We discuss one approach to determining the hazard threat to a locale due to a large volcanic avalanche. The methodology employed includes large-scale numerical simulations, field data reporting the volume and runout of flow events, and a detailed statistical analysis of uncertainties in the modeling and data. The probability of a catastrophic event impacting a locale is calculated, together with a estimate of the uncertainty in that calculation. By a careful use of simulations, a hazard map for an entire region can be determined. The calculation can be turned around quickly, and the methodology can be applied to other hazard scenarios.

Out-of-equilibrium behavior is the characteristic feature of the long-time dynamics of nonlinear dispersive equations on compact domain. This means that solutions typically do not exhibit any form of long-time stability near equilibrium solutions or configurations. We shall survey several aspects of this behavior both from a dynamical systems and statistical mechanics point of view.

In this talk, I will survey the recent understandings on the motion of water waves obtained via rigorous mathematical tools, this includes the evolution of smooth initial data and some typical singular behaviors. In particular, I will present our recently results on gravity water waves with angled crests.

A remarkable model of stochastic coalescence arises from considering shock statistics in scalar conservation laws with random initial data. While originally rooted in the study of Burgers turbulence, the model has deep connections to statistics, kinetic theory, random matrices, and completely integrable systems. The evolution takes the form of a Lax pair which, in addition to yielding interesting conserved quantities, admits some rather intriguing exact solutions. We will describe several distinct derivations for the evolution equation and, time-permitting, discuss properties of the corresponding kinetic system. This talk consists of joint work with Govind Menon (Brown).

Developing algorithms for solving high-dimensional partial differential equations (PDEs) and forward-backward stochastic differential equations (FBSDEs) has been an exceedingly difficult task for a long time, due to the notorious difficulty known as the curse of dimensionality. In this talk we introduce a new type of algorithms, called "deep BSDE method", to solve general high-dimensional parabolic PDEs and FBSDEs. Starting from the BSDE formulation, we approximate the unknown Z-component by neural networks and design a least-squares objective function for parameter optimization. Numerical results of a variety of examples demonstrate that the proposed algorithm is quite effective in high-dimensions, in terms of both accuracy and speed. We furthermore provide a theoretical error analysis to illustrate the validity and property of the designed objective function.

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.

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.

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.

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.

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.

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.

We discuss the geometry of Laplacian eigenfunctions on compact manifolds and combinatorial graphs. The `dual' geometry of Laplacian eigenfunctions is well understood on the torus and euclidean space, and is of tremendous importance in various fields of pure and applied mathematics. The purpose of this talk is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of `similarity' between eigenfunctions is given by a global average of local correlations, and its relationship to pointwise products. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result. This talk will also focus on the applications of discovering such a dual geometry, namely in constructing anisotropic graph wavelet packets and anisotropic graph cuts.

We will consider some inverse coefficient problems to system of linear and semilinear diffusion equations where the aim is to recover unknown parameters in the equations from partial information on the solutions to the systems. We present some recent theoretical and numerical results, and point out possible applications of such problems in imaging.

The upper respiratory tract is the portal of entry for inhaled air and anything we breath in with it. For most of us, the nasal passages do most of the work cleansing, humidifying, and warming inhaled air using a lining of highly vascularized tissue coated with mucus. This tissue is susceptible to damage from inhaled material, can adversely affect life quality if deformed or diseased, and is a potential route of systemic exposure via circulating blood. To understand nasal physiology and the effects of inhalants on nasal tissue, information on airflow, gas uptake and particle deposition patterns is needed for both laboratory animals and humans. This information is often difficult to obtain in vivo but may be estimated with three-dimensional computational fluid dynamics (CFD) models. At CIIT Centers for Health Research (CIIT-CHR), CFD models of nasal airflow and inhaled gas and particle transport have been used to test hypotheses about mechanisms of toxicity, help extrapolate laboratory animal data to people, and make predictions for human health risk assessments, as well as study surgical interventions and nasal drug delivery. In this talk an overview of CIIT-CHR's nasal airflow modeling program will be given with the goal of illustrating how CFD modeling can help researchers clarify, organize, and understand the complex structure, function, physiology, pathobiology, and utility of the nasal airways.

