### Queueing Theory - M1 - 6EC

Prerequisites

Basic knowledge of probability at the level:

S.M. Ross, Introduction to probability models, 9th edition, Academic Press, 2007 (chapters 1-3).

Aim of the course

To provide insight in the theory of queueing models. The following subjects will be treated:

• Fundamental queueing relations (Little's law, PASTA property)
• Markovian queues (M/M/1 queue, M/M/c queue, M/E_r/1 queue)
• M/G/1 queue and G/M/1 queue
• Mean value technique
• Priority queues
• Variations of the M/G/1 queue
• Insensitive queues (M/G/c/c queue and M/G/infinity queue)

Lecturer

Jacques Resing (TU/e) and Sindo Nunez-Queija (UvA)

### Advanced Linear Programming - M1 - 6EC

Aim of the course

To provide insight in the theory of linear optimization and in the design of advanced practical methods for solving (integer) linear optimization problems.

Part 1: Basic theory and algorithms of linear optimization: - Linear optimization - polyhedra and polytopes - the simplex algorithm - duality - linear inequalities and Farkas' lemma – network flow problems – ellipsoid method and separation

Part 2: Advanced linear optimization methods - the revised simplex method and column generation - Dantzig-Wolfe and Benders' decomposition - integer programming formulations and solution methods- valid inequalities- branch-and-cut

Prerequisites

• Parts on Linear Programming in any book Introduction to Operations Research.

• A course on Linear Algebra based on e.g. David C. Lay, Stephen R. Lay and Judi J. McDonald, Linear Algebra and its Applications . For mathematics students their knowledge of Linear Algebra should suffice to match up rather easily.

• The course is organized by the Dutch Network for the Mathematics of Operations Research and as such meant, among others for Master students in Operations Research.

Lecture notes/Literature

The course is covered by the book `Introduction to Linear Optimization', Bertsimas and Tsitsiklis, Athena Scientific 1997, ISBN 1-886529-19-1. According to some students the book cannot be ordered through ball.com or similar companies. It can be ordered directly from the publisher Dynamic Ideas. Since the past has taught that many students do not have the book yet at the start of the course, I make a copy available of the first five chapters (see the section on Lecture Notes/Literature a bit further).

Next to the book, we make as much as possible lecture notes available that you find in the week descriptions. There, but also in the lecture notes, you also find the Exercises for the week. The notes you see there now are those of last year and may be subject to small changes.

For a more basic course on linear programming I refer to books titled Introduction to Operations Research , e.g. by Hillier and Lieberman, or by Taha. Another very good book on linear programming is written by Chvatal. Everything you ever wish to know about the Theory of Linear and Integer Programming is found in the more advanced book with that title by Lex Schrijver.

Lecturers

L. Stougie (VU/CWI)

### Scheduling - M1 - 6EC

Introduction

The term scheduling represents the assignment of resources over time to perform tasks, jobs or activities. Feasible schedules are compared with respect to a given optimality criterion. Mostly, the optimization problem is combinatorial and very complex. From a computational point of view, many of these problems are hard (NP-hard). In this course, an overview on the most classical scheduling models is given and exact as well as some heuristic solution methods are discussed for these models.

In detail, the following issues are treated:
- Classification of scheduling models
- Single-machine models
- Parallel-machines models
- Open shop, flow shop and job shop models

Aim of the course

In this course, students will learn techniques for a broad variety of scheduling problems. In particular, it is expected that after this course students will be able to construct mathematical models for the basic problems, classify them, address the questions on computational complexity of the problems, and apply standard algorithmic techniques to solve the problems.

Prerequisites

Basic knowledge (bachelor level) of analysis and linear algebra. Linear programming (modelling, not necessarily solving, see e.g. Chapter 1 of Linear Programming: Foundations and Extensions by Robert J. Vanderbei) and dynamic programming (see e.g. Chapter 5 of Integer Programming by Laurence A. Wolsey).

