Crystals play a critical role in the design of novel devices. The surface of a crystal can evolve with time and give rise to a variety of interesting structures used in applications. In this evolution, several length and time scales, from the atomistic to the continuum, are implicated. The description of their linkages poses challenging questions. How can surface evolution at large scales, possibly described by PDEs, emerge from the motion of mesoscale crystal defects? And how does the description of such defects arise from atomistic motion? In this talk, I will address recent progress and open challenges in answering these questions. In particular, I will discuss how facets, macroscopically flat surface regions, lead to PDE free boundary problems with nontrivial microstructures.

A very classical shape optimization problem consists in optimizing the topology and geometry of an elastic structure subjected to fixed boundary loads. One typically aims to minimize a weighted sum of material volume, structure perimeter, and structure compliance (a measure of the inverse structure stiffness). This task is not only of interest for optimal designs in engineering, but e.g. also helps to better understand biological structures. The high nonconvexity of the problem makes it impossible to find the globally optimal design; if in addition the weight of the perimeter is chosen small, very fine material structures are optimal that cannot even be resolved numerically. However, one can prove an energy scaling law that describes how the minimum of the objective functional scales with the model parameters. Part of such a proof involves the construction of a near-optimal design, which typically exhibits fine-scale structure in the form of branching and which gives an idea of how optimal geometries look like. (Joint with Robert Kohn)

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.

We begin with an introduction to time-dependent quantum mechanics and the role of Planck's constant. We then describe some mathematical results about solutions to the Schr\"odinger equation for small values of the Planck constant. Finally, we discuss two new numerical techniques for semiclassical quantum dynamics, including one that is a work in progress.

A dielectric boundary in a biomolecular system is a solute-solvent (e.g., protein-water) interface that defines the dielectric coefficient to be one value in the solute region and another in solvent. The inhomogeneous dielectric medium gives rise to the dielectric boundary force that is crucial to the biomolecular conformation and dynamics. This talk begins with a review of the Poisson-Boltzmann theory commonly used for electrostatic interactions in biomolecular interactions and then focuses on the mathematical description of dielectric boundary force. A precise definition and explicit formula of such force are presented. The motion of a cylindrical dielectric boundary driven by the competition between the dielectric boundary force and the reduction of surface energy is then studied. Implications of the mathematical findings to biomolecular conformational stabilities are discussed.

The problem of finding transition paths in the Lennard-Jones cluster of 38 atoms became a benchmark problem in chemical physics due to its beauty and complexity. The two deepest potential minima, the face-centered cubic truncated octahedron and an icosahedral structure with 5-fold rotational symmetry, are far away from each other in the configuration space, which makes problem of finding transition paths between them difficult. D. Wales's group created a network of minima and transition states associated with this cluster. I will present two approaches to analyze this network. The first one, a zero-temperature asymptotic approach, is based on the Large Deviation Theory and Freidlin's cycles. I will show that in the gradient case the construction of the hierarchy of cycles can be simplified dramatically and present a computational algorithm for building a hierarchy of only those Freidlin's cycles associated with the transition process between two given local equilibria. The second approach is the Discrete Transition Path Theory, a finite temperature tool. This approach allows us to establish the range of validity of the zero-temperature asymptotic and describe the transition process at still low but high enough temperatures where the zero-temperature asymptotic approach is no longer valid.

In this talk we investigate the relationship between Hele-Shaw evolution with a drift and a transport equation with a drift potential, where the density is transported with a constraint on its maximum. The latter model, in a simplified setting, describes the congested crowd motion with a density constraint. When the drift potential is convex, the crowd density is likely to aggregate, and thus if the initial density starts as a patch (i.e. if it is a characteristic function of some set) then it is expected that the density evolves as a patch. We show that the evolving patch satisfies a Hele-Shaw type equation. This is joint work with Damon Alexander and Yao Yao.

