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.

Further information can be downloaded here.

# 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.

# Scans of my lecture notes

Chapter I

Chapter II.1 and II.2

Chapter II.3

Chapter II.4 and II.5

Chapter III.1

Chapter III.2

Chapter III.3

Chapter III.4

Chapter IV

Illustration of sparse grids (slides)

Illustration of Haar wavelets (slides)