Coarsening is the phenomenon where many objects (water drops, molecular islands, particles in a freezing liquid) becoming smaller in number but larger in size in an orderly way. This talk will examine modeling one such system, nanoscopic liquid drops, through three models: a PDE for the fluid, a coarsening dynamical system for the drops, and an LSW-type ensemble model for the distribution of drops. We will find self-similar solutions for the drop population valid for intermediate times and discuss transient effects that can delay the self-similar scaling.

The idea of using nonlinear approximations as a tool for solving equations is as natural as that of using bases and, in fact, was proposed in 1960 in the context of quantum chemistry. The usual approach to solving partial differential and integral equations is to select a basis (possibly a multiresolution basis) or a grid, project equations onto such basis and solve the resulting discrete equations. The nonlinear alternative is to look for the solution within a large lass of functions (larger than any basis) by constructing optimal or near optimal approximations at every step of an algorithm for solving the equations. While this approach can theoretically be very efficient, the difficulties of constructing optimal approximations prevented any significant use of it in practice. However, during the last 15 years, nonlinear approximations have been successfully used to approximate operator kernels via exponentials or Gaussians to any user-specified accuracy, thus enabling a number of multidimensional multiresolution algorithms. In a new development several years ago, we constructed a fast and accurate reduction algorithm for optimal approximation of functions via exponentials or Gaussians (or, in a dual form, by rational functions) than can be used for solving partial differential and integral equations equations. We present two examples of the resulting solvers: one for the viscous Burgers' equation and another for solving the Hartree-Fock equations of quantum chemistry. Burgers' equation is often used as a testbed for numerical methods: if the viscosity \vu; is small, its solutions develop sharp (moving) transition regions of width O (\vu) presenting significant challenges for numerical methods. Using nonlinear approximations for solving the Hartree-Fock equations is the first step to a wider use of the approach in quantum chemistry. We maintain a functional form for the spatial orbitals as a linear combinations of products of decaying exponentials and spherical harmonics entered at the nuclear cusps. While such representations are similar to the classial Slater-type orbitals, in the course of computation we optimize both the exponents and the coefficients in order to achieve an efficient representation of solutions and to obtain guaranteed error bounds.

In radio signal encoding and decoding, frequency modulation (FM) has several advantages over amplitude modulation (AM) - we all enjoy the high fidelity and nearly static free reception of FM radio. Sensing and imaging can also be approached with AM or FM modalities. All imaging methods practiced today are AM implementations, and FM for imaging has never been explored or its advantages exploited. In this talk I'll introduce the FM approach to sensing and imaging in its infant form. I'll show that the FM approach is closely related to design of Gaussian quadratures for bandlimited functions. I'll demonstrate the superiorities of the FM approach over AM by proposing three FM methods to deal with Gibbs phenomenon encountered in imaging.

For a more detailed abstract, see http://www.math.duke.edu/~jonm/yuChen.html

The kernel-free boundary integral method is a Cartesian grid based method for solving elliptic partial differential equations (PDEs). It solves elliptic PDEs in the framework of boundary integral equations (BIEs). The method evaluates boundary and volume integrals by solving equivalent simple interface problems on Cartesian grids. It takes advantages of the well-conditioning properties of the BIE formulation, the convenience of grid generation with Cartesian grids and the availability of fast and efficient elliptic solvers for the simple interface problems. In this talk, I will present recent developments of the method for the reaction-diffusion equations in computational cardiology, the nonlinear Poisson-Boltzmann equation in biophysics, the Stokes equation in fluid dynamics as well as some free boundary and moving interface problems.

Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\mathbb{R}^n$. We show that the random vector $Y=T(X)$ satisfies $$ \mathbb{E} \sum_{j=1}^k j\mbox{-}\min_{i \leq n} {X_{i}}^2 \leq C \mathbb{E} \sum_{j=1}^k j\mbox{-}\min_{i\leq n}{Y_{i}}^2 $$ for all $k \leq n$, where $ j\mbox{-}\min$ denotes the $j$-th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the Karhunen--Lo`eve basis for the nonlinear reconstruction. We also show some relations for order statistics of random vectors (not only Gaussian), which are of independent interest. This is a joint work with Konstantin Tikhomirov.