An efficient method for simulating the propagation of a localized solution of the Schroedinger equation near the semiclassical limit is presented. The method is based on a time dependent transformation closely related to Gaussian wave packets and yields a Schroedinger type equation that is very ammenable to numerical solution in the semi-classical limit. The wavefunction can be reconstructed from the transformed wavefunction whereas expectation values can easily be evaluated directly from the transformed wavefunction. The number of grid points needed per degree of freedom is small enough that computations in dimensions of up to 4 or 5 are feasible without the use of any basis thinning procedures. This is joint work with Giovanni Russo.

In Computerized Tomography a 2D or 3D object is reconstructed from projection data (Radon transform data) from multiple directions. When the X-ray beams are sufficiently wide to fully embrace the object and when the beams from a sufficiently dense set of directions around the object can be used, this problem and its solution are well understood. When the data are more limited the image reconstruction problem becomes much more challenging; in the figure below only the region within the circle of the Field Of View is illuminated from all angles. In this talk we consider a limited data problem in 2D Computerized Tomography that gives rise to a restriction of the Hilbert transform as an operator HT from L2(a2,a4) to L2(a1,a3) for real numbers a1 < a2 < a3 < a4. We present the framework of tomographic reconstruction from limited data and the method of differentiated back-projection (DBP) which gives rise to the operator HT. The reconstruction from the DBP method requires recovering a family of 1D functions f supported on compact intervals [a2,a4] from its Hilbert transform measured on intervals [a1, a3] that might only overlap, but not cover [a2, a4]. We relate the operator HT to a self-adjoint two-interval Sturm-Liouville prob- lem, for which the spectrum is discrete. The Sturm-Liouville operator is found to commute with HT , which then implies that the spectrum of HT∗ HT is discrete. Furthermore, we express the singular value decomposition of HT in terms of the so- lutions to the Sturm-Liouville problem. We conclude by illustrating the properties obtained for HT numerically.

The periodic generalized Korteweg-de Vries equation (gKdV) is a canonical dispersive partial differential equation with numerous applications in physics and engineering. In this talk we present the invariance of the Gibbs measure under the flow of the gauge transformed periodic quartic gKdV. The proof relies on probabilistic arguments which exhibit nonlinear smoothing when the initial data are randomized. As a corollary we obtain almost sure global well-posedness for the (ungauged) quartic gKdV at regularities where this PDE is deterministically ill-posed.

In this talk I will give an overview of my recent work with H. Knuepfer on the analysis of a class of geometric problems in the calculus of variations. I will discuss the basic questions of existence and non-existence of energy minimizers for the isoperimetric problem with a competing non-local term. A complete answer will be given for the case of slowly decaying kernels in two space dimensions, and qualitative properties of the minimizers will be established for general Riesz kernels.

Kohn-Sham density functional theory (KSDFT) is the most widely used electronic structure theory for molecules and condensed matter systems. Although KSDFT is often stated as a nonlinear eigenvalue problem, an alternative formulation of the problem, which is more convenient for understanding the convergence of numerical algorithms for solving this type of problem, is based on a nonlinear map known as the Kohn-Sham map. The solution to the KSDFT problem is a fixed point of this nonlinear map. The simplest way to solve the KSDFT problem is to apply a fixed point iteration to the nonlinear equation defined by the Kohn-Sham map. This is commonly known as the self-consistent field (SCF) iteration in the condensed matter physics and chemistry communities. However, this simple approach often fails to converge. The difficulty of reaching convergence can be seen from the analysis of the Jacobian matrix of the Kohn-Sham map, which we will present in this talk. The Jacobian matrix is directly related to the dielectric matrix or the linear response operator in the condense matter community. We will show the different behaviors of insulating and metallic systems in terms of the spectral property of the Jacobian matrix. A particularly difficult case for SCF iteration is systems with mixed insulating and metallic nature, such as metal padded with vacuum, or metallic slabs. We discuss how to use these properties to approximate the Jacobian matrix and to develop effective preconditioners to accelerate the convergence of the SCF iteration. In particular, we introduce a new technique called elliptic preconditioner, which unifies the treatment of large scale metallic and insulating systems at low temperature. Numerical results show that the elliptic preconditioner can effectively accelerate the SCF convergence of metallic systems, insulating systems, and systems of mixed metallic and insulating nature. (Joint work with Chao Yang)

