Fall 2021

Abstract:  We will talk about how the continuous counterpart of Nesterov's acceleration leads to not only intuitive understanding of the scheme and also to its practical generalization to locally Lipschitz objectives. Then, we talk about how the scheme makes implicit schemes attractive from a practical point of view with two examples of real world problems: phase field crystal and functionalized Cahn-Hilliard equations.  

Abstract: Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the $3D$ viscous primitive equations are globally well-posed in Sobolev spaces. In this talk, I will discuss the ill-posedness in Sobolev spaces, the local well-posedness in the space of analytic functions, and finite-time blowup of solutions to the  inviscid PEs (the hydrostatic Euler equations) with rotation (Coriolis force). Moreover, I will also show, in the case of ``well-prepared" analytic initial data, the regularizing effect of the Coriolis force by providing a lower bound for the life-span of the solutions that grows toward infinity with the rotation rate. If time permits, I will also show some recent progress on the study of the PEs with only vertical viscosity (the hydrostatic Navier-Stokes equations).

Abstract: We consider a general framework for obtaining uniform-in-time rates of convergence for numerical approximations of SPDEs in suitable Wasserstein distances. The framework is based on two general results under an appropriate set of assumptions: a Wasserstein contraction result for a given Markov semigroup; and a uniform-in-time weak convergence result for a parametrized family of Markov semigroups. We provide an application to a suitable space-time discretization of the 2D stochastic Navier-Stokes equations in vorticity formulation. Specifically, we obtain that the Markov semigroup induced by this discretization satisfies a Wasserstein contraction result which is independent of any discretization parameters. This allows us to obtain a corresponding weak convergence result towards the Markov semigroup induced by the 2D SNSE. The proof required technical improvements from the related literature regarding finite-time error estimates. Finally, our approach does not rely on standard gradient estimates for the underlying Markov semigroup, and thus provides a flexible formulation for further applications. This is a joint work with Nathan Glatt-Holtz (Tulane U).

Abstract: Combinatorial geometry can also play a role in foundations of data science. Here I present two concrete examples:

In statistical inference we wish to find the properties or parameters of a distribution or model through sufficiently many samples. A famous example is logistic regression, a popular non-linear model in multivariate statistics and supervised learning. Users often rely on optimizing of maximum likelihood estimation, but how much training data do we need, as a function of the dimension of the covariates of the data, before we expect an MLE to exist with high probability? 

Similarly, for unsupervised learning and non-parametric statistics, one wishes to uncover the shape and patterns from samples of a measure or measures. We use only the intrinsic geometry and topology of the sample. A famous example of this type of method is the $k$-means clustering algorithm. A fascinating challenge is to explain the variability of behavior of $k$-means algorithms with distinct random initializations and the shapes of the clusters. 

In this talk we present new stochastic combinatorial theorems, that give bounds on the probability of existence of maximum likelihood estimators in multinomial logistic regression and also quantify to the variability of clustering initializations. Along the way we will see fascinating connections to the coupon collector problem, topological data analysis, and to the computation of Tukey centerpoints of data clouds (a high-dimensional generalization of median). This is joint work with T. Hogan, R.  D. Oliveros, E. Jaramillo-Rodriguez, and A. Torres-Hernandez.

Abstract: In this work, we derive a Dirac equation in curved space-time, modeling electron dynamics on general smooth strained graphene surfaces. Rather than working in Cartesian coordinates which would lead to a very complex equation, we use isothermal coordinates. The latter are obtained from quasi-conformal transformations and are characterized by the Beltrami equation, whose solution gives a mapping between isothermal and Cartesian coordinate systems.

We will then discuss i) the well-posedness on the overall Beltrami-Dirac system and ii) its numerical approximation by a least square finite element method (Beltrami) and an accurate pseudo-spectral method (Dirac). If time permits, we will also discuss the inclusion of Perfectly Matched Layers (PML) within the pseudo-spectral method. The analysis of the numerical methods and some experiments will be presented to illustrate the model.

Abstract: Many scientific and engineering disciplines depend on efficient numerical methods to model complex physical systems. