A wide range of viral infections, such as HIV or influenza, can now be treated using antiviral drugs. Since viruses can evolve rapidly, the emergence and spread of drug resistant virus strains is a major concern. We shall describe within and between host models that can help indicate settings in which resistance is more or less likely to be problematic. In particular, we shall discuss the potential for the emergence of resistance in the context of human rhinovirus infection, an acute infection that is responsible for a large fraction of 'common cold' cases.

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.

Nonlocal models are experiencing a firm upswing recently as more realistic alternatives to the conventional local models for studying various phenomena from physics and biology to materials and social sciences. In this talk, I will describe our recent effort in taming the computational challenges for nonlocal models. I will first highlight a family of numerical schemes -- the asymptotically compatible schemes -- for nonlocal models that are robust with the modeling parameter approaching an asymptotic limit. Second, fast algorithms will be presented to reduce the high computational cost from the numerical implementation of the nonlocal operators. Although new nonlocal models have been gaining popularity in various applications, they often appear as phenomenological models, such as the peridynamics model in fracture mechanics. Here we will try to provide better perspectives of the origin of nonlocality from multiscale modeling and homogenization, which in turn may help the development of more effective numerical methods for homogenization.

We will discuss recent approximations to boundary value problems to non-linear systems of Boltzmann or Landau (for Coulombic interactions) equations coupled to the Poisson equation. The proposed approximation methods involve hybrid schemes of spectral and Galerkin type, were conservation of flow invariants are achieved by a constrain minimization problem. We will discuss some analytical and computational issues related to these approximations.

We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. Technically, the heart of the argument are new laws of large numbers at the exponential scale, proved through an explicit combinatorics approach. Our result only requires weak regularity on the interaction kernel in negative Sobolev spaces, thus including the Biot-Savart law and the point vortices dynamics for the 2d incompressible Navier-Stokes. For dissipative gradient flows, we may allow any singularity lower than the Poisson kernel. This talk corresponds to a joint work with Z. Wang and an upcoming work with D. Bresch and Z. Wang.

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.

We will discuss a class of binary measurement matrices having a simple discrete incoherence property. These matrices can be shown to have both useful analytic (i.e., restricted isometry and l1-approximation properties) and combinatorial (i.e., group testing and expander graph related) structure which allows them to be utilized for sparse signal approximation in the spirit of compressive sensing. In addition, their structure allows the actual signal recovery process to be carried out by highly efficient algorithms once measurements have been taken. One application of these matrices and their related recovery algorithms is their application to the development of sublinear-time Fourier methods capable of accurately approximating periodic functions using far fewer samples and run time than required by standard Fourier transform techniques.

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 this talk we discuss an algorithm to overcome the curse of dimensionality, in possibly non-convex/time/state-dependent Hamilton-Jacobi partial differential equations. They may arise from optimal control and differential game problems, and are generally difficult to solve numerically in high dimensions.
A major contribution of our works is to consider an optimization problem over a single vector of the same dimension as the dimension of the HJ PDE instead. To do so, we consider the new approach using Hopf-type formulas. The sub-problems are now independent and they can be implemented in an embarrassingly parallel fashion. That is ideal for perfect scaling in parallel computing.
The algorithm is proposed to overcome the curse of dimensionality when solving high dimensional HJ PDE. Our method is expected to have application in control theory, differential game problems, and elsewhere. This approach can be extended to the computational of a Hamilton-Jacobi equation in the Wasserstein space, and is expected to have applications in mean field control problems, optimal transport and mean field games.

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 first discuss a general method to derive macroscopic laws from certain microscopic interactions that can be applied to a large class of particle systems. In particular we consider a broad class of systems that are linear in a special algebra, so-called tropical algebra. Some natural connections among the scaling limits of these random systems, the solutions to specific partial differential equations (Hamilton-Jacobi type) and combinatorial optimization problems have been noticed. If time allows, we further discuss a useful application of the variational formula (microscopic version of Hopf-Lax formula) to a well-known interacting particle system, totally asymmetric simple exclusion process.

In this talk we will introduce a new class of flocking models that feature emergence of global alignment via only local communication. Such models have been sought for since the introduction of Cucker-Smale dynamics which showed global unconditional alignment for models with substantially strong non-local interaction kernels. We introduce a set of new structural components into the communication protocol, including singular kernel, and topological adaptive diffusion, that enhance alignment mechanisms with purely local interactions. We highlight some challenges that arise in the problem of global well-posendess and stability of flocks.

Enhacement of diffusion and reaction processes by advection is a classical subject that has been extensively studied by both physicists and mathematicians. In this talk I will consider reaction-diffusion on a bounded domain in the presence of a fast incompressible flow. I will also consider convective systems in a bounded domain, in which viscous fluids described by the Stokes system are coupled using the Boussinesq approximation to a reaction-advection-diffusion equation for the temperature. For such systems enhacement of diffusion implies, in particular, that the explosion threshold for power-like nonlinearities tends to infinity in the large Rayleigh number limit.

Porous, deformable membranes are encountered in a wide range of applications including red blood cells, vesicles, porous wave makers, and parachutes. The "immersed boundary method" has already proven to be a versatile and robust approach for simulating the interaction of impermeable, elastic structures with an incompressible fluid flow. We demonstrate how to extend the method to handle porous boundaries by incorporating an explicit porous slip velocity that is determined by Darcy's law. We derive a simple, radially-symmetric exact solution, which is then used to validate numerical simulations of porous membranes in two dimensions.

