Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches. In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs. We present recent results on properties of these equilibria and long-time asymptotics of solutions in the setting where attractive and repulsive forces are in competition. In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.

The study of various questions about networks have increased dramatically in recent years across a number of areas of application, including communications, sociology, and phylogenetic biology. Important questions about communities and groupings in networks have led to a number of competing techniques for identifying communities, structures and hierarchies. We discuss results about the networks of (1) NCAA Division I-A college football matchups and (2) committee assignments in the U.S. House of Representatives. In college football, the underlying structure of the network strongly influences the computer rankings that contribute to the Bowl Championship Series standings. In Congress, the changes of the hierarchical structure from one Congress to the next can be used to investigate major political events, such as the "Republican Revolution" of 1994 and the introduction of the Select Committee on Homeland Security following September 11th. While many structural elements in each case are seemingly robust, we include attention to variations across identification algorithms as we investigate the roles of such structures.

Coupling between free fluid flow and flow through porous media is important in many industrial applications, such as filtration, underground water flow in hydrology, oil recovery in petroleum engineering, fluid flow through body tissues in biology, to name a few.

Stokes flows appear in many applications where the fluid viscosity is high and/or the velocity and length scales are small. The flow through a porous medium can be described by Darcy's law. A region that contains both requires a careful coupling of these different systems at the interface through appropriate boundary conditions.

Our objective is to develop a method based on the boundary integral formulation for computing the fluid/porous medium problem with higher accuracy using fundamental solutions of Stokes and Darcy's equations. We regularize the kernels to remove the singularity for stability of numerical calculations and eliminate the largest error for higher accuracy.

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.

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.

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 complexity of solving the time-dependent wave equation via traditional methods scales faster than linearly in the complexity of the initial data. This behavior is mostly due to the necessity of timestepping at the CFL level, and is hampering the resolution of large-scale inverse scattering problems such as reflection seismology, where massive datasets need to be processed. In this talk I will report on some algorithmic progress toward time upscaling of the wave equation using discrete symbol calculus for pseudodifferential and Fourier integral operators. Joint work with Lexing Ying from UT Austin.

I will present several extremely fast algorithms for recovering a compressible signal from a few linear measurements. These examples span a variety of orthonormal bases, including one large redundant dictionary. As part of the presentation of these algorithms, I will give an explanation of the crucial role of group testing in each algorithm.

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.

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

Parkinson's Disease (PD) is the most common movement disorder in the United States, with symptoms due to progressive loss of neurons within the basal ganglia. In the first part of the talk, we present and analyze a minimal model for the lack of cross-correlations in neuronal activity in the healthy basal ganglia.

The second part of the talk focuses on experimentally-observed changes in neuronal firing patterns that accompany PD and that may result in the motor symptoms. We have constructed a neuronal network model for the increases in correlated activity within the basal ganglia following the onset of PD. We then apply dynamical systems methods to understand transitions between irregular and rhythmic, correlated firing in the model. Geometric singular perturbation theory and one-dimensional maps are used to understand how an excitatory-inhibitory neuronal network with fixed architecture can generate both activity patterns for possibly different values of the intrinsic and synaptic parameters. We discuss hypotheses arising from the model as well as ongoing experiments to test these predictions.

We present several versions of non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for (a) multiple scattering configurations, (b) bounded composite scatterers with piecewise constant material properties, and (c) layered media. We show that DDM solvers give rise to important computational savings over other existing solvers, especially in the challenging high-frequency regime.

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.

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

The nascent technique of 4D printing has the potential to revolutionize manufacturing in fields ranging from organs-on-a-chip to architecture to soft robotics. By expanding the pallet of 3D printable materials to include the use stimuli responsive inks, 4D printing promises precise control over patterned shape transformations. With the goal of creating a new manufacturing technique, we have recently introduced a biomimetic printing platform that enables the direct control of local anisotropy into both the elastic moduli and the swelling response of the ink.

We have drawn inspiration from nastic plant movements to design a phytomimetic ink and printing process that enables patterned dynamic shape change upon exposure to water, and possibly other external stimuli. Our novel fiber-reinforced hydrogel ink enables local control over anisotropies not only in the elastic moduli, but more importantly in the swelling. Upon hydration, the hydrogel changes shape accord- ing the arbitrarily complex microstructure imparted during the printing process.

To use this process as a design tool, we must solve the inverse problem of prescribing the pattern of anisotropies required to generate a given curved target structure. We show how to do this by constructing a theory of anisotropic plates and shells that can respond to local metric changes induced by anisotropic swelling. A series of experiments corroborate our model by producing a range of target shapes inspired by the morphological diversity of flower petals.

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.

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.

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.

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)

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.

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.

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.

For any quantity of interest in a system governed by nonlinear differential equations it is natural to seek the largest (or smallest) long-time average among solution trajectories. Upper bounds can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization. We show that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on maximal time averages. They also provide volumes in phase space where maximal trajectories must lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming which we illustrate using the Lorenz and Kuramoto-Sivashinsky equations. This is joint work with Ian Tobasco and David Goluskin, part of which appears in Physics Letters A 382, 382Â?386 (2018).

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.

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.

The usual pseudospectrum acquires an additional feature when restricted to matrices with a certain symmetry. The new feature is a simple form of K-theory which can be used to compute the index of some one-dimensional topological insulators. The usual pseudospectrum applies to a single matrix, or equivalently to two Hermitian matrices. Generalized to apply to more Hermitian matrices, the nature of the pseudospectrum changes radically, often having interesting geometry. Examples come from D-branes and higher-dimensional topological insulators. The algorithm to compute the pseudospectrum also produces common approximate eigenvectors for a collection of almost commuting Hermitian matrices. Applied to a basic model of a finite volume topological insulator it produces vectors that are approximately stationary and somewhat localized in position.

Many experimental systems can spend extended periods of time relative to their natural time scale in localized regions of phase space, transiting infrequently between them. This display of metastability can arise in stochastically driven systems due to the presence of large energy barriers, or in deterministic systems due to the presence of narrow passages in phase space. To investigate metastability in these different cases, we take the Langevin equation and determine the effects of small damping, small noise, and dimensionality on the dynamics and mean transition time. In finite dimensions, we show the limit of small noise and small damping do not interchange. In the limit of infinite dimensions, we argue the equivalence of the finitely-damped system and the zero-damped infinite energy Hamiltonian system.

Starting with an introduction to multiphase image segmentation, this talk will focus on inpainting and illusory contour using variational models with curvature terms. Recent developments of fast algorithms, based on operator splitting, augmented Lagrangian, and alternating minimization, enabled us to efficiently solve functional with higher order terms. Main ideas of the models and algorithms, some analysis and numerical results will be presented.

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).