations with Random Coefficients**

Partial differential equations with random coefficients present many challenges. The dimensionality of the problem can be high or even infinite, and if the spatial variation of the field is rapid the quality of a finite-dimensional approximation will be poor, unless a very high dimensional approximation is allowed.

**Multilevel Algorithms **

The stochastic multi-level technique was invented about 15 years ago by Stefan Heinrich as a part of his theoretical work on the complexity of local and global solutions of integral equations. We are expecting further substantial progress on multi-level algorithms, both with respect to theory and to applications in more diverse areas.

**Computational Stochastic Processes**

We focus on quadrature problems for stochastic processes, which are given by ordinary and partial stochastic differential equations. While the majority of results in this area deals with globally Lipschitz coefficients and smooth integrands, such assumptions are typically not met for real world applications from biology, chemistry or computational finance.

**Tractability of Multivariate Problems**

This is a topic included in previous Dagstuhl Seminars. Many new questions will be discussed.

There will be another Seminar during the same week, 15392 "Measuring the Complexity of Computational Content: Weirauch Reducibility and Reverse Analysis" See www.dagstuhl.de/15392 for their Motivation text.