The method of geometric optics for incompressible Euler equations is used to study localized shortwave instabilities in ideal fluids. In linear approximation evolution of the shortwave ansatz can be described by a finite dimensional skew-product flow which determines all the linear instabilities coming from essential spectrum. In this talk we will discuss mathematical description of the method, aspects related to vanishing viscosity limit and application to the problem of inherent instability of ideal fluid flows.

The incompressible Euler equation of fluid mechanics has been derived in 1755. It is one of the central equations of applied analysis, yet due to its nonlinearity and non-locality many fundamental properties of solutions remain poorly understood. In particular, the global regularity vs finite time blow up question for incompressible three dimensional Euler equation remains open. In two dimensions, it has been known since 1930s that solutions to Euler equation with smooth initial data are globally regular. The best available upper bound on the growth of derivatives of the solution has been double exponential in time. I will describe a construction showing that such fast generation of small scales can actually happen, so that the double exponential bound is qualitatively sharp. This work has been motivated by numerical experiments due to Hou and Luo who propose a new scenario for singularity formation in solutions of 3D Euler equation. The scenario is axi-symmetric. The geometry of the scenario is related to the geometry of 2D Euler double exponential growth example and involves hyperbolic points of the flow located at the boundary of the domain. If time permits, I will discuss some recent attempts to gain insight into the three-dimensional fluid behavior in this scenario.

Analyzing and inferring the underlying global intrinsic structures of data from its local information are critical in many fields. In practice, coherent structures of data allow us to model data as low dimensional manifolds, represented as point clouds, in a possible high dimensional space. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data sets are far more challenging due to their complicated geometry. For example, the same geometric object can take very different coordinate representations due to the variety of embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). These ambiguities form an infinite dimensional isometric group and make higher-level tasks in manifold-structured data analysis and understanding even more challenging. To overcome these ambiguities, I will first discuss modeling based methods. This approach uses geometric PDEs to adapt the intrinsic manifolds structure of data and extracts various invariant descriptors to characterize and understand data through solutions of differential equations on manifolds. Inspired by recent developments of deep learning, I will also discuss our recent work of a new way of defining convolution on manifolds and demonstrate its potential to conduct geometric deep learning on manifolds. This geometric way of defining convolution provides a natural combination of modeling and learning on manifolds. It enables further applications of comparing, classifying and understanding manifold-structured data by combing with recent advances in deep learning.

In this talk, we will explore stochastic motion of cellular cycles inside CW complexes. This serves as a generalization of random walks on graphs, and a discretization of stochastic flows on smooth manifolds. We will define a notion of stochastic current, connect it to classical electric current, and show it satisfies a quantization result. Along the way, we will define the main combinatorial objects of study, namely spanning trees and spanning co-trees in higher dimensions. We will relate these to stochastic current, as well as discrete Hodge theory.

Fluid velocities and Brownian effects at nanoscales in the near-wall r egion of microchannels can be experimentally measured in an image plane parallel to the wall, using for example, an evanescent wave illumination technique combi ned with particle image velocimetry [R. Sadr et al., J. Fluid Mech. 506, 357-367 (2004)]. Tracers particles are not only carried by the flow, but they undergo r andom fluctuations, the details of which are affected by the proximity of the wa ll. We study such a system under a particle based stochastic approach (Langevin) . We present the modeling assumptions and pay attention to the details of the si mulation of a coupled system of stochastic differential equations through a Mils tein scheme of strong order of convergence 1. Then we demonstrate that a maximum likelihood algorithm can reconstruct the out-of-plane velocity profile, as spec ified velocities at multiple points, given known mobility dependence and perfect mean measurements. We compare this new method with existing cross-correlation t echniques and illustrate its application for noisy data. Physical parameters are chosen to be as close as possible to the experimental parameters.