Lecturers

Theresia van Essen (Delft University of Technology)
Johann Hurink (University of Twente)

Video Recordings

Video recordings are available on Vimeo: https://vimeo.com/showcase/6774759

### Applied Finite Elements - M1 - 6EC

FEM_Error_Analysis.pdf

Prerequisites

• Linear algebra (eigenvectors, eigenvalues, matrix algebra, Gershgorin's circle theorem, inner products, projections)
• Calculus (differentiation, integration, integration over lines, surfaces and domains, integral theorems (Gauss, Green))
• Partial differential equations (definition, heat, Laplace, Poisson, wave equation))
• Introductory numerical analysis (numerical time integration, interpolation, finite differences, quadrature, approximation methods for nonlinear equations)

Aim of the course

After completion of the course, the particpant will be able to construct and to use finite-element methods to solve partial differential equations. Furthermore, the student will be able to assess the quality of the obtained numerical approximations.

The course aims at learning how to apply and construct finite-element methods to various kinds of partial differential equations. The emphasis will be on the application and implementation of the finite-element methods. The finite-element formalisms due to Ritz and Galerkin will be treated. The course will include linear, quadratic, bilinear Lagrangian elements for time-independent and time dependent problems. Next to Galerkin frameworks, Petrov-Galerkin frameworks will be considered for the treatment of convection-dominated cases. Theoretical issues will be assessed in terms of convergence and error analysis, although this will not be the main focus of the course. Several lab assignments will be helpful in gaining understanding in the development of finite-element methods.

After completion of the course, the participant will be able to construct and to use finite-element methods to solve partial differential equations. Furthermore, the student will be able to assess the quality of the obtained numerical approximations.

Lecturers

Fred Vermolen (TUD), Jaap van der Vegt (UT)

### Inverse Problems in Imaging - M1 - 6EC

Aim of the course

This course is about inverse problems in imaging. The mathematical reconstruction and processing of images is of fundamental importance in state of the art applications in health and geosciences, e.g. in medical tomography, in high-resolution microscopy or in geophysical inversion. In many cases, underlying inverse problems can be formulated and solved using variational methods and partial differential equations. This course offers a theoretical as well as an applied insight into inverse problems and variational methods for mathematical imaging. It addresses reconstruction problems of different imaging modalities (e.g. CT or PET) in biomedicine and geophysics. The course covers the full chain of solving inverse problems in imaging, namely

Problem identification → Modeling and discretization → Analysis → Numerical optimization

Where variational principles, regularization theory and numerical optimization (scientific computing) form the underlying joint core. The course connects and extends upon the main concepts of basic courses on differential equations and numerics. The main learning goal for the students is to model, analyze and use state-of-the-art variational methods, PDEs and optimization techniques to solve challenging inverse problems in imaging. Upon completing this course, students achieved the following learning goals:

• Problem identification: Identification of imaging problems as mathematical inverse operator problems (e.g. integral equations, dynamical systems);
• Modeling and discretization: Problem formulation arising in applications using the language of nonlinear variational methods and partial differential equations; use Bayesian modeling to take data and model uncertainty into account; continuous versus discrete modeling;
• Analysis: Understanding the main concepts of nonlinear regularization theory in an analogous way in PDEs as well as in variational methods, and how it influences existence (duality, weak topologies, Theorem of Banach-Alaoglu) and uniqueness results;
• Numerical optimization: Formulating optimality conditions (variations) for constrained convex variational methods (saddle point problems) and to solve them via primal-dual methods or discretized higher-order methods.

At the end of the course participants will be able to tackle inverse problems for imaging in biomedicine or geophysics with a new repertoire of state-of-the-art mathematical tools.

Prerequisites

The course is aimed at Master and starting PhD students in Mathematics (and Technical Medicine) at the comprehensive as well as the technical universities. Solid knowledge of linear algebra and calculus are essential. Furthermore, knowledge of differential equations and numerics is desirable. All at the Bachelor level.

Course material

We will use (digital) lectures notes which will be made available at the start of the course.

Lecturers

Christoph Brune (Mathematics, University of Twente) c.brune@utwente.nl

Tristan van Leeuwen (Mathematics, University of Utrecht) t.vanleeuwen@uu.nl