It is often desirable to derive an effective stochastic model for the physical process from observational and/or numerical data. Various techniques exist for performing estimation of drift and diffusion in stochastic differential equations from discrete datasets. In this talk we discuss the question of sub-sampling of the data when it is desirable to approximate statistical features of a smooth trajectory by a stochastic differential equation. In this case estimation of stochastic differential equations would yield incorrect results if the dataset is too dense in time. Therefore, the dataset has to sub-sampled (i.e. rarefied) to ensure estimators' consistency. Favorable sub-sampling regime is identified from the asymptotic consistency of the estimators. Nevertheless, we show that estimators are biased for any finite sub-sampling time-step and construct new bias-corrected estimators.

Non-equilibrium steady states for chains of oscillators interacting with stochastic heat baths at different temperatures have been the subject of several studies. In this talk I will discuss how to generalize these results to multidimensional networks of oscillators. I will first introduce the model and motivate it from a physical point of view. Then, I will present conditions on the topology of the network and on the interaction potentials which imply the existence and uniqueness of the non-equilibrium steady state, as well as exponential convergence to it. The two main ingredients of the proof are (1) a controllability argument using Hörmander's bracket criterion and (2) a careful study of the high-energy dynamics which leads to a Lyapunov-type condition. I will also mention cases where the non-equilibrium steady state is not unique, and cases where its existence is an open problem. This is joint work with J.-P. Eckmann, M. Hairer and L. Rey-Bellet, Electronic Journal of Probability 23(55): 1-28, 2018 (arXiv:1712.09413).

Hilbert space frames have traditionally been used in signal/image processing. Recently, there have arisen a variety of new applications to speeding up the internet, producing cell phones which won't fade, quantum information theory, distributed processing and more. We will review the fundamentals of frame theory and then look at the myriad of applications of frames.

Recently there has been extensive development of numerical methods for fluid flow interacting with moving boundaries or interfaces, using regular finite difference grids which do not conform to the boundaries. Simulations at low Reynolds number have demonstrated that, with certain choices in the design of the method, the velocity can be accurate to about O(h^2) while discretizing near the interface with truncation error as large as O(h). We will describe error estimates which verify that such accuracy can be achieved in a simple prototype problem, even near the interface, using corrections to difference operators as in the immersed interface method. We neglect errors in the interface location and derive uniform estimates for the fluid velocity and pressure. We will first discuss maximum norm estimates for finite difference versions of the Poisson equation and diffusion equation with a gain of regularity. We will then describe the application to the Navier-Stokes equations.

In computational physics, molecular dynamics refers to the computer simulation of a material at the atomic level. I will consider classical deterministic molecular dynamics in which large Hamiltonian systems of ordinary differential equations are used, though many of the same issues arise with other models. Given its scientific importance there is very little rigorous justification of molecular dynamics. From the viewpoint of numerical analysis it is surprising that it works at all. The problem is that individual trajectories computed by molecular dynamics are accurate for only small time intervals, whereas researchers trust the results over very long time intervals. It has been conjectured that molecular dynamics trajectories are accurate over long time intervals in some weak statistical sense. Another conjecture is that numerical trajectories satisfy the shadowing property: that they are close over long time intervals to exact trajectories with different initial conditions. I will explain how these two views are actually equivalent to each other, after we suitably modify the concept of shadowing.

One of the most fundamental questions in partial differential equations is that of regularity and the possible breakdown of solutions. We will discuss this question for solutions to a canonical example of a geometric wave equation; energy critical wave maps. Break-through works of Krieger-Schlag-Tataru, Rodnianski-Sterbenz and Rapha ̈el-Rodnianski produced examples of wave maps that develop singularities in finite time. These solutions break down by concentrating energy at a point in space (via bubbling a harmonic map) but have a regular limit, away from the singular point, as time approaches the final time of existence. The regular limit is referred to as the radiation. This mechanism of breakdown occurs in many other PDE including energy critical wave equations, Schro ̈dinger maps and Yang-Mills equations. A basic question is the following: • Can we give a precise description of all bubbling singularities for wave maps with the goal of finding the natural unique continuation of such solutions past the singularity? In this talk, we will discuss recent work (joint with J. Jendrej and A. Lawrie) which is the first to directly and explicitly connect the radiative component to the bubbling dynamics by constructing and classifying bubbling solutions with a simple form of prescribed radiation. Our results serve as an important first step in formulating and proving the following Radiative Uniqueness Conjecture for a large class of wave maps: every bubbling solution is uniquely characterized by it’s radiation, and thus, every bubbling solution can be uniquely continued past blow-up time while conserving energy.

Large complex networks naturally represent relationships in a variety of settings, e.g. social interactions, computer/communication networks, and genomic sequences. A significant challenge in analyzing these networks has been understanding the “intermediate structure” – those properties not captured by metrics which are local (e.g. clustering coefficient) or global (e.g. degree distribution). It is often this structure which governs the dynamic evolution of the network and behavior of diffusion-like processes on it. Although there is a large body of empirical evidence suggesting that complex networks are often “tree-like” at intermediate to large size-scales (e.g. work of Boguna et al in physics, Kleinberg on internet routing, and Chung & Lu on power-law graphs), it remains a challenge to take algorithmic advantage of this structure in data analysis. We discuss several approaches and heuristics for quantifying and elucidating tree-like structure in networks, including various tree-decompositions and Gromov's delta hyperbolicity. These approaches were developed with very different "tree-like" applications in mind, and thus we discuss the strengths and short-comings of each in the context of complex networks and how each might aid in identifying intermediate-scale structure in these graphs.

After a short introduction presenting our group and our main research topics, I will address the problem of audio-visual speech recognition, i.e. a typical example of multimodal signal analysis, when we want to extract and exploit information coming from two different but complementary signals: an audio and a video channel. We will discuss two important aspects of this analysis. We will first present a new feature extraction algorithm based in information theoretical principles, and show its performances, compared to other classical approaches, in our multimodal context. Then we will discuss multimodal information fusion, i.e. how to combine information from those two channels for optimal classification.

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.

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.

We present a collection of results, based on a novel operator maximal inequality approach, providing precise relations between the time it takes a Markov chain to converge to equilibrium and the time required for it to exit from small sets. These refine results of Aldous and Lovasz & Winkler. Among the applications are: (1) A general characterization of an abrupt convergence to equilibrium phenomenon known as cutoff. Specializing this to Ramanujan graphs and trees. (2) Proving that the return probability decay is not geometrically robust (resolving a problem of Aldous, Diaconis - Saloff-Coste and Kozma). (3) Random walk in evolving environment

Consider wavefront propagation in the plane, in a medium whose propagation speed is doubly periodic. Think of a "wavefront" as the moving boundary between "light" and "darkness." There are "macroscopic plane wavefronts", for which the wavefront is everywhere and always bounded close to a moving line with constant normal velocity. The normal velocity depends on direction. Some nice function of angle, and we ought to compute it, no? To contemplate this calculation, visualize an infinite planar vineyard, with a square lattice of grape stakes. Gaze outward from the the grape stake at the origin. In some directions, the line of sight is blocked by a stake, and we'll call these directions "rational." The rational directions are dense, but of zero measure on the unit circle. The simplest formal asymptotics of the direction dependent speed produces a strange result: A formal series, one term for each rational direction. The graph of an individual term as a function of angle looks like the amplitude response of a forced oscillator, with a plus infinity vertical asymptote as you scan through the rational direction. Nevertheless, the series converges absolutely for almost all directions on the unit circle. (In the vineyard, almost all lines of sight escape to infinity.) At the outset, the prognosis seems: "Difficult and obscure." The problem of direction dependent speed does not LOOK like a venue for formal classical asymptotics, but that's what this talk proposes. The ingredients are standard matched asymptotic expansions, and baby number theory concerning period cells of the vineyard.

This talk will focus on probability questions arising in the geophysical and biological sciences concerning dispersion in highly heterogeneous environments, as characterized by abrupt changes (discontinuities) in the diffusion coefficient. Some specific phenomena observed in laboratory and field experiments involving breakthrough curves (first passage times), occupation times, and local times will be addressed. This is based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, and Brian Wood at Oregon State University.

In this talk, we will introduce a dynamic mathematical model that describes the interaction of the immune system with the human immunodeficiency virus (HIV). Using optimal control theory, we will illustrate that optimal dynamic multidrug therapies can produce a drug dosing strategy that exhibits structured treatment interruption, a regimen in which patients are cycled on and off therapy. In addition, sensitivity analysis of the model including both classical sensitivity functions and generalized sensitivity functions will be presented. Finally, we will describe how stochastic estimation can be used to filter and estimate states and parameters from noisy data. In the course of this analysis it will be shown that automatic differentiation can be a powerful tool for this type of study.

Although “big data” is ubiquitous in data science, one often faces challenges of “small data,” as the amount of data that can be taken or transmitted is limited by technical or economic constraints. To retrieve useful information from the insufficient amount of data, additional assumptions on the signal of interest are required, e.g. sparsity (having only a few non-zero elements). Conventional methods favor incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent. I will present a nonconvex approach that works particularly well in the coherent regime. I will also address computational aspects in the nonconvex optimization. Various numerical experiments have demonstrated advantages of the proposed method over the state-of-the-art. Applications, ranging from super-resolution to low-rank approximation, will be discussed.

In vertebrates, ovulation is triggered by a surge of luteinizing hormone (LH) from the pituitary. The precise mechanism by which rising estradiol (E2) from the ovaries initiates the LH surge in the human menstrual cycle remains a mystery. The mystery is due in part to the bimodal nature of estradiol feedback action on LH secretion, and in part to disagreement over the site of the feedback action.

We will describe a differential equations model in which the mysterious bimodality of estradiol action arises from the electrical connectivity of a network of folliculo-stellate cells in the pituitary. The mathematical model is based as closely as possible on current experimental data, and is being used to design and conduct new experiments. No biological background will be assumed.

Chemotaxis is a movement of micro-organisms or cells towards the areas of high concentration of a certain chemical, which attracts the cells and may be either produced or consumed by them. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It is well-known that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxissystems extremely delicate and challenging task. In this talk, I will present a new family of high-order finite-volume finite-difference methods for the Keller-Segel chemotaxis system and several related models. Applications of the proposed methods to the classical Patlak-Keller-Segel model, its extensions to the two-species case as well as to the coupled chemotaxis–fluid system will also be discussed.

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.

We consider a coupled system of Schroedinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. In the first case, the classical limit is only taken in one of the two equations, leading to a mixed quantum-classical model which is closely connected to the well-known Ehrenfest method in molecular dynamics. In the second case, the classical limit of the full system is rigorously established, resulting in a system of coupled Vlasov-type equations. In the second part of our work, we provide a numerical study of the coupled semiclassically scaled Schroedinger equations and of the mixed quantum-classical model obtained via Ehrenfest's method. A second order (in time) method is introduced for each case. We show that the proposed methods allow time steps independent of the semi-classical parameter(s) while still capturing the correct behavior of physical observables. It also becomes clear that the order of accuracy of our methods can be improved in a straightforward way.

I will present the proof of the random homogenization of general coercive Hamiltonian in 1d with the form as H(p,x,\omega)=H(p)+V(x, \omega). Some interesting and complex phenomena associated with non-convex Hamiltonian will also be discussed. This is a joint work with Scott Armstrong and Hung Tran.

Accounting for system-parameter and model uncertainties in computational models is a highly topical issue at the interface of computational mechanics, materials science and probability theory. In addition to the construction of efficient (e.g. Galerkin-type) stochastic solvers, the construction, calibration and validation of probabilistic representations are now widely recognized as key ingredients for performing accurate and robust simulations. This talk is specifically focused on the modeling and simulation of spatially-dependent properties in both linear and nonlinear frameworks. Information-theoretic models for matrix-valued random fields are first introduced. These representations are typically used, in solid mechanics, to define tensor-valued coefficients in elliptic stochastic partial differential operators. The main concepts and tools are illustrated, throughout this part, by considering the modeling of elasticity tensors fluctuating over nonpolyhedral geometries, as well as the modeling and identification of random interfaces in polymer nanocomposites. The latter application relies, in particular, on a statistical inverse problem coupling large-scale Molecular Dynamics simulations and a homogenization procedure. We then address the probabilistic modeling of strain energy functions in nonlinear elasticity. Here, constraints related to the polyconvexity of the potential are notably taken into account in order to ensure the existence of a stochastic solution. The proposed framework is finally exemplified by considering the modeling of various soft biological tissues, such as human brain and liver tissues.

We construct explicit approximate Green's functions of time-dependent, linear Fokker-Planck equations in terms of Dyson series, Taylor expansions, and exact commutator formulas. Our method gives an approximate solution that is accurate to arbitrary order in time in the short-time limit, and it can be extended to large time by bootstrapping. I will also present some numerical results showing that our algorithm works well also for degenerate equations such as those arising in pricing of contingent claims. This is joint work with Victor Nistor and Wen Cheng.

Given two probability measures and a transportation cost, the optimal transport problem asks to find the most cost efficient way to transport one measure to the other. Since its introduction in 1781 by Gaspard Monge, the optimal transport problem has found applications in logistics, economics, physics, PDEs, and more recently data science. However, despite sustained attention from the numerics community, solving optimal transport problems has been a notoriously difficult task. In this talk I will introduce the back-and-forth method, a new algorithm to efficiently solve the optimal transportation problem for a general class of strictly convex transportation costs. Given two probability measures supported on a discrete grid with n points, the method computes the optimal map in O(n log(n)) operations using O(n) storage space. As a result, the method can compute highly accurate solutions to optimal transportation problems on spatial grids as large as 4096 x 4096 and 384 x 384 x 384 in a matter of minutes. If time permits, I will demonstrate an extension of the algorithm to the simulation of a class of gradient flows. This talk is joint work with Flavien Leger.

We propose a general framework to design asymptotic preserving schemes for the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We will show how this idea can be applied to other collision operators, such as the Landau-Fokker-Planck operator, Ullenbeck-Urling model, and in the kinetic-fluid model of disperse multiphase flows, and can be implemented in the Monte-Carlo framework in which is time step is not limited by the possibly small mean free time.

When we want to understand a geometric picture, finding the zero set of a polynomial hiding in it can be very helpful: it can reveal structure and allow computations. Polynomial partitioning, developed by Guth and Katz, is a technique to find such a nice algebraic hypersurface. Polynomial partitioning has revolutionised discrete incidence geometry in the recent years, thanks to the fact that interaction of lines with algebraic hypersurfaces is well-understood. Recently, however, Guth discovered agreeable interaction between tubes and algebraic hypersurfaces, and thus used polynomial partitioning to improve on the 3-dim restriction problem. In this talk, we will present polynomial partitioning via a discrete analogue of the Kakeya problem, and discuss its potential to be extensively used in harmonic analysis.

Regularity of solutions in the optimal transport problem requires very rigid hypotheses (e.g., convexity of certain sets). When such conditions are not available, one can consider the question of partial regularity, in other words, the in-depth analysis of the structure of singular sets. In this talk, I will discuss the regularity of the set of ``free singularities`` which arise in an optimal transport problem with inner product cost, from a connected set to a disconnected set, along with the stability of such sets under suitable perturbations of the data involved. Some of these results are proven via a non-smooth implicit function theorem for convex functions, which is of independent interest. This talk is based on joint work with Robert McCann.

Interpolatory matrix factorizations provide alternatives to the singular value decomposition (SVD) for obtaining low-rank approximations; this class includes the CUR factorization, where the C and R matrices are subsets of columns and rows of the target matrix. While interpolatory approximations lack the SVD's optimality, their ingredients are easier to interpret than singular vectors: since they are copied from the matrix itself, they inherit the data's key properties (e.g., nonnegative/integer values, sparsity, etc.). We shall provide an overview of these approximate factorizations, describe how they can be analyzed using interpolatory projectors, and introduce a new method for their construction based on the Discrete Empirical Interpolation Method (DEIM). (This talk describes joint work with Dan Sorensen (Rice).)

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.