Abstract: I will discuss a family of recently developed stochastic techniques for linear algebra problems involving massive matrices.Â These methods can be used to, for example, solve linear systems, estimate eigenvalues/vectors, and apply a matrix exponential to a vector, even in cases where the desired solution vector is too large to store.Â The first incarnations of this idea appear for dominant eigenproblems arising in statistical physics and in quantum chemistry and were inspired by the real space diffusion Monte Carlo algorithm which has been used to compute chemical ground states for small systems since the 1970's.Â I will discuss our own general framework for fast randomized iterative linear algebra as well share a (very partial) explanation for their effectiveness.Â I will also report on the progress of an ongoing collaboration aimed at developing fast randomized iterative schemes specifically for applications in quantum chemistry.Â This talk is based on joint work with Lek-Heng Lim, Timothy Berkelbach, and Sam Greene.
Abstract: Nonlinear dynamic phenomena often require a large number of
dynamical variables to model, only a small fraction of which
are of direct interest. Reduced models that use only the
relevant dynamical variables can be very useful in such
situations, both for computational efficiency and insights
into the dynamics. Recent work has shown that the NARMAX
(Nonlinear Auto-Regressive Moving-Average with eXogenous
inputs) representation of stochastic processes provides an
effective basis for parametric model reduction in a number
of concrete settings [Chorin-Lu PNAS 2015]. In this talk, I
will review these developments as well as a general
theoretical framework for model reduction due to Mori and
Zwanzig. I will then explain how the NARMAX method can be
seen as a special case of the Mori-Zwanzig formalism, and
discuss some general implications and technical issues that
arise. These ideas will be illustrated on a prototypical
model of spatiotemporal chaos.
Abstract: This talk reviews numerical and asymptotic analysis methods for models that describe heterogeneous or composite materials. In particular, we focus on high contrast two-phase dispersed composites that are described by PDEs with rough coefficients, e.g. the case of highly conducting particles that are distributed in the matrix of finite conductivity. In our numerical studies, we assume that particles are located at distances comparable with their sizes, while in our asymptotic methods, we consider densely packed composites where particles are almost touching one another. The proposed numerical procedure yields robust iterative methods whose numbers of iterations are independent of the contrast parameter and the discretization scale. Discrete models constructed using our asymptotic procedures are used for capturing and characterizing of various blow-up phenomena that occur in dense high contrast materials.
Abstract: Consider a smooth connected closed two-dimensional Riemannian manifold Î£ with positive Gauss curvature. If u is a C^2 isometric embedding of Î£, then u(Î£) is convex. On the other hand, in the fifties Nash and Kuiper showed, astonishingly, that this conclusion is in general false for C^1 isometric embeddings. It is expected that the threshold at which isometric embeddings âchange natureâ is the 1/2-Hoelder continuity of their derivatives, a conjecture which shares a striking similarity with another famous one in the theory of fully developed turbulence.
In my talk I will review several plausible reasons for the threshold and a very recent work, joint with Dominik Inauen, which indeed shows a suitably weakened form of the conjecture.
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