# Teaching

# Seminar numerical analysis

Gaps in the bachelor-master program in numerically oriented research topics in the widest sense are being dealt with in this seminar, as well as topics relating to the theses of the participating students. The seminar format aims at a balance between presentations by the participating students and the responsible research groups.

Several guest speakers will also be invited.

## Practical information

Students |
Master of Mathematics: Financial Mathematics (part 1 or 2) |

Period |
1st/2nd term 2017-2018 |

Contact hours |
Monday 16.00-18.00, room M.G.016check the time schedule for the right dates and times |

Tutors |
prof. dr. Annie Cuyt prof. dr. Karel in 't Hout prof. dr. Wim Vanroose prof. dr. Uwe Peter Wystup |

**Important:** your reports (with illustrative examples/code/...) need to be handed in on 26 January 2018. Send them to both professor Cuyt and Christophe Segers.

## Time schedule

**2 October 2017 — Introduction**

**16 October 2017 — Two paths to sparse interpolationâ€‹: One-shot algorithms**

Talk by Daniel Roche (United States Naval Academy).Polynomial interpolation has been well understood for centuries and is a crucial tool in many areas of mathematics and computing. Sparse polynomial interpolation seeks to extend these techniques - where possible - to handle multivariate polynomials, or polynomials with large degrees but few nonzero terms. Specifically, sparse interpolation is the challenge of discovering the coefficients and exponents of an unknown polynomial, given some way to sample evaluations of that polynomial at chosen points. After understanding the meaning of the problem and the challenges it poses, we look at an important class of solutions that recovers the unknown polynomial "all at once" from a single series of evaluations. The algorithm for this recovery depends strongly on solving structured linear systems, and we will briefly examine some of the numerical stability issues that can arise. Finally, we will see how the problem translates to wildly different areas, from coding theory to signal processing, and some of the unique challenges in each domain.

**17 October 2017 — Two paths to sparse interpolationâ€‹: Homomorphic imagingâ€‹**

Talk by Daniel Roche (United States Naval Academy).

This talk is scheduled for Tuesday, 17 October, 10.45 in room M.G.017.We continue our study of sparse polynomial interpolation by first examining how to recover an unknown polynomial in many variables. There are two primary techniques for this, both of which reduce the problem to sparse interpolation over a single variable. Next, we recall some shortcomings of the "all at once" approach, namely some issues of numerical instability and the computational expense of discrete logarithms in certain domains. We then look at a second path to sparse interpolation, which involves recovering the unknown polynomial exponents and coefficients from a sequence of low-degree views of the polynomial. This over-sampling technique improves the issues with numerical instability and discrete logarithms, but introduces a new challenge in minimizing the number of required evaluations. The methods here depend heavily on randomization and have been an area of considerable recent work. We conclude by seeing how these various algorithms behave in practice.

**13 November 2017 — Validated Exponential Analysis**

Talk by Matteo Briani (Universiteit Antwerpen).In signal processing data are traditionally sampled according to the Shannon-Nyquist theorem in order to prevent aliasing effects. Here we focus on parametric methods and introduce a procedure that allows these methods to work with sub-sampled data. We actually make use of the aliasing effect to regularize the problem statement rather than that we avoid it.

The new approach adds a number of features to a standard exponential analysis, among which output validation, the automatic detection of the exponential model order, robustness against outliers, and the possibilty to parallellize the analysis.

**20 November 2017 — Singular Value Decomposition**

Talk by Roel Weyn.

**4 December 2017 — Interval Linear Systems**

Talk by Thomas Smeets.

**18 December 2017 — Generalised Eigenvalue Problems**

Talk by Daniel Frenkel.

## Project subjects

- Roel Weyn: Singular Value Decomposition
- Matrix Computations, Chapter 8.6: Computing the SVD (Golub, Van Loan)
- Numerical Recipes in C, Chapter 2.6: Singular Value Decomposition (Press, Teukolsky, Vetterling, Flannery)

- Thomas Smeets: Interval Linear Systems
- Introduction to Interval Computations, Chapter 1: Real Interval Arithmetic (Alefeld, Herzberger)
- Modeling, Design, and Simulation of Systems with Uncertainties, Chapter 2: A New Method for Inner Estimation of Solution Sets to Interval Linear Systems (Shary)
- Systems of Linear Interval Equations (Rohn)

- Daniel Frenkel: Generalised Eigenvalue Problems
- Numerical Methods for General and Structured Eigenvalue Problems, Chapter 1: The QR Algorithm & Chapter 2: The QZ Algorithm (Kressner)
- Matrix Computations, Chapter 7.7: The QZ Algorithm for Ax = λBx (Golub, Van Loan)

## Extra

Spreken voor een volle zaal (Bob de Groof)