PDE-Applied Math Archives for Academic Year 2018

Fast randomized iterative numerical linear algebra for quantum chemistry and other applications

When: Thu, November 1, 2018 - 3:30pm
Where: John S. Toll Physics Building 1412
Speaker: Jonathan Weare (New York University) - https://cims.nyu.edu/~weare/
Abstract: I will discuss a family of recently developed stochastic techniques for linear algebra problems involving massive matrices.  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.  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.  I will discuss our own general framework for fast randomized iterative linear algebra as well share a (very partial) explanation for their effectiveness.  I will also report on the progress of an ongoing collaboration aimed at developing fast randomized iterative schemes specifically for applications in quantum chemistry.  This talk is based on joint work with Lek-Heng Lim, Timothy Berkelbach, and Sam Greene.

Mori-Zwanzig formalism and discrete-time modeling of chaotic dynamics

When: Thu, November 8, 2018 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Kevin Lin (University of Arizona) - http://math.arizona.edu/~klin/index.php
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.

Efficient Analytical and Numerical Treatment of High Contrast Composites

When: Tue, February 5, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Yulia Gorb (University of Houston) - https://www.math.uh.edu/~gorb/about.html
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.

Rigidity and flexibility of isometric embeddings

When: Thu, February 7, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Camillo De Lellis (Institute for Advanced Studies Princeton) -
Abstract: Consider a smooth connected closed two-dimensional Riemannian manifold Σ with positive Gauss curvature. If u is a C^2 isometric embedding of Σ, then u(Σ) 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 “change nature” 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.

Speeds and Homogenization for Fisher-KPP Reaction-Diffusion Equations in Random Media

When: Thu, March 7, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Jessica Lin (McGill University) -
Abstract: The study of spreading speeds, front speeds, and homogenization for Fisher-KPP reaction-diffusion equations in random heterogeneous media is of interest for many applications to mathematical modelling. However, all pre-existing approaches rely upon linearizing the reaction term. In this talk, we will discuss a new approach to such problems which does not utilize linearization. This talk is based on joint work with Andrej Zlatos

Long time dynamics for nonlinear Schrodinger equations at critical regularity

When: Thu, April 4, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Zehua Zhao (John Hopkins University) - http://www.math.jhu.edu/~zzhao25/
Abstract: In this talk, we will talk about the long time dynamics for critical nonlinear Schrodinger equations (NLS). First, we will introduce the background, some important results and some famous conjectures in this area. Moreover, two specific results of the speaker will be addressed, i.e. ``Scattering for the high dimensional inter-critical NLS'' (joint with C. Gao) and ``Dynamics of subcritical threshold solutions for energy-critical NLS'' (joint with Q. Su).

A non-local one-phase free boundary problem, from obstacle to cavitation

When: Thu, April 11, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Yiling Wu (UT Austin) -
Abstract: Properties of a non-local free boundary problem which is an intermediate case of thin obstacle problem and fractional cavitation problem will be discussed. First there will be a general introduction to free boundary problems including the local and non-local obstacle and cavitation problems. Then I will prove in the non-local intermediate case, the blow-up profiles are homogeneous by a Weiss type monotonicity formula,  and the flatness condition of the free boundary of the viscosity solutions implies C^{1,\theta} regularity.

Latent factor models (continued)

When: Thu, April 18, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: David Bindel (Cornel University) - http://www.cs.cornell.edu/~bindel/
Abstract: Approximate low-rank factorizations pervade matrix data analysis, often interpreted in terms of latent factor
models. After discussing the ubiquitous singular value decomposition (aka PCA), we turn to factorizations
such as the interpolative decomposition and the CUR factorization that offer advantages in terms of inter-
pretability and ease of computation. We then discuss constrained approximate factorizations, particularly
non-negative matrix factorizations and topic models, which are often particularly useful for decomposing
data into sparse parts. Unfortunately, these decompositions may be very expensive to compute, at least in
principal. But in many practical applications one can make a separability assumption that allows for rela-
tively inexpensive algorithms. In particular, we show how to the separability assumption enables efficient
linear-algebra-based algorithms for topic modeling, and how linear algebraic preprocessing can be used to
“clean up” the data and improve the quality of the resulting topics.