A new numerical method being developed with Charles Peskin is described which simulates interacting fluid, membrane, and particle systems in which thermal fluctuations play an important role. This method builds on the "Immersed Boundary Method" of Peskin and McQueen, which simplifies the coupling between the fluid and the immersed particles and membranes in such a way as to avoid complex boundary problems. Thermal fluctuations are introduced in the fluid through the theory of statistical hydrodynamics. We discuss some approximate analytical calculations which indicate that the immersed particles should exhibit some physically correct properties of Brownian motion. Our intended use of this numerical method is to simulate microphysiological processes; one advantage this method would have over Langevin particle dynamics approaches is its explicit tracking of the role of the fluid dynamics.

In phase retrieval, one aims to recover a signal from magnitude measurements. In the literature, an effective SDP algorithm, referred to as PhaseLift, was proposed with numerical success as well as strong theoretical guarantees. In this talk, I will first introduce some recent theoretical developments for PhaseLift, which demonstrate the applicability and adaptivity of this convex method. Although convex methods are provably effective and robust, the computational complexity may be relatively high. Moreover, there is often an issue of storage to solve the lifted problem. To address these issues, we introduce a nonconvex optimization algorithm, named Wirtinger flow, with theoretically guaranteed performance. It is much more efficient than convex methods in terms of computation and memory. Finally, I will introduce how to modify Wirtinger flow when the signal is known to be sparse, in order to improve the accuracy of the recovery.

In this talk, we settle a long-standing problem on the connectivity of spaces of finite unit norm tight frames (FUNTFs), essentially affirming a conjecture first appearing in Dykema and Strawn (2003). Our central technique involves continuous liftings of paths from the polytope of eigensteps (see Cahill et al. (2012)) to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense, and that generic FUNTFs are full spark.

In this talk, we will present some recent work on dispersive estimates for wave equations on Schwarzschild black hole backgrounds. We in particular will discuss Strichartz estimates and localized energy estimate. This is from a joint work with Jeremy Marzuola, Daniel Tataru, and Mihai Tohaneanu.

$\ell_1$ regularization for sparsity has played important role in recent developments in many fields including signal processing, statistics, optimization. The concept of sparsity is usually for the coefficients (i.e., only a small set of coefficients are nonzero) in a well-chosen set of modes (e.g. a basis or dictionary) for representation of the corresponding vectors or functions. In this talk, I will discuss our recent work on a new use of sparsity-promoting techniques to produce “compressed modes/compress plane waves" - modes that are sparse and localized in space - for efficient solutions of constrained variational problems in mathematics and physics. In particularly, I will focus on L1 regularized variational Schrodinger equations for creating spatially localized modes and orthonormal basis, which can efficiently represent localized functions and has promising potential to a variety of applications in many fields such as signal processing, solid state physics, materials science, etc. (This is a joint work with Vidvuds Ozolins, Russel Caflisch and Stanley Osher)

(joint with Mboyo Esole) String theory considerations lead to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. After giving a very sketchy idea of the physics arguments leading to this identity, I will present a Chern class identity which confirms it, generalizing it to arbitrary dimension and to varieties that are not necessarily Calabi-Yaus. The relevant loci are singular, and this plays a key role in the identity.

In this talk, I will present the optimal transport theory on discrete spaces. Various recent developments related to free energy, Fokker-Planck equations, as well as Wasserstein distance on graphs will be presented, some of them are rather surprising. Applications in robotics as well as Schrodinger equation on graphs will be demonstrated.

We present an overview of experiments, numerical simulations, and mathematical analysis of the breakup of a low viscosity drop in a viscous fluid, and consider the role of surface contaminants, or surfactants, on the dynamics near breakup. As part of our study, we address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant that occurs in the important limit of very large values of bulk Peclet number Pe. At the high values of Pe in typical fluid-surfactant systems, there is a narrow transition layer near the drop surface or interface in which the surfactant concentration varies rapidly, and its gradient at the interface must be determined accurately to find the drop’s dynamics. Accurately resolving the layer is a challenge for traditional numerical methods. We present recent work that uses the narrowness of the layer to develop fast and accurate `hybrid’ numerical methods that incorporate a separate analytical reduction of the dynamics within the transition layer into a full numerical solution of the interfacial free boundary problem.

Nonparametric density estimation is a challenging problem in theoretical statistics -- in general the maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints allows a path forward. This talk offers an invitation to non-parametric density estimation under total positivity (i.e. log-supermodularity) and log-concavity. Totally positive random variables are ubiquitous in real world data and possess appealing mathematical properties. Given i.i.d. samples from such a distribution, we prove that the maximum likelihood estimator under these shape constraints exists with probability one. We characterize the domain of the MLE and show that it is in general larger than the convex hull of the observations. If the observations are 2-dimensional or binary, we show that the logarithm of the MLE is a tent function (i.e. a piecewise linear function) with "poles" at the observations, and we show that a certain convex program can find it. In the general case the MLE is more complicated. We give necessary and sufficient conditions for a tent function to be concave and supermodular, which characterizes all the possible candidates for the MLE in the general case.

Many partial differential equation models arising in applications generate complex patterns evolving with time which are hard to quantify due to the lack of any underlying regular structure. Such models often include some element of stochasticity which leads to variations in the detail structure of the patterns and forces one to concentrate on rougher common geometric features. From a mathematical point of view, algebraic topology suggests itself as a natural quantification tool. In this talk I will present some recent results for both the deterministic and the stochastic Cahn-Hilliard equation, both of which describe phase separation in alloys. In this situation one is interested in the geometry of time-varying sub-level sets of a function. I will present theoretical results on the pattern formation and dynamics, show how computational homology can be used to quantify the geometry of the patterns, and will assess the accuracy of the homology computations using probabilistic methods.

Finite difference methods are awkward for solving boundary value problems, such as the Dirichlet problem, with general boundaries, but they are well suited for interface problems, which have prescribed jumps in an unknown across a general interface or boundary. The two problems can be connected through potential theory: The Dirichlet boundary value problem is converted to an integral equation on the boundary, and the integrals can be thought of as solutions to interface problems. Wenjun Ying et al. have developed a practical method for solving the Dirichlet problem, and more general ones, by solving interface problems with finite difference methods and iterating to mimic the solution of the integral equation. We will describe some analysis which proves that a simplified version of Ying's method works. A recent view of classical potential theory leads to a finite difference version of the theory in which, remarkably, the crude discrete operators have much of the structure of the exact operators. This simplified method produces the Shortley-Weller solution of the Dirichlet problem. Details can be found at arxiv.org/abs/1803.08532 .

Our knowledge of the long-time behavior of 2d inviscid flows is quite limited. There are some appealing conjectures based on ideas in Statistical Mechanics, but they appear to be beyond reach of the current methods. We will discuss some partial results concerning the dynamics, as well as some results for variational problems to which the Statistical Mechanics methods lead.

A chain of pendula connected by nearest neighbor coupling is a near-- integrable system if the coupling is weak. As a consequence of KAM, for most initial data the energy of each pendulum stays near its initial value for all time. We show that these KAM motions coexist with ``diffusing" motions for which the energy can leak from any pendulum to any other pendulum, and it can do so with a prescribed itinerary. This is joint work with Vadim Kaloshin.

Granular materials are common in everyday life but are historically difficult to model. This has direct ramifications owing to the prominent role granular media play in multiple industries and terrain dynamics. One can attempt to track every grain with discrete particle methods, but realistic systems are often too large for this approach and a continuum model is desired. However, granular media display unusual behaviors that complicate the continuum treatment: they can behave like solid, flow like liquid, or separate into a "gas", and the rheology of the flowing state displays remarkable subtleties that have been historically difficult to model. To address these challenges, in this talk we develop a family of continuum models and solvers, permitting quantitative modeling capabilities for a variety of applications, ranging from general problems to specific techniques for problems of intrusion, impact, driving, and locomotion in grains.

To calculate flows in general cases, a rather significant nonlocal effect is evident, which is well-described with our recent nonlocal model accounting for grain cooperativity within the flow rule. This model enables us to capture a number of seemingly disparate manifestations of particle size-effects in granular flows including: (i) the wide shear-band widths observed in many inhomogeneous flows, (ii) the apparent strengthening exhibited in thin layers of grains, and (iii) the fluidization observed due to far-away motion of a boundary. On the other hand, to model only intrusion forces on submerged objects, we will show, and explain why, many of the experimentally observed results can be captured from a much simpler tension-free frictional plasticity model. This approach gives way to some surprisingly simple general tools, including the granular Resistive Force Theory, and a broad set of scaling laws inherent to the problem of granular locomotion. These scalings are validated experimentally and in discrete particle simulations suggesting a new down-scaled paradigm for granular locomotive design, on earth and beyond, to be used much like scaling laws in fluid mechanics.

We consider the basis pursuit problem: find the solution of an underdetermined system $Ax=y$ that minimizes the $l_1$-norm. We formulate a min-max principle (that, as we learned, actually goes back to 1970's) based on a Largange multiplier, and propose an iterative shrinkage-thresholding type algorithm that seems to work quite well. We show that the numerical algorithm converges to the exact solution of the basis pursuit problem. We also discuss its application to array imaging in wave propagation. The analysis is based on ODE techniques, regularization and energy methods. This is a joint work with M. Moscoso, A. Novikov and G. Papanicolaou.

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.

Electrophysiological recordings of neurons in the cortex of mammals reveal a ubiquitous high degree of irregularity of single neuron activity. The mechanisms and functional role of this irregular activity remain the subject of debate. Here, I will describe simplified models of networks of neurons, and analytical tools that can be used to understand their dynamics. Under some conditions, such networks can be described using a system of coupled Fokker-Planck equations (one for each class of neurons composing the network), in which the drift and diffusion terms depend on the probability flux at firing threshold. Provided specific conditions on network connectivity are satisfied, these models reproduce some of the landmark features observed in experiments (highly irregular firing at low rates, weak correlations between neurons, wide distributions of firing rates). Interestingly, these networks show a rich diversity of irregular states (chaotic or not, asynchronous or synchronous).

When a fluid layer is heated from below with temperature larger than a certain critical value, the convective motion appears in the fluid. The convection caused by the thermocapillary effect is called Benard-Marangoni heat convection. The thermocapillary effect is the dependence of the surface tension on the temperature. Near a hot spot on a free surface of fluid a thermocapillary tangential stress generates a fluid motion. In this talk the mathematical model system for this convection is explained. The Oberbeck-Boussinesq approximation is used for the system and the upper boundary is a free surface with surface tension which depends on the temperature. After formulating the linearized problem around the conductive state, we derive the resolvent estimates which guarantee the sectorial property. Stationary and Hopf bifurcations (periodic solutions) are proved to exist depending on the parameters (Raylegh and Marangoni numbers).

The Fermi-Pasta-Ulam (FPU) model of coupled anharmonic oscillators has long been of interest in nonlinear science. It is only recently (Friesecke and Wattis 1994, Frieseck and Pego 1999-2003, and Mizumachi (submitted)) that the existence and stability of solitary waves in FPU has been completely understood. In light of the fact that the Korteweg-deVries (KdV) equation may recovered as a long wave limit of FPU and that the theory of soliton interaction is both beautiful and completely understood in KdV, it is of interest to describe the interaction of two colliding solitary waves in the FPU model. We show that the FPU model contains an open set of solutions which remain close to the linear sum of two long wave low amplitude solitions as time goes to either positive or negative infinity.

We are interested in large systems of agents collectively looking for a consensus (about e.g. their direction of motion, like in bird flocks). In spite of the local character of the interactions (only a few neighbours are involved), these systems often exhibit large scale coordinated structures. The understanding of how this self-organization emerges at the large scale is still poorly understood and offer fascinating challenges to the modelling science. We will discuss a few of these issues among (time permitting) phase transitions, propagation of chaos and the derivation of macroscopic models.