High dimension matters!

Speaker: Dongwoo Sheen

Venue: 210 Haina Complex Building 2

Abstract: High dimensional problems (of dimension >> 4) occur in many fields.

Typical examples include
(1) String theory (10D, 11D), Kaluza-Klein theories (5D),
(2) Financial Mathematics: Multi-asset options pricing (high-dimensional PDEs)
(3) Parametric Studies: Physical space + parameter space dimensions
(4) Uncertainty Quantification: 3D space + 1D time + N stochastic dimensions
(5) Machine Learning: Feature spaces with hundreds of dimensions

Some high dimensional problems can be reduced to lower dimension, by using suitable dimension reduction techniques. However, some other challenging problems should be solved as accurate as possible without dimension reduction.
In this talk, we review several dimension reduction strategies briefly. Then we move to discuss several issues in essentially high dimensional problems, with the aim of accurate computing. In order to challenge the curse of dimensionality, sometimes we need to break our common senses, looking at problems from a completely different directions. Some examples from the finite element methods will be given.