We have witnessed a lot of exciting development of data science in recent years. From the perspective of optimization, many modern data-science problems involve some basic ``non’’-properties that lack systematic treatment by the current approaches for the sake of the computation convenience. These non-properties include the coupling of the non-convexity, non-differentiability and non-determinism. In this talk, we present rigorous computational methods for solving two typical non-problems: the piecewise linear regression and the feed-forward deep neural network. The algorithmic framework is an integration of the first order non-convex majorization-minimization method and the second order non-smooth Newton methods. Numerical experiments demonstrate the effectiveness of our proposed approach. Contrary to existing methods for solving non-problems which provide at best very weak guarantees on the computed solutions obtained in practical implementation, our rigorous mathematical treatment aims to understand properties of these computed solutions with reference to both the empirical and the population risk minimizations. This is based on joint work with Jong-Shi Pang, Bodhisattva Sen and Ziyu He.

We will discuss two issues in the context of applying deep learning methods to multi-scale molecular modelling: 1) how to construct symmetry-preserving neural network models for scalar and tensorial quantities; 2) how to efficiently explore the relevant configuration space and generate a minimal set of training data. We show that by properly addressing these two issues, one can systematically develop deep learning-based models for electronic properties and interatomic and coarse-grained potentials, which greatly boost the ability of ab-initio molecular dynamics; one can also develop enhanced sampling techniques that are capable of using tens or even hundreds of collective variables to drive phase transition and accelerate structure search

Boundary value and interface problems for the Laplace equation are often solved by boundary integral methods due to the reduction of dimensionality and its flexibility in domain geometry. However, there are two well-known computational issues with the boundary integral method: (a) evaluation of boundary integrals at points close to domain boundaries usually has low order accuracy; (b) the method typically yields dense coefficient matrices in the resulting discrete systems, which makes the matrix vector multiplication very expensive when the size of the system is very large. In this talk, I will describe a fast accurate boundary integral method for the Laplace boundary value and interface problems. The algorithm uses the high order accurate method proposed by (Beale and Lai 2001) for evaluation of the boundary integrals and applies the fast multipole method for the dense matrix vector multiplication. Numerical results demonstrating the efficiency and accuracy of the method will be presented.

The problem of mixing via incompressible flows is classical and rich with connections to several branches of analysis including PDE, ergodic theory, and topological dynamics. In this talk I will discuss some recent developments in the area and then present a construction of universal mixers - incompressible flows that asymptotically mix arbitrarily well general solutions to the corresponding transport equation - in all dimensions. This mixing is in fact exponential in time (i.e., essentially optimal) for any initial condition with at least some degree of regularity, while there exists no uniform mixing rate for all measurable initial conditions.

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.

The purpose of this presentation is to communicate some mathematical and computational issues in sediment transport for open channels. The main topics are: (1) mathematical simulation for surface and subsurface fluid flow, (2) mathematical modeling of sediment transport in open channels as a 2D problem, and (3) numerical methods for fluid flow and sediment transport.

Perspectives from computational algebra and numerical optimization are brought to bear on a scientific application and a data science application. In the first part of the talk, I will discuss cryo-electron microscopy (cryo-EM), an imaging technique to determine the 3-D shape of macromolecules from many noisy 2-D projections, recognized by the 2017 Chemistry Nobel Prize. Mathematically, cryo-EM presents a particularly rich inverse problem, with unknown orientations, extreme noise, big data and conformational heterogeneity. In particular, this motivates a general framework for statistical estimation under compact group actions, connecting information theory and group invariant theory. In the second part of the talk, I will discuss tensor rank decomposition, a higher-order variant of PCA broadly applicable in data science. A fast algorithm is introduced and analyzed, combining ideas of Sylvester and the power method.

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.

The beautiful work of Borgs, Chayes, Lovasz, Sos, Szegedy, Vesztergombi, and many others on dense graph limits has received quite a bit of attention in pure math as well as statistics and machine learning. In this talk we will review some of the previous work on dense graph limits and then present recent work on providing a more robust mathematical framework for proving the statistical consistency of graph partitioning algorithms such as spectral clustering. A striking feature of our approach is that it is model-free, compared to the popular iid paradigm. Our results are thus broadly applicable in real-world settings, where it is notoriously difficult to obtain relevant models for network data, and observations are not independent. At the end, I will discuss implications for how mathematical foundations can be developed for other modern data analysis techniques. This is joint work with Dominique Guillot, Apoorva Khare, and Bala Rajaratnam. Preprint available at https://arxiv.org/abs/1608.03860.

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 optimal transport (Monge-Kantorovich) problem is a variational problem involving transportation of mass subject to minimizing some kind of energy, and it arises in connection with many parts of math, both pure and applied. In this talk, I will discuss a numerical algorithm to approximate solutions in the semi-discrete case. We propose a damped Newton algorithm which exploits the structure of the associated dual problem, and using geometric implications of the regularity theory of Monge-Amp{\`e}re equations, we are able to rigorously prove global linear convergence and local superlinear convergence of the algorithm. This talk is based on joint work with Quentin M{\’e}rigot and Boris Thibert.