CASC 2017 will feature for the first time tutorials on the topics related to the CASC universe. They will take place on the first day of the conference. They will be presented by leading experts in the respective field and targeted at graduate students and non-specialists. Participation is open to all registered participants of CASC 2017 without any additional fee.

Wen-Shin Lee

University of Antwerp, Belgium

Sparse interpolation and its connections to Padé approximation, signal processing, and tensor decomposition: 

A mathematical model is called sparse if it is a combination of only a few non-zero terms. In sparse interpolation, the aim is to determine both the support of the sparse linear combination and the scalar coefficients in the representation, from a small or minimal amount of data samples. Sparse techniques solve the problem statement from a number of samples proportional to the number of terms in the representation rather than the number of available data points or available generating elements. Sparse representations reduce the complexity in several ways: data collection, algorithmic complexity, model complexity.

In this tutorial, we introduce sparse interpolation. We indicate the connections between sparse interpolation, generalized eigenvalue computation, exponential analysis, rational approximation, and tensor decomposition. In the past few years, insight gained from the computer algebra community combined with methods developed by the numerical analysis community, has led to significant progress in several very practical and real-life signal processing applications. We make use of tools such as the singular value decomposition and various convergence results for Padé approximants to regularize an otherwise inverse problem. Classical resolution limitations in signal processing with respect to frequency and decay rates, are overcome. 

We particularly focus on multi-exponential models. These models appear, for instance, in transient detection, motor fault diagnosis, electrophysiology, magnetic resonance and infrared spectroscopy, vibration analysis, fluorescence lifetime imaging, music signal processing, direction of arrival estimation in wireless communication systems, dynamic spectrum management such as in cognitive radio, and so on. The connection with tensor decomposition leads to new possibilities to exploit sparsity in analyzing tensor-structured datasets.

Jan Verschelde

University of Illinois at Chicago, USA

Numerical Algebraic Geometry in the Cloud

Solving a problem in algebraic geometry with numerical algorithms involves embedding the problem in a family of problems with the same structure. The solver then proceeds from an easier to solve instance in the family to the given problem.

As polynomial homotopy methods have progressed from computing approximations for isolated solutions into the manipulation of positive solution sets, the technology (both hardware and software) has evolved as well. Cloud computing offers the user both algorithms and computers, which has implications for what it means to solve a problem. The tutorial will introduce numerical algebraic geometry and the demonstration will give participants hands on experience with our web interface, which builds on PHCpack and phcpy. This is joint work with Nathan Bliss and Jeff Sommars.