In this talk, I will present a kernel-free boundary integral (KFBI) method for the variable coefficient elliptic partial differential equation on complex domains. The KFBI method is a generalization of the standard boundary integral method. But, unlike the standard boundary integral method, the KFBI method does not need to know an analytical expression for the kernel of the boundary integral operator or the Green's function associated with the elliptic PDE. So it is not limited to the constant-coefficient PDEs. The KFBI method solves the discrete integral equations by an iterative method, in which only part of the matrix vector multiplication involves the discretization of the boundary integral. With the KFBI method, the evaluation of the boundary integral is replaced by interpolation from a structured grid based solution to an equivalent interface problem, which is solved quickly by a Fourier transform or geometric multigrid based fast elliptic solver. Numerical examples for Dirichlet and Neumann BVPs, interface problems with different conductivity constants and the Poisson-Boltzmann equations will be presented.

Historically, catalytic research and many areas of surface science have used phenomenological rate equations to build kinetic models for surface processes. The models treat surfaces as a lattice of sites, track the probability of finding a site in a particular state, and use maximal-entropy/well-mixed assumption to reconstruct spatially correlated information. This well-mixed assumption, however, often fails. This talk will develop a hierarchy of models that are able to take into account short range spatial correlations. The hierarchy is developed in the context of averaging an underlying master equation. The talk will continue with some simple examples, an example in catalysis, and conclude with ideas on several other applications for this framework.

Conservation equations are at the heart of many interesting and important problems. Examples come from physics, chemistry, biology, traffic and many more. Analytically, hyperbolic equations have a beautiful structure due to the existence of characteristics. These provide the possibility of transforming a conservation PDE into a system of ODE and thus greatly reducing the computational effort required to solve such problems. However, even in one dimension, one encounters problems after a short time.

The most obvious difficulty that needs to be dealt with has to do with the creation of shocks, or in other words, the crossing of characteristics. With a particle based method one would like to avoid a situation when one particle overtakes a neighboring one. However, since shocks are inherent to many hyperbolic equations and relevant to the problems that one would like to solve, it would be good not to ``smooth away'' the shock but rather find a good representation of it and a good solution for the offending particles.

In this talk I will present a new particle based method for solving (one dimensional, scalar) conservation law equations. The guiding principle of the method is the conservative property of the underlying equation. The basic method is conservative, entropy decreasing, variation diminishing and exact away from shocks. A recent extension allows solving equations with a source term, and also provides ``exact'' solutions to the PDE. The method compares favorably to other benchmark solvers, for example CLAWPACK, and requires less computation power to reach the same resolution. A few examples will be shown to illustrate the method, with its various extensions. Due to the current limitation to 1D scalar, the main application we are looking at is traffic flow on a large network. Though we still hope to manage to extend the method to either systems or higher dimensions (each of these extensions has its own set of difficulties), I would be happy to discuss further possible applications or suggestions for extensions.

Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such models leads to the question of representing general L^2-data as the divergence of uniformly bounded vector fields.
We use a multi-scale approach to construct uniformly bounded solutions of div(U)=f for general f’s in the critical regularity space L^2(T^2). The study of this equation and related problems was motivated by results of Bourgain & Brezis. The intriguing critical aspect here is that although the problems are linear, construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations $U = \sum_j u_j$ which we introduced earlier in the context of image processing, yielding a multi-scale decomposition of “images” U.

We consider large Prandtl number behavior of the Boussinesq system for Rayleigh-B\'enard convection at large time. We first show that the global attractors of the Boussinesq system converge to that of the infinite Prandtl number model. This is accomplished via a generalization of upper semi-continuity property with respect to parameters of dissipative dynamical systems to the case of singular limit of two time scale problems of relaxation type. We then show that stationary statistical properties (in terms of invariant measures) of the Boussinesq system converge to that of the infinite Prandtl number model. In particular, we derive a new upper bound on heat transport in the vertical direction (the Nusselt number) for the Boussinesq system. The new upper bound agrees with the recent physically optimal upper bound on the infinite Prandtl number model at large Prandtl number. We will also comment on possible noise induced stability and its relation to the E-Mattingly-Sinai theory.