We study the large-time behavior of hydrodynamic model which describes the collective behavior of continuum of agents, driven by pairwise alignment interactions with additional external potential forcing. The external force tends to compete with alignment which makes the large time behavior very different from the original Cucker-Smale (CS) alignment model, and far more interesting. Here we focus on uniformly convex potentials. In the particular case of \emph{quadratic} potentials, we are able to treat a large class of admissible interaction kernels, $\phi(r) \gtrsim (1+r^2)^{-\beta}$ with `thin' tails $\beta \leq 1$ --- thinner than the usual `fat-tail' kernels encountered in CS flocking $\beta\leq\nicefrac{1}{2}$: we discover unconditional flocking with exponential convergence of velocities \emph{and} positions towards a Dirac mass traveling as harmonic oscillator. For general convex potentials, we impose a necessary stability condition, requiring large enough alignment kernel to avoid crowd dispersion. We prove, by hypocoercivity arguments, that both the velocities \emph{and} positions of smooth solution must flock. We also prove the existence of global smooth solutions for one and two space dimensions, subject to critical thresholds in initial configuration space. It is interesting to observe that global smoothness can be guaranteed for sub-critical initial data, independently of the apriori knowledge of large time flocking behavior.

We consider the heat equation with a random potential in dimensions d>=3, and show that the large scale random fluctuations are described by the Edwards-Wilkinson model with the renormalized diffusivity and variance. This is based on a joint work with Lenya Ryzhik and Ofer Zeitouni.

In this talk, I will start with a simple algorithm for community detection in stochastic block models and discuss its statistical optimality. After that, we will discuss two related issues. One is model selection for stochastic block models. The other is the extension to community detection in degree-corrected block models. We shall pay close attention to the achievability of statistical optimality by computationally feasible procedures throughout the talk.

The talk is concerned with a matrix-valued Boltzmann equation derived from the Fermi-Hubbard or Bose-Hubbard model for weak interactions. The quantum analogue of the classical distribution function is the Wigner function, which is matrix-valued to accommodate the spin degree of freedom. Conservation laws and the H-theorem can be proven analytically, and numerical simulations illustrate the time dynamics.

In this talk, we attempt to derive some experimentally verifiable physical laws of nature based only on a few fundamental first principles. First, we present two basic principles, the principle of interaction dynamics (PID) and the principle of representation invariance (PRI). Intuitively, PID takes the variation of the action under energy-momentum conservation constraint. PID offers a completely different and natural way of introducing Higgs fields. For gravity, we show that PID is the direct consequence of Einstein’s principle of general relativity and the presence of dark matter and dark energy. PRI requires that the SU(N) gauge theory be independent of representations of SU(N). PRI has remarkably rich physical consequences. Second, we show that the physical laws of the four fundamental forces—gravity, electromagnetic force, weak and strong forces—are dictated by 1) the Einstein principle of general relativity, 2) the principle of gauge symmetry, 3) PID, and 4) PRI. The new theory will lead to solutions to a number of longstanding problems in particle physics and cosmology. The talk is based on recent joint work with Tian Ma.

The Vicsek model, describing alignment and self-organisation in large systems of self-propelled particles, such as fish schools or flocks of birds, has attracted a lot of attention with respect to its simplicity and its ability to reproduce complex phenomena. We consider here a time-continuous version of this model, in the spirit of the one proposed by P. Degond and S. Motsch, but where the rate of alignment is proportional to the mean speed of the neighboring particles. In the hydrodynamic limit, this model undergoes a phase transition phenomenon between a disordered and an ordered phase, when the local density crosses a threshold value. We present the two different macroscopic limits we can obtain under and over this threshold, namely a nonlinear diffusion equation for the density, and a first-order non-conservative hydrodynamic system of evolution equations for the local density and orientation. (joint work with Pierre Degond and Jian-Guo Liu).

Density Functional Theory (DFT) and methods based on it are the primary production methods for electronic-structure based "first principles" simulations in materials science today. This talk focuses on the anatomy of the FHI-aims code: an all-electron implementation of DFT that makes no a priori shape approximations to the potential or solutions (orbitals), yet implements the necessary algorithms in a way that scales up to thousands of atoms and on massively parallel computers with (ten)thousands of cores for routine simulations. Particularly important developments include a scalable, massively parallel dense eigenvalue solver "ELPA" and a framework to expand the (expensive) two-electron Coulomb operator in a linear-scaling localized resolution of identity framework for large-scale calculations.