Consider wavefront propagation in the plane, in a medium whose propagation speed is doubly periodic. Think of a "wavefront" as the moving boundary between "light" and "darkness." There are "macroscopic plane wavefronts", for which the wavefront is everywhere and always bounded close to a moving line with constant normal velocity. The normal velocity depends on direction. Some nice function of angle, and we ought to compute it, no? To contemplate this calculation, visualize an infinite planar vineyard, with a square lattice of grape stakes. Gaze outward from the the grape stake at the origin. In some directions, the line of sight is blocked by a stake, and we'll call these directions "rational." The rational directions are dense, but of zero measure on the unit circle. The simplest formal asymptotics of the direction dependent speed produces a strange result: A formal series, one term for each rational direction. The graph of an individual term as a function of angle looks like the amplitude response of a forced oscillator, with a plus infinity vertical asymptote as you scan through the rational direction. Nevertheless, the series converges absolutely for almost all directions on the unit circle. (In the vineyard, almost all lines of sight escape to infinity.) At the outset, the prognosis seems: "Difficult and obscure." The problem of direction dependent speed does not LOOK like a venue for formal classical asymptotics, but that's what this talk proposes. The ingredients are standard matched asymptotic expansions, and baby number theory concerning period cells of the vineyard.

The heart is a coupled electro-fluid-mechanical system. The contractions of the cardiac muscle are stimulated and coordinated by the electrophysiology of the heart; these contractions in turn affect the electrical function of the heart by altering the macroscopic conductivity of the tissue and by influencing stretch-activated transmembrane ion channels. In this talk, I will present mathematical models and adaptive numerical methods for describing cardiac mechanics, fluid dynamics, and electrophysiology, as well as applications of these models and methods to cardiac fluid-structure and electro-mechanical interaction. I will also describe novel models of cardiac electrophysiology that go beyond traditional macroscopic (tissue-scale) descriptions of cardiac electrical impulse propagation by explicitly incorporating details of the cellular microstructure into the model equations. Standard models of cardiac electrophysiology, such as the monodomain or bidomain equations, account for this cellular microstructure in only a homogenized or averaged sense, and we have demonstrated that such homogenized models yield incorrect results in certain pathophysiological parameter regimes. To obtain accurate model predictions in these parameter regimes without resorting to a fully cellular model, we have developed an adaptive multiscale model of cardiac conduction that uses detailed cellular models only where needed, while resorting to the more efficient macroscale equations where those equations suffice. Applications of these methods will be presented to simulating cardiac and cardiovascular dynamics in whole heart models, as well as in detailed models of cardiac valves and novel models of aortic dissection. Necessary physiological details will be introduced as needed.

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.

Given a number of labeled and unlabeled images, it is possible to determine the class membership of each unlabeled image by creating a sequence of such image transformations that connect it, through other unlabeled images, to a labeled image. In order to measure the total transformation, a robust and reliable metric of the path length is proposed, which combines a local dissimilarity between consecutive images along the path with a global connectivity-based metric. For the local dissimilarity we use a symmetrized version of the zero-order image deformation model (IDM) proposed by Keysers et al. For the global distance we use a connectivity-based metric proposed by Chapelle and Zien in [2]. Experimental results on the MNIST benchmark indicate that the proposed classifier out-performs current state-of-the-art techniques, especially when very few labeled patterns are available.

In seismic and SAR imaging, fitting cross-correlations of wavefields rather than the wavefields themselves can result in much improved robustness vis-a-vis model uncertainties. This approach however raises two challenges: (i) new spurious local minima may complicate the inversion, and (ii) one must find a good subset of cross-correlations to make the problem well-posed. I will explain how to address these two problems with lifting, semidefinite relaxation, and expander graphs. This mix of ideas has recently proved to be the right approach in other contexts as well, such as angular synchronization (Singer et al.) and phase retrieval (Candes et al.). Joint work with Vincent Jugnon.