Scalable kernel methods (continued)

When: Thu, April 25, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: David Bindel (Cornell University) - http://www.cs.cornell.edu/~bindel/
Abstract: Kernel methods are used throughout statistical modeling, data science, and approximation theory. Depend-
ing on the community, they may be introduced in many different ways: through dot products of feature
maps, through data-adapted basis functions in an interpolation space, through the natural structure of a
reproducing kernel Hilbert space, or through the covariance structure of a Gaussian process. We describe
these various interpretations and their relation to each other, and then turn to the key computational bot-
tleneck for all kernel methods: the solution of linear systems and the computation of (log) determinants for
dense matrices whose size scales with the number of examples. Recent developments in linear algebra make
it increasingly feasible to solve these problems efficiently even with millions of data points. We discuss some
of these techniques, including rank-structured factorization, structured kernel interpolation, and stochastic
estimators for determinants and their derivatives. We also give a perspective on some open problems and
on approaches to addressing the constant challenge posed by the curse of dimensionality.

Nonlocal minimal surfaces

When: Thu, May 2, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Enrico Valdinoci (University of Western Australia) - https://research-repository.uwa.edu.au/en/persons/enrico-valdinoci
Abstract: We discuss some recent results on a geometric problem of nonlocal type, consisting in the minimization of a fractional version of the perimeter. In particular, we will present asymptotics and regularity results, discuss the rather unexpected behavior of the minimizers at the boundary, and some findings in the directions of geometric flows, stressing similarities and differences with respect to the classical case.

Invariant measure and global flow for the SQG equation

When: Wed, May 8, 2019 - 11:00am
Where: Kirwan Hall 3206
Speaker: Mouhamadou Sy (U Virginia, Charlottesville) - https://sites.google.com/site/sycergy/home
Abstract: We consider the surface quasi-geostrophic equation and construct a measure on H^3, we show that the equation is globally well-posed on the support of this measure, and the induced flow leaves the measure invariant along the time. We show qualitatives properties for the measure that is constructed. Our method relies on a fluctuation-dissipation argument. An important ingredient is the proof of new set of `statistical orthogonalities'. A byproduct of this work is a solution of a conjecture formulated by Glatt-Holtz, Sverak and Vicol about the support of an analogous measure constructed by Kuksin for the 2D Euler equation. This is a work in collaboration with Juraj Földes (UVa).

Numerically simulating sticky particles

When: Thu, May 9, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Miranda Holmes-Cerfon (New York University) - https://cims.nyu.edu/~holmes/
Abstract: Colloids, particles with diameters of nanometres to micrometres, form the building blocks of many of the materials around us, and are widely studied both to understand existing materials, and to design new ones. One challenge in simulating such particles is they are “sticky”: the range over which they interact attractively, is often much shorter than their diameters, so the SDEs describing the particles’ dynamics are stiff, and take a long time simulate up to the timescales of interest. I will introduce methods aimed at accelerating these simulations, which simulate instead the limiting equations as the range of the attractive interaction goes to zero. In this limit a system of particles is described by a diffusion process on a collection of manifolds connected by so-called “sticky” boundary conditions. A canonical example is a (root-2) reflecting Brownian motion that is sticky at the origin, whose forward and backward equations are identically f_t = f_{xx} with sticky boundary condition f_{x} = kf_{xx} at x=0, where k>0. I will show that discretizing such an equation in space, rather than in time, gives a numerical method that is orders of magnitude faster than resolving the short-range potential directly, at least in Euclidean spaces. Furthermore, I will introduce a Monte-Carlo method to sample a probability density on a manifold, which can handle arbitrarily large timesteps. Combining these two methods is future work that is hoped to yield more efficient simulations of attractive colloidal particles, though I will point out some ongoing challenges arising from singularities in the geometry of the particles’ configuration space. (The first part of the talk is joint work with Nawaf Bou-Rabee, and the second part with Jonathan Goodman.)