Stochastic Systems Seminar
Abstract: A random environment (in $Z^d$) is an iid collection, indexed by $Z^d$, of probability measures on $\{e\in Z^d: |e|=1\}$. A random walk in random environment is the Markov chain on $Z^d$ that, when at location $z$, uses the probability measure indexed by $z$ to generate its next move. Many open problems relate to RWRE. In dimension $d=1$, non standard limit laws occur. It is conjectured that in dimensions $d\geq 2$, one always has a CLT (after proper centering), but this has not yet been established. Even when the transitions are small perturbation of simple random walk, and the expected local drift vanishes, examples have been constructed for which the limiting velocity does not vanish. Under an extra isotropy in law assumption, however, this is not the case, and in fact local limit results for the exit from a ball can be derived. I will explain the approach in dimension $d\geq 3$ and hint at recent extensions to $d=2$. (Joint work with Erwin Bolthausen).
Scientific Computing Seminar
Abstract: Estuarine systems are home to some of the most biological diverse life on the planet yet they are one of the most anthropogenically-degraded habitat-types on Earth. The behaviour of these systems is complex and dependent on hydrodynamic flows, sediment transport and geochemical interactions. To date the majority of mechanistic estuarine modelling has focused on simulating eutrophication based upon `representative' realizations of model parameters and initial and boundary conditions. Yet these systems are subject to a wide range of uncertainty in initial and boundary conditions, model coefficients, forcing terms and geometry. Subsequently computational methods and software tools are needed to facilitate analysis of model sensitivity and uncertainty. Traditional Monte-Carlo techniques are often infeasible due to the large CPU-time needed to run the model in question. Stochastic Collocation is an efficient alternative. I will present a locally adaptive Sparse Grid Collocation method for quantifying uncertainty in differential equations with rapidly varying or discontinuous solutions. The approach is based upon a Sparse grid approximation of the solution surface using a piecewise multi-linear basis. The hierarchical surplus is used as an error indicator to detect non-smooth regions in which a higher level of interpolation is needed.
PDE Seminar
Abstract: We prove well-posedness for compressible flow with free-boundary in physical vacuum, modeled by the 3D compressible Euler equations. The vanishing of the density at the vacuum boundary induces degenerate hyperbolic equations that become characteristic, requiring a separate analysis of time, normal, and tangential derivatives to handle the manifest 1/2-derivative loss. Unfortunately, the methods for incompressible flow do not work for the degenerate compressible regime; a priori nonlinear estimates are obtained using the geometric structure of the Euler equations, and an existence theory is developed using a novel approximation scheme employing an artificial phase.
Lefschetz Center for Dynamical Systems Seminar
***Special note: Joint with algebra seminar at McMillan 115***
Abstract: Arithmetic dynamics is the study of dynamical systems from a viewpoint derived from the classical theory of Diophantine equations and arithmetic geometry. I will explain this correspondence and describe some of the main results and conjectures in the area, in particular those related to rationality of periodic points and integrality of wandering points. Additional topics, as time permits, will include dynamical canonical heights, dynamical analogues of theorems of Faltings and Raynaud, reduction modulo p, and dynamical analogues of cyclotomic fields and cyclotomic units.
Center for Computational Molecular Biology Seminar
Abstract: Database homology searching may be the most important application in computational molecular biology, and since the 1990s, BLAST has been our main workhorse. Since BLAST's introduction, theoretical advances have been made in applying full probabilistic inference to homology searches by using hidden Markov model (HMM) approaches. These methods have been deployed in some important niches, notably in protein domain analysis (as in the Pfam and SMART databases). More general adoption has been limited by the fact that the popular HMM implementations (including my HMMER software) are slow; they use dynamic programming algorithms without heuristic acceleration, which results in running times comparable to Smith/Waterman as opposed to BLAST. I will describe progress on HMMER3, a new generation of HMMER that aims to more fully deploy probabilistic inference technology on homology searches, while at the same time attaining BLAST's speed. I will describe HMMER3's statistical inference framework, its probabilistic model of local sequence alignment, new statistical theory for log-likelihood ratio scores summed over all alignments that extends Karlin/Altschul theory for optimal alignment scores, and an implementation of HMMER3's core algorithms that has accelerated HMMER3 200-fold relative to HMMER2. HMMER3's prototypes are currently faster than WU-BLAST, while being more sensitive than HMMER2.
Scientific Computing Seminar
Abstract: We introduce a second order compact scheme for the streamfunction formulation, which is enhanced to a fourth-order scheme. We construct fourth order approximations for the Laplacian, the biharmonic and the nonlinear convective operators. The scheme is compact (nine-point stencil) for the Laplacian and the biharmonic operators, which are both treated implicitly in the time-stepping scheme. The approximation of the convective term, which is treated explicitly in the time-stepping scheme, is nearly compact (thirteen points stencil). However, no ghost points or artificial boundary conditions are needed for our scheme. We prove stability and convergence properties of these schemes. Numerical results demonstrate that the fourth order accuracy is actually obtained for several test-cases.
PDE Seminar
Abstract: See website: http://www.math.brown.edu/~joyko/pdeSeminar/classial_soft_potential.pdf
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