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.

From a mathematical point of view self-organization can be described as the formation of patterns, where certain dynamical systems modeling social dynamics tend autonomously to converge. The fascinating mechanism to be revealed by such a modeling is how to connect the microscopical and usually binary rules or social forces of interaction between individuals with the eventual global behavior or group pattern, forming as a superposition in time of the different microscopical effects. In this talk we explore mechanisms to go beyond self-organization, in particular how to externally control such dynamical systems in order to eventually enforce pattern formation also in those situations where this wished phenomenon does not result from spontaneous and autonomous convergence. Our focus is on dynamical systems of Cucker-Smale type, modeling consensus emergence, and we question the existence of stabilization and optimal control strategies which require the minimal amount of external intervention for nevertheless inducing consensus in a group of interacting agents. On the one hand and formally, our main result realizes the connection between certain variational problems involving L1-norm terms and optimal sparse controls. On the other hand, our findings can be informally stated in terms of the general principle for which "A policy maker should always consider more favorable to intervene with stronger actions on the fewest possible instantaneous optimal leaders than trying to control more agents, with minor strength".

Exterior calculus generalizes vector calculus to manifolds. For numerical solutions of PDEs on meshes this language has been discretized as finite element exterior calculus and discrete exterior calculus. I'll first give a very brief introduction to these discretizations. Tools from geometry and topology, such as Hodge theory, and basic ideas from cohomology and homology will be seen to be an integral part of these discretizations. A specific example I'll describe will be the computation of harmonic forms. This is a crucial first step in a finite element solution of even the most basic elliptic PDE like Poisson's equation. I'll show how the availability of a homology basis allows one to find a basis for discrete harmonic forms using least squares. When viewed appropriately, the concepts, language, and software for these PDE discretizations can be easily used to solve some interesting problems in data analysis. A slight generalization also leads to some problems in computational topology. Specifically, this involves moving from 2-norms to 1-norms. In some sense, this is an example of how work in numerical PDEs can lead to a very combinatorial and classical problem in computational topology.

Sampling and reconstruction of functions is a central tool in science. A key result is given by the classical sampling theorem for bandlimited functions. We describe the recently developed sampling theory for operators. We call operators bandlimited if their Kohn-Nirenberg symbols are band limited. The addresses engineers and mathematicians and should be accessible for those who have some education in linear algebra and calculus. The talk reviews sampling of functions and introduces some terminology from the theory of pseudodifferential operators. We will also discuss sampling theorems for stochastic operators.

The nonlinear Schrodinger equation

As time permits the talk will also discuss the energy - critical problem in R^d \ W,

This talk will focus on probability questions arising in the geophysical and biological sciences concerning dispersion in highly heterogeneous environments, as characterized by abrupt changes (discontinuities) in the diffusion coefficient. Some specific phenomena observed in laboratory and field experiments involving breakthrough curves (first passage times), occupation times, and local times will be addressed. This is based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, and Brian Wood at Oregon State University.

Large complex networks naturally represent relationships in a variety of settings, e.g. social interactions, computer/communication networks, and genomic sequences. A significant challenge in analyzing these networks has been understanding the “intermediate structure” – those properties not captured by metrics which are local (e.g. clustering coefficient) or global (e.g. degree distribution). It is often this structure which governs the dynamic evolution of the network and behavior of diffusion-like processes on it. Although there is a large body of empirical evidence suggesting that complex networks are often “tree-like” at intermediate to large size-scales (e.g. work of Boguna et al in physics, Kleinberg on internet routing, and Chung & Lu on power-law graphs), it remains a challenge to take algorithmic advantage of this structure in data analysis. We discuss several approaches and heuristics for quantifying and elucidating tree-like structure in networks, including various tree-decompositions and Gromov's delta hyperbolicity. These approaches were developed with very different "tree-like" applications in mind, and thus we discuss the strengths and short-comings of each in the context of complex networks and how each might aid in identifying intermediate-scale structure in these graphs.

