Content

Deel 1 : Approximation techniques for integrals.
Approximating integrals is an important step in many numerical models. This course starts with an overview of the classical deterministic approach, after which Monte Carlo methods are introduced. These methods are used in many simulations and are traditionally introduced via methods to approximate multidimensional integrals. Deterministic and stochastic methods are put next to each other to highlight their advantages and limitations. The focus lies on the understanding of basic concepts, the implemenation of integration methods, including adaptivity and error estimation, and the application of the methods.
1. Deterministic approach for the approximation of Riemann integrals. This is not limited to integration rules that are exact for polynomials. Also quasi-Monte Carlo methods are considered, as are techniques for integrals with specific problems, such as strong oscillations and singularities. Practical error estimation and adaptive algorithms are treated, with attention for different types of adaptivity.
2. Monte Carlo methods. Error estimation depends on stochastic error estimators that depend on the variance of the problem. Several techniques for variance reduction are treated, as is the generation of pseudo-random numbers distributed according to a given probability distribution.

Deel 2: Stochastische processen en simulatie
Not all phenomena can be well modelled by deterministic models (ordinary and partial differential equations). When uncertainty is added to such a model, one quickly ends up with a stochastic differential equation. We will see that such problems can be addressed with Monte Carlo methods. In this part, an introduction is given to stochastic integrals and differential equations, along with some numerical solution techniques. All modelling techniques are illustrated with different applications (biology, chemistry, finance, physics). Emphasis lies on the understanding of basic concepts, the implementation of simple numerical techniques, and the application to practical examples.
1. Stochastic differential equations. Concepts, applications and numerical simulation. (Relation between stochastic differential equations and an advection diffusion equation for the corresponding probability density.) First introduction of Monte Carlo simulation.
2. Markov-Chain Monte Carlo (Metropolis-Hastings). This is one of the most influential algorithms of the 20th century. Monte Carlo method based on the properties of stochastic processes.
3. Alternative representations of stochastic processes: jump processes, Markov chains, and their application in biology, physics and chemistry.
4. Equilibria of stochastic systems. Invariant probability distribution. Applications: determination of material properties. Formulation as an integral computation.

Course material

Study cost: 1-10 euros (The information about the study costs as stated here gives an indication and only represents the costs for purchasing new materials. There might be some electronic or second-hand copies available as well. You can use LIMO to check whether the textbook is available in the library. Any potential printing costs and optional course material are not included in this price.)

Articles and notes, provided via Toledo.

Format: more information

When the size of the group allows, the lectures will be organised in an alternative way.

• Before every lecture, two students are assigned (in mutual agreement) to prepare the next lecture thoroughly. The other students prepare the lecture less thoroughly.
• During the lecture, the assigned students discuss the material, and explain it to their fellow students, who can then ask questions.
• The lecturer moderates the discussion and guides where needed.

Content

Exercises and tasks related to the lectures on Deterministic and Stochastic Integration Techniques.