Numerical Modeling of Dynamic Random Processes
Schedule
Lecture: Tuesday, 10:00–12:00, Room 05-426
Tutorial: Tuesday, 14:00–16:00, Room 05-426
Oral presentations:
Final presentation:
(each: 30-minute presentation + 10 minutes for questions)
Lecturer:
Prof. Dr. Mária Lukáčová
Assistant:
Simon Schneider
Content
The aim of the modeling lab is to formulate, analyze, and numerically simulate dynamic random processes. We discuss various approaches to modeling uncertainties in dynamic systems. In particular, we focus on:
- Stochastic ordinary differential equations (e.g., used in statistical physics to describe particle dynamics)
- Partial differential equations with random coefficients (suitable for modeling uncertain measurement data and parameters)
- Partial differential equations with spatio-temporal stochastic noise terms
As part of the projects, which are carried out in small groups, we will investigate practical problems from mathematical biology, fluid dynamics, and materials science, and develop concrete solutions.
Organization
The modeling practicum is the second part of the module Scientific Computing (NUM-004). This module is: an elective requirement in mathematical Master’s programs, and a compulsory module in the interdisciplinary Master’s program Computational Sciences. Basic knowledge of numerical methods for ordinary and partial differential equations is required. Background knowledge in physics, biology, or chemistry is not strictly necessary.
During the Modeling Lab Course, specific project tasks are completed independently in small groups (3–4 students). The tasks include:
- Formulating (modeling) a real-world problem mathematically
- Solving it using mathematical methods with computational support
- Interpreting the results in the context of the original problem
Groups will regularly present their progress in oral presentations. At the end of the semester, a final presentation and a written project report are required.
Literature
- G.J. Lord, C.E. Powell, T. Shardlow: An Introduction to Computational Stochastic PDEs, Cambridge University Press, 2014.
- D.J. Higham: An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review 43(3), 2001. https://epubs.siam.org/doi/pdf/10.1137/S0036144500378302
“A practical and accessible introduction to numerical methods for stochastic differential equations is given. The reader is assumed to be familiar with Euler's method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable; however, no knowledge of advanced probability theory or stochastic processes is assumed...” - D.J. Higham: Stochastic Ordinary Differential Equations in Applied and Computational Mathematics, IMA Journal of Applied Mathematics, 76(3), 2011.
https://doi.org/10.1093/imamat/hxr016