We construct explicit approximate Green's functions of time-dependent, linear Fokker-Planck equations in terms of Dyson series, Taylor expansions, and exact commutator formulas. Our method gives an approximate solution that is accurate to arbitrary order in time in the short-time limit, and it can be extended to large time by bootstrapping. I will also present some numerical results showing that our algorithm works well also for degenerate equations such as those arising in pricing of contingent claims. This is joint work with Victor Nistor and Wen Cheng.

Marine snow, composed of organic and inorganic aggregates, plays a major role in marine carbon cycling. Most of these macroscopic particles are extremely porous, allowing diffusion of salt from the ambient fluid to affect the density and therefore the settling of these particles. In a first approximation, these particles can be modeled as spheres. This talk will present a study of the effect of porosity and salt diffusion in the dynamics of a sphere settling under gravity in a salt-stratfied fluid analytically and semi-analytically (depending on the ambient density gradient) in viscosity dominated regimes. For linear stratification, an explicit solution for the sphere's position in time is derived. For more general ambient fluid stratification, the sphere's position can be solved for numerically, under the asymptotic assumptions about the typical time scales of diffusion and settling. A parametric study of the settling behaviors and preliminary comparisons with experiments will be presented.

This talk will survey my efforts with coworkers to develop and analyze Monte Carlo sampling algorithms for complex (usually high dimensional) probability distributions. These sampling problems are typically difficult because they have multiple high probability regions separated by low probability regions and/or they are badly scaled in the sense that there are strong unknown relationships between variables. I'll begin the talk by discussing a simple modification of the standard diffusion Monte Carlo algorithm that results in a more efficient and much more flexible tool for use, for example, in rare event simulation. If time permits I'll discuss a few other ensemble based sampling tools designed to directly address energy barriers and scaling issues.

When iterative methods are used to solve a discretized linear system for partial differential equations, the key issue is how to make the convergence fast. For different type of problems convergence mechanism can be quite different. In this talk, I will present an efficient iterative method, the fast sweeping method, for a class of nonlinear hyperbolic partial differential equation, Hamilton-Jacobi equation, which is widely used in optimal control, geometric optics, geophysics, classical mechanics, image processing, etc. We show that the fast sweeping method can converge in a finite number of iterations when monotone upwind scheme,  Gauss-Seidel iterations with causality enforcement and proper orderings are used. We  analyze its convergence, which is very different from that for iterative method for elliptic problems. If time permit I will present a new formulation to compute effective Hamiltonians for homogenization of a class of Hamilton-Jacobi equations. Both error estimate and stability analysis will be shown.

We investigate two kinetic models for active suspensions of rod-like and ellipsoidal particles, and passive suspensions of dumbbell beads dimmers, which couple a Fokker-Planck equation to the incompressible Navier-Stokes or Stokes equation. By applying cut-off techniques in the approximate problems and using compactness argument, we prove the existence of the global weak solutions with finite (relative) entropy for the two and three dimensional models. For the second model, we establish a new compact embedding theorem of weighted spaces which is the key in the compactness argument. (Joint work with Jian-Guo Liu)

Consistent reconstruction is a linear programming technique for reconstructing a signal $x\in\RR^d$ from a set of noisy or quantized linear measurements. In the setting of random frames combined with noisy measurements, we prove new mean squared error (MSE) bounds for consistent reconstruction. In particular, we prove that the MSE for consistent reconstruction is of the optimal order $1/N^2$ where $N$ is the number of measurements, and we prove bounds on the associated dimension dependent constant. For comparison, in the important case of unit-norm tight frames with linear reconstruction (instead of consistent reconstruction) the mean squared error only satisfies a weaker bound of order $1/N$. Our results require a mathematical analysis of random polytopes generated by affine hyperplanes and of associated coverage processes on the sphere. This is joint work with Alex Powell.