## CHALLENGES IN COMPUTATIONAL MATHEMATICS

### January 23, Department of Mathematics, Texas A&M

#### Mitchell Institute for Fundamental Physics & Astronomy (MIST) Room M102

### Lecturers

- Wolfgang Dahmen (University of South Carolina)
- Albert Cohen (Laboratoire Jacques-Louis Lions - Sorbonne)
- Eitan Tadmor (Center for Scientific Computation and Mathematical Modeling - University of Maryland)

### Abstracts

**Wolfgang Dahmen: Error controlled Computation - the merit of residuals**

Rigorous approaches to Uncertainty Quantification very much rely on
stable variational formulations of the underlying mathematical model, typically given
in terms of parametric families of PDEs. This is well understood for linear elliptic
problems. However, for "less friendly" problem classes (e.g. transport dominated,
unsymmetric, singularly perturbed, indefinite, or nonlinear problems) even the quantification of discretization errors effecting any forward simulation is far less understood. This talk discusses some concepts towards certifiable error control for
such problem types. The main conceptual constituents are the identification of suitable stable variational formulations and related residual based a posteriori error bounds. The merit of these principles and the key mechanisms are illustrated by some examples.

**Albert Cohen: Optimal non-intrusive methods in high-dimension**

Motivated by non-intrusive approaches for high-dimensional parametric
PDEs, we consider the approximation of an unknown arbirary function
in any dimension from the data of point samples, where the approximants
are picked from given or adaptively chosen finite dimensional spaces.
One principal objective is to obtain an approximation which
performs as good as the best possible using a sampling budget
that is linear in the dimension of the approximating space.
Using a particular sampling measure, this objective turns out to be
met by both least-squares and pseudo-spectral methods, however
with some notable distinctions that will be discussed in this talk.

**Eitan Tadmor: Emergent behavior in collective dynamics**

Collective dynamics is driven by alignment that tend to self-organize the crowd and different external forces that keep the crowd together.
I will overview recent results on the hydrodynamics of large-time, large-crowd collective behavior, driven by different “rules of engagement”.
In particular, I address the question how short-range interactions lead, over time, to the emergence of long-range patterns,
comparing geometric vs. topological interactions.

### Tentative Schedule

Wed, Jan 23 | |
---|---|

13:00 -- 13:50 | Dahmen: Error controlled computation - the merit of residuals |

13:50 -- 14:15 | BREAK |

14:15 -- 15:05 | Cohen: Optimal non-intrusive methods in high dimension |

15:05 -- 15:30 | BREAK |

15:30 -- 16:30 | Tadmor: Emergent behavior in collective dynamics |