Geometric problems - such as finding corresponding points over a collection of shapes, or computing shape deformation under geometric constraints - pose various computational challenges. I will show that despite the very different nature of these two highly non-convex problems, Semidefinite Programming (SDP) can be leveraged to provide a tight convex approximation in both cases. A different approach is used for each problem, demonstrating the versatility of SDP: (i) For establishing point correspondences between shapes, we devise an SDP relaxation. I will show it is a hybrid of the popular spectral and doubly-stochastic relaxations, and is in fact tighter than both. (ii) For the computation of piecewise-linear mappings, we introduce a family of maximal SDP restrictions. Solving a sequence of such SDPs enables the optimization of functionals and constraints expressed in terms of singular values, which naturally model various geometry processing problems.

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.

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.

The derivation of nonlinear dispersive PDE, such as the nonlinear Schr\"{o}dinger (NLS) from many particle quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE that describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE. We will discuss the process of going from a quantum many particle system of bosons to the NLS via the GP. The most involved part in such a derivation of NLS consists in establishing uniqueness of solutions to the GP. In the talk we will focus on approaches to the uniqueness step that are motivated by the perspective coming from nonlinear dispersive PDE, including the approach that we developed with Chen, Hainzl and Seiringer based on the quantum de Finetti's theorem. Also we will look into what else the nonlinear PDE such as the NLS can tell us about the GP hierarchy, and will present recent results on infinitely many conserved quantities for the GP hierarchy that are obtained with Mendelson, Nahmod and Staffilani.

At the beginning of this talk, We will introduce the background of tensor network states (TNS) in various areas such as quantum physics, quantum chemistry and numerical partial differential equations. Famous TNS including tensor trains (TT), matrix product states (MPS), projected entangled pair states (PEPS) and multi-scale entanglement renormalization ansatz (MERA). Then We will explain how to define TNS by graphs and we will define tensor network ranks which can be used to measure the complexity of TNS. We will see that the notion of tensor network ranks is an analogue of tensor rank and multilinear rank. We will discuss basic properties of tensor network ranks and the comparison among tensor network ranks, tensors rank and multilinear rank. If time permits, we will also discuss the dimension of tensor networks and the geometry of TNS. This talk is based on papers joined with Lek-Heng Lim.

The Boltzmann equation models the evolution of the probability density of gas particles that interact through predominantly binary collisions. The equation consists of a transport operator and a collision operator. The latter is a bilinear integral with a non-integrable angular kernel. For a long time the equation was simplified by assuming that the kernel is integrable (so called Grad's cutoff), with a belief that such an assumption does not affect the equation significantly. Recently, however, it has been observed that a non-integrable singularity carries regularizing properties, which motivates further analysis of the equation in this setting. We study the behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime, by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. We show how the singularity rate of the angular kernel affects the order of tails that can be propagated. The result uses Mittag-Leffler functions, which are a generalization of exponential functions. This is based on joint works with Alonso, Gamba, Pavlovic and with Gamba, Pavlovic.

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.

I will present a survey of recent results on geometry-based image processing. The topics will include wavelet-based diffuse interface methods, pan sharpening and hyperspectral sharpening, and sparse image representation.

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.

The efficient parallelization of numerical methods for ordinary or partial differential equations in the temporal direction is an intriguing possibility that has of yet not been fully realized despite decades of investigation. For partial differential equations, virtually all large scale computations now employ parallelization across space, and there are freely available computational tools and libraries to aid in the development of spatially parallelized codes. Conversely, parallelization in the temporal direction is rarely even considered. I will discuss a relatively recent parallel strategy called the parareal algorithm that has generated a renewed wave of interest in time parallelization. I will show how the iterative structure of the parareal algorithm can be interpreted as a particular form of deferred corrections and then present a modified parareal strategy based on spectral deferred corrections that can significantly reduce the computational cost of the method. Finally I will make some observations as to why parallel in time methods may be attractive in the future.

We consider the vanishing viscosity limit of the Navier-Stokes equations in a half space, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the Navier-Stokes solutions remain bounded in $L^2_t L^\infty_x$ independently of the kinematic viscosity, and if they are equicontinuous at $x_2 = 0$. These conditions imply that there is no boundary layer separation: the Lagrangian paths originating in a boundary layer, stay in a proportional boundary layer during the time interval considered. We then give a proof of the (numerical) conjecture of vanDommelen and Shen (1980) which predicts the finite time blowup of the displacement thickness in the Prandtl boundary layer equations. This shows that the Prandtl layer exhibits separation in finite time.

There has been high scientific interest to understand the behavior of the surface quasi-geostrophic (SQG) equation because it is a possible model to explain the formation of fronts of hot and cold air and because it also exhibits analogies with the 3D incompressible Euler equations. It is not known at this moment if this equation can produce singularities or if solutions exist globally. In this talk I will discuss some recent works on the existence of global solutions.

A neuron receives thousands of synaptic inputs from other neurons and integrates them to process information. Many experimental results demonstrate this integration could be highly nonlinear, yet few theoretical analyses have been performed to obtain a precise quantitative characterization. Based on asymptotic analysis of an idealized cable model, we derive a bilinear spatiotemporal integration rule for a pair of time-dependent synaptic inputs. Note that the above rule is obtained from idealized models. However, we have confirmed this rule both in simulations of a realistic pyramidal neuron model and in electrophysiological experiments of rat hippocampal CA1 neurons. Our results demonstrate that the integration of multiple synaptic inputs can be decomposed into the sum of all possible pairwise integration with each paired integration obeying a bilinear rule.

We discuss some recent results on stability of channel flows and boundary layers expansions in the inviscid limit.

In this talk we present some observations regarding the pressure conditions leading to the vanishing of velocity in the Euler and the Navier-Stokes equations. In the case of axisymmetric 3D Euler equations with special initial data we find that the unformicity condition for the derivatives of the pressure is not consistent with the global regularity.

The theory of homogenization for equations with random coefficients is now quite well-developed. What is less studied is the theory for the correctors to homogenization, which asymptotically characterize the randomness in the solution of the equation and as such are important to quantify in many areas of applied sciences. I will present recent results in the theory of correctors for elliptic and parabolic problems and briefly mention how such correctors may be used to improve reconstructions in inverse problems. Homogenized (deterministic effective medium) solutions are not the only possible limits for solutions of equations with highly oscillatory random coefficients as the correlation length in the medium converges to zero. When fluctuations are sufficiently large, the limit may take the form of a stochastic equation and stochastic partial differential equations (SPDE) are routinely used to model small scale random forcing. In the very specific setting of a parabolic equation with large, Gaussian, random potential, I will show the following result: in low spatial dimensions, the solution to the parabolic equation indeed converges to the solution of a SPDE, which however needs to be written in a (somewhat unconventional) Stratonovich form; in high spatial dimension, the solution to the parabolic equation converges to a homogenized (hence deterministic) equation and randomness appears as a central limit-type corrector. One of the possible corollaries for this result is that SPDE models may indeed be appropriate in low spatial dimensions but not necessarily in higher spatial dimensions.

The Explicit Jump Immersed Interface Method (EJIIM) is a finite difference method for elliptic partial differential equations that, like all Immersed Interface Methods, works on a regular grid in spite of non-grid aligned discontinuities in equation parameters and solution. The specific idea is to introduce jumps in function and its derivatives explicitely as additional variables. We present a finite difference based EJIIM for the stationary Stokes flow in saddle point formulation. Challenges related to staggered grid, fast Stokes solver and non-simply connected domains will be discussed.

In the first part of the talk, we will discuss a numerical method for wave propagation in inhomogeneous media. The Trefftz method relies on basis functions that are solution of the homogeneous equation. In the case of variable coefficients, basis functions are designed to solve an approximation of the homogeneous equation. The design process yields high order interpolation properties for solutions of the homogeneous equation. This introduces a consistency error, requiring a specific analysis.
In the second part of the talk, we will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudo-spectral algorithms were developed for inverting the Laplace-Beltrami operator and computing the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, well-conditioned solvers for the Maxwell equations will rely on these algorithms.

Mathematical models of complex physical processes often involve large number of degrees of freedom as well as events occurring on different time scales. Therefore, direct simulations based on these models face tremendous challenge. This focus of this talk is on the Mori-Zwanzig (MZ) projection formalism for reducing the dimension of a complex dynamical system. The goal is to mathematically derive a reduced model with much fewer variables, while still able to capture the essential properties of the system. In many cases, this formalism also eliminates fast modes and makes it possible to explore events over longer time scales. The models that are directly derived from the MZ projection are typically too abstract to be practically implemented. We will first discuss cases where the model can be simplified to generalized Langevin equations (GLE). Furthermore, we introduce systematic numerical approximations to the GLE, in which the fluctuation-dissipation theorem (FDT) is automatically satisfied. More importantly, these approximations lead to a hierarchy of reduced models with increasing accuracy, which would also be useful for an adaptive model refinement (AMR). Examples, including the NLS, atomistic models of materials defects, and molecular models of proteins, will be presented to illustrate the potential applications of the methods.

Phylogenetics is a science that aims at reconstructing the history of evolution. Phylogenetic tree models are generalizations of well-known Markov chains. In my talk I will present so-called group-based models and their relations to algebra and combinatorics. To a model of evolution one associates an algebraic variety that is the Zariski closure of points corresponding to probability distributions allowed by the model. Many important varieties arise by this construction, e.g. secant varieties of Segre products of projective spaces. It turns out that group-based models provide toric varieties. In particular, they may be studied using tools from toric geometry relating to combinatorics of lattice polytopes.

An introduction to the topic of multi-dimensional optical solitons ("light bullets"), localized simultaneously in the direction of propagation (as temporal solitons) and in one or two transverse directions (as spatial solitons) will be given, including a review of basic theoretical and experimental results. Also considered will be connection of this topic to the problem of the creation of multidimensional solitons in Bose-Einstein condensates. In both settings (optical and BEC), the main problem is stabilization of the multidimensional solitons against the spatiotemporal collapse. The stabilization may be provided in various ways (in particular, by means of an optical lattice in BEC). The talk will partly based on a review article: B.A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons", J. Optics B: Quant. Semics. Opt. 7, R53-R72 (2005).

The forces generated by moving interfaces usually include the parts due to tangential stretching. We derive the evolution equation for the tangential stretching, which then forms the basis for an Eulerian formulation based on level set functions. The jump conditions are then derived using the level set and stretch functions. The derived jump conditions under this Eulerian formulation are clean. This makes possible a local level set method for immersed interface method to simulate membranes or vesicles where the tangential forces are present. This is a continuation of one piece of my work during my Ph.D. study.

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.

We study a two-parameter family of solutions of the surface Euler equations in which solutions return to a spatial translation of their initial condition at a later time. Pure standing waves and pure traveling waves emerge as special cases at fixed values of one of the parameters. We find many examples of wave crests that nearly sharpen to a corner, with corner angles close to 120 degrees near the traveling wave of greatest height, and close to 90 degrees for large-amplitude pure standing waves. However, aside from the traveling case, we do not believe any of these solutions approach a limiting extreme wave that forms a perfect corner. We also compute nonlinear wave packets, or breathers, which can take the form of NLS-type solitary waves or counterpropagating wave trains of nearly equal wavelength. In the latter case, an interesting phenomenon occurs in which the pressure develops a large DC component that varies in time but not space, or at least varies slowly in space compared to the wavelength of the surface waves. These large-scale pressure zones can move very rapidly since they travel at the envelope speed, and may be partially responsible for microseisms, the background noise observed in earthquake seismographs.

In many physical and biological systems, there are some competing effects such as focus and de-focus, attraction and repulsion, spread and concentration. These competing effects usually are represented by terms with different signs in a free energy. The dynamics of the physical system sometimes can be described by a gradient flow driven by the free energy. Some functional inequalities can be used to determine the domination among these competing effects in the free energy, and provided sharp conditions on initial data or coefficients in the system for the global existence. In this talk, we will introduce some important relations between functional inequalities and sharp conditions for the global existence. For example, the Hardy-Littlewood-Sobolev inequality vs parabolic-elliptic Keller-Segel model, Onofri's inequality vs parabolic-parabolic Keller-Segel model, and Sz. Nagy inequality vs 1-D thin film equation, and provide the results on the global existence and blow-up for above models under sharp conditions. Moreover, we obtain the uniqueness of the weak solution for the linear diffusion Keller-Segel model using the refined hyper-contractivity of the $L^p$ of the solution under the sharp initial condition, and prove the $L^{\infty}$ estimate of the solution utilizing the bootstrap method. We also provide some results on existence of the global smooth solution.

Epitaxial growth is a process in which a thin film is grown above a much thicker substrate. In the simplest case, no deposition is considered, and all the interactions are assumed to be purely elastic. However, since the film may potentially have different rigidity constant from the substate, such growth leads to a nonuniform film thickness. The equations governing epitaxial growth are high order (generally fourth order), nonlocal, and highly nonlinear. In this talk I will present some recent results about the regularity of solutions to several equations arising from epitaxial growth. Joint work with I. Fonseca and G.Leoni.

PART 1: I will construct a mathematical model of a defect embedded in an infinite homogeneous crystal. I will then establish a regularity result for minimisers, which given the crucial information on which approximation schemes are based. As an elementary application of this framework I will prove convergence rates for two computational schemes: (1) clamped far-field and (2) coupling to harmonic far-field model.
PART 2: The conditions under which the theory of Part 1 holds are separability and locality of the total energy. In Part 2 I will show how for a tight-binding model (a minimalistic electronic structure model) these two condition arise. This analysis raises some interesting (open) questions.
PART 3: Finally, I will use the theory developed in PART 1 and PART 2 to construct and analyse a new family of QM/MM embedding schemes with rigorous error estimates.