 ### High-dimensional approximation (Winter Semester 2009/10)

• Lecturer:
• Classes: Lecture (1097)
• Weekly hours: 2
• Audience: Mathematics, physics (from 5. semester)
Schedule
Lecture: Tuesday 8:00-9:30 1C-04 Begin: 20.10.2009
Lecturers
Lecturer Prof. Dr. Tobias Jahnke
Office hours: Monday, 10 am - 11 am
Room 3.042 Kollegiengebäude Mathematik (20.30)
Email: tobias.jahnke@kit.edu

Many important applications in financial mathematics, systems biology, chemistry, or physics require
the solution of high-dimensional partial differential equations. Such problems are particularily challenging
because they cannot be solved with traditional methods. The main reason is that the number
of unknowns grows exponentially with the dimension such that the computational workload exceeds
the capacity of most computers. For example, an equidistant discretization of the unit interval 0, 1
by mesh points with distance 0.1 has only 11 points (0, 0.1, ...0.9, 1), but a similar discretization of
the unit cube requires 11^3 = 1331 mesh points, and a corresponding mesh on the 10-dimensional
hypercube contains 11^10 = 25, 937, 424, 601 mesh points. This exponential growth of the size of the
problem is known as the curse of dimesionality.

In this lecture, we will give examples for applications which lead to high-dimensional problems
and discuss properties of the corresponding equations (Master equation, Fokker-Planck equation,
Schrödinger equation). Then, three strategies to avoid the curse of dimensionality will be introduced
and analyzed: sparse grids, wavelet compression, and variational approximation. Special emphasis
will be devoted to the question why these approaches work and which assumptions have to be made.
The main goal of this lecture, however, is to convince the audience week by week that it pays to get
up early to attend a lecture at 8 a.m.

The lecture will be given in English. It will be suited for students in mathematics, physics and other
sciences with a basic knowledge in ordinary and partial differential equations and the corresponding
numerical methods.

# References

H.-J. Bungartz and M. Griebel.
Sparse grids.
Acta Numerica, 13:1-123, 2004.
http://wissrech.ins.uni-bonn.de/research/pub/griebel/sparsegrids.pdf

A. Cohen, W. Dahmen, and R. DeVore.
Adaptive wavelet methods for elliptic operator equations: Convergence rates.
Math. Comput. 70(233):27-75, 2001.
http://www.ams.org/mcom/2001-70-233/S0025-5718-00-01252-7/home.html

Ch. Lubich.
From quantum to classical molecular dynamics: Reduced models and numerical analysis.
Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS),
2008.