Lefschetz Center for Dynamical Systems Seminar (CANCELLED)
Brown Analysis Seminar
CCMB - Postdoctoral Fellow Interview Seminar
Abstract: Analysis of human genetic variation has gained significant momentum due to the success of many large-scale sequencing projects. Within the last five years, millions of single nucleotide polymorphisms (SNPs) have been genotyped over thousands of individuals belonging to several different ethnic groups. The large-scale efforts have now made it possible to analyze genetic variation within humans, at very fine-scales. In the talk, we outline two research directions to analyze SNP variation within humans. Firstly, we develop maximum parsimony phylogenetic tree reconstruction algorithms that are specifically catered to work on SNP data. Such a phylogeny should cluster closely related individuals (perhaps an ethnic group) together. Therefore, these techniques are widely used to answer questions of human migration. A second method to understand genetic variation is to infer population substructure, i.e., find clusters of individuals based on the property that individuals within a cluster would share similar genetic sequences. Mathematically, we work with two variants of the phylogeny reconstruction problem, both of which are NP-complete. The first variant is equivalent to finding a Steiner minimum tree on a hypercube and the second is a generalization of this problem. We solve the two variants in polynomial time when the size of the phylogeny (Steiner minimum tree) is 'small' with respect to the dimensions of the hypercube (near-perfect). For detecting substructure, we reduce the problem to finding the max-cut of a graph, a well-known NP-complete problem. We can show that if the graph is generated with the clusters 'planted', then our algorithm finds the max-cut in polynomial time given enough data. We provide extensive empirical results to show the methods' effectiveness for both problems.
(CANCELLED) Boston University/Brown University PDE Seminar
Scientific Computing Seminar
Abstract:
We discuss reduced basis approximation and associated a posteriori error
estimation for reliable real-time solution of parametrized partial differential
equations.
The crucial ingredients are rapidly convergent Galerkin approximations over a
space spanned by "snapshots" on the parametrically induced solution manifold;
effective constructions for stability-constant lower bounds; rigorous and sharp
a posteriori error estimators for the outputs/quantities of interest; efficient
Greedy (in parameter) or POD (in time) Greedy (in parameter) selection of
quasi-optimal samples; and Offline-Online computational procedures for rapid
calculation in the many-query and real-time contexts.
We consider linear and nonlinear elliptic problems, linear and nonlinear
parabolic equations, and linear hyperbolic equations. Examples are drawn from
heat transfer (steady and unsteady conduction and convection), solid mechanics
(e.g., crack stress intensity factors), and fluid dynamics (the incompressible
Navier-Stokes equations and Boussinesq natural convection).
PDE Seminar
<--- 2008 Index