The Radial Basis Functions (RBF) method is not immune from the disastrous effects of the Gibbs phenomenon. When interpolating or solving PDEs whose solutions are piecewise smooth functions, the RBF method loses its notorious spectral accuracy. In this talk, a new method will be presented, based on the RBF method, which incorporates singularities using Heaviside functions and which keeps track of their location using the level set method. The resulting sharp interface method will be shown to recover the lost spectral accuracy and thus overcome the Gibbs phenomenon altogether.

In mathematics, the study of multilinear algebra is largely limited to properties of a whole space of tensors --- tensor products of k vector spaces, modules, vector bundles, Hilbert spaces, operator algebras, etc. There is also a tendency to take an abstract coordinate-free approach. In most applications, instead of a whole space of tensors, we are often given just a single tensor from that space; and it usually takes the form of a hypermatrix, i.e.\ a k-dimensional array of numerical values that represents the tensor with respect to some coordinates/bases determined by the units and nature of measurements. How could one analyze this one single tensor then? If the order of the tensor k = 2, then the hypermatrix is just a matrix and we have access to a rich collection of tools: rank, determinant, norms, singular values, eigenvalues, condition number, etc. This talk is about the case when k > 2. We will see that one may often define higher-order analogues of common matrix notions rather naturally: tensor ranks, hyperdeterminants, tensor norms (Hilbert-Schmidt, spectral, Schatten, Ky Fan, etc), tensor eigenvalues and singular values, etc. We will discuss the utility as well as difficulties of various tensorial analogues of matrix problems. In particular we shall look at how tensors arise in a variety of applications including: computational complexity, control engineering, mathematical biology, neuroimaging, quantum computing, signal processing, spectroscopy, and statistics.

One of the original motivations for the development of stochastic partial differential equations traces it's origins to the study of turbulence. In particular, invariant measures provide a canonical mathematical object connecting the basic equations of fluid dynamics to the statistical properties of turbulent flows. In this talk we discuss some recent results concerning inviscid limits in this class of measures for the stochastic Navier-Stokes equations and other related systems arising in geophysical and numerical settings. This is joint work with Peter Constantin, Vladimir Sverak and Vlad Vicol.

Interpolatory matrix factorizations provide alternatives to the singular value decomposition (SVD) for obtaining low-rank approximations; this class includes the CUR factorization, where the C and R matrices are subsets of columns and rows of the target matrix. While interpolatory approximations lack the SVD's optimality, their ingredients are easier to interpret than singular vectors: since they are copied from the matrix itself, they inherit the data's key properties (e.g., nonnegative/integer values, sparsity, etc.). We shall provide an overview of these approximate factorizations, describe how they can be analyzed using interpolatory projectors, and introduce a new method for their construction based on the Discrete Empirical Interpolation Method (DEIM). (This talk describes joint work with Dan Sorensen (Rice).)

A variety of problems in modeling of large biomolecules and nonlinear optics lead to large, stiff, mildly nonlinear systems of ODEs that have Hamiltonian form. This talk describes a discrete calculus approach to constructing unconditionally stable, time-reversal symmetric discrete gradient conservative schemes for such Hamiltonian systems (akin to the methods developed by Simo, Gonzales, et al), an iterative scheme for the solution of the resulting nonlinear systems which preserves unconditional stability and exact conservation of quadratic first integrals, and methods for increasing the order of accuracy. Some comparisons are made to the more familiar momentum conserving symplectic methods. As an application, some models of pulse propagation along protein and DNA molecules and related numerical observations will be described, with some consequences for the search for continuum limit PDE approximations.

The use of PDE models to describe complex systems in the social sciences, such as socio-economic segregation and crime, has been popularized during the past decade. In this talk I will introduce some PDE models which can be seen as basic models for a variety of social phenomena. I will then discuss how these models can be used to explore and gain understanding of the real-world systems they describe. For example, we learn that a populations innate views toward criminal activity can play a significant role in the prevention of crime-wave propagation.

Elliptic boundary value problems are well-understood in the case when the boundary, the data, and the coefficients exhibit smoothness. However, it has been long recognized in physics and engineering that irregularities (non-smooth boundary, abrupt change of media, noise or disorder) can decisively influence the properties of the solutions and give rise to completely new phenomena. 

The analysis of general non-smooth elliptic PDEs gives rise to decisively new challenges: possible failure of maximal principle and positivity, breakdown of boundary regularity, lack of the classical L^2 estimates, to mention just a few. Further progress builds on an involved blend of harmonic analysis, potential theory and geometric measure theory techniques. In this talk we are going to discuss some highlights of the history, conjectures, paradoxes, and recent discoveries such as the higher-order Wiener criterion and maximum principle for higher order PDEs, solvability of rough elliptic boundary problems, harmonic measure, as well as an intriguing phenomenon of localization of eigenfunctions -- within and beyond the limits of the famous Anderson localization.

I will describe a recently discovered equivalence between the first two objects mentioned in the title. The stationary Schr\"odinger's equation, a.k.a. Hill’s equation, is ubiquitous in mathematics, physics, engineering and chemistry. Just to mention one application, the main idea of the Paul trap (for which W. Paul earned the 1989 Nobel Prize in physics) amounts to a certain property of Hill's equation. I will show that Hill's equation is equivalent to a seemingly unrelated problem of “tire tracks”. There is, in addition, a yet another connection between the ``tire tracks” problem and the high frequency forced vibrations which I will also outline briefly.

I will present various results on the Langevin dynamics, both from theoretical and numerical perspectives. This dynamics is quite popular for sampling purposes in computational statistical physics. It can be seen as a Hamiltonian dynamics perturbed by an Ornstein-Uhlenbeck process on the momenta. I will start on the theoretical side with an account of the hypocoercive approach by Dolbeault, Mouhot and Schmeiser, which is a key technique to prove that the asymptotic variance of time averages is well defined, and also to obtain quantitative bounds on it. I will then discuss various extensions/modifications of the standard Langevin dynamics, such as replacing the standard quadratic kinetic energy by a more general one, constructing control variates relying on a simplified Poisson equation, proving the convergence of nonequilibrium versions such as the one encountered in the Temperature Accelerated Molecular Dynamics method, etc.

The diffusion or adoption of new products (such as fax machines, skype, facebook, Ipad, etc.) is one of the key problems in Marketing research. In recent years, this problem was often studied numerically, using agent-based models (ABMs). In this talk I will focus on analysis of the aggregate diffusion dynamics in ABMs with a spatial structure. In one-dimensional ABMs, the aggregate diffusion dynamics can be explicitly calculated, without using the mean-field approximation. In multidimensional ABMs, we introduce a clusters-dynamics approach, and use it to derive an analytic approximation of the aggregate diffusion dynamics. The clusters-dynamics approximation shows that the aggregate diffusion dynamics does not depend on the average distance between individuals, but rather on the expansion rate of clusters of adopters. Therefore, the grid dimension has a large effect on the aggregate adoption dynamics, but a small-world structure and heterogeneity among individuals have only a minor effect. Our results suggest that the one-dimensional model and the fully-connected Bass model provide a lower bound and an upper bound, respectively, for the aggregate diffusion dynamics in agent-based models with "any" spatial structure. This is joint work with Ro'i Gibori and Eitan Muller

In the event of a vascular injury, a blood clot will form to prevent bleeding. This response involves two intertwined processes: platelet aggregation and coagulation. Activated platelets are critical to coagulation in that they provide localized reactive surfaces on which many of the coagulation reactions occur. The final product from the coagulation cascade directly couples the coagulation system to platelet aggregation by acting as a strong activator of platelets and cleaving blood-borne fibrinogen into fibrin which then forms a mesh to help stabilize platelet aggregates. Together, the fibrin mesh and the platelet aggregates comprise a blood clot, which in some cases, can grow to occlusive diameters. Transport of coagulation proteins to and from the vicinity of the injury is controlled largely by the dynamics of the blood flow. It is crucial to learn how blood flow affects the growth of clots, and how the growing masses, in turn, feed back and affect the fluid motion. We have developed the first spatial-temporal model of platelet deposition and blood coagulation under flow that includes detailed descriptions of the coagulation biochemistry, chemical activation and deposition of blood platelets, as well as the two-way interaction between the fluid dynamics and the growing platelet mass.

This talk focuses on two instances in which scientific fields outside mathematics benefit from incorporating the geometry of the data. In each instance, the application area motivates the need for new mathematical approaches and algorithms and leads to interesting new questions. (1) A method to determine and predict drug treatment effectiveness for patients based off their baseline information. This motivates building a function adapted diffusion operator on high-dimensional data X when the function F can only be evaluated on large subsets of X, and defining a localized filtration of F and estimation values of F at a finer scale than it is reliable naively. (2) The current empirical success of deep learning in imaging and medical applications, in which theory and understanding is lagging far behind. By assuming the data lie near low-dimensional manifolds and building local wavelet frames, we improve on existing theory that breaks down when the ambient dimension is large (the regime in which deep learning has seen the most success).

I will show results of molecular dynamics simulations of hard disks with non-classical collision rules. In particular, I will focus on how local interactions at the microscopic level between these particles can lead to large-scale coherent dynamics at the mesoscopic level.

This work is inspired by collective behaviors, in the form of vortices and jets, recently observed in bacterial colonies. I will start with a brief summary of basic experimental facts and review a hydrodynamic model developed in collaboration with Thierry Passot (Observatoire de la Cote d'Azur, Nice, France). I will then motivate the need for a complementary approach that includes microscopic considerations, and describe the principal computational issues that arise in molecular dynamics simulations, as well as the standard ways to address them. Finally, I will discuss how classical collision rules that conserve energy and momentum may be modified to describe ensembles of live particles, and will show results of numerical simulations in which such rules have been implemented. Randomness, included in the form of random reorientation of the direction of motion of the particles, plays an important role in the type of collective behaviors that are observed.

Regularity of solutions in the optimal transport problem requires very rigid hypotheses (e.g., convexity of certain sets). When such conditions are not available, one can consider the question of partial regularity, in other words, the in-depth analysis of the structure of singular sets. In this talk, I will discuss the regularity of the set of ``free singularities`` which arise in an optimal transport problem with inner product cost, from a connected set to a disconnected set, along with the stability of such sets under suitable perturbations of the data involved. Some of these results are proven via a non-smooth implicit function theorem for convex functions, which is of independent interest. This talk is based on joint work with Robert McCann.

In this talk, we shall look at discrepancy theory through the prism of harmonic and functional analysis. Discrepancy theory deals with finding optimal approximations of continuous objects by discrete sets of points and quantifying the inevitably arising errors (irregularities of distribution). This field lies at the interface of several areas of mathematics: approximation, probability, discrete geometry, number theory. Historically, methods of analysis (Fourier techniques, Riesz product, wavelet expansions etc) played a pivotal role in the development of the subject.

A number of exciting new connections of discrepancy theory to other fields were discovered recently and are not yet fully understood. These include approximation theory (metric entropy of spaces with mixed smoothness, hyperbolic approximations), probability (small deviations of Gaussian processes, empirical processes), harmonic analysis (small ball inequality, Sidon theorem), compressed sensing etc.

We shall describe some of the recent results in the field, the main ideas and methods, and numerous relations to other areas of mathematics.

The equations governing the motion of a system consisting of a deformable body attached to a rigid body are the partial differential equations for the deformable body subject to boundary conditions that are the equations of motion for the rigid body. (For the ostensibly elementary problem of a mass point on a light spring, the dynamics of the spring itself is typically ignored: The spring is reckoned merely as a feedback device to transmit force to the mass point.) If the inertia of a deformable body is small with respect to that of a rigid body to which it is attached, then the governing equations admit an asymptotic expansion in a small inertia parameter. Even for the simple problem of the spring considered as a continuum, the asymptotics is tricky: The leading term of the regular expansion is not the usual equation for a mass on a massless spring, but is a curious evolution equation with memory. Under very special physical circumstances, an elementary but not obvious process shows that the solution of this equation has an attractor governed by a second-order ordinary differential equation. (This survey of background material is based upon joint work with Michael Wiegner, J. Patrick Wilber, and Shui Cheung Yip.) This lecture describes the rigorous asymptotics and the dimensions of attractors for the motion in space of light nonlinearly viscoelastic rods carrying heavy rigid bodies and subjected to interesting loads. (The motion of the rod is governed by an 18th-order quasilinear parabolic-hyperbolic system.) The justification of the full expansion and the determination of the dimensions of attractors, which gives meaning to these curious equations, employ some simple techniques, which are briefly described (together with some complicated techniques, which are not described). These results come from work with Suleyman Ulusoy.