CHALLENGES IN COMPUTATIONAL MATHEMATICS
January 23, Department of Mathematics, Texas A&M
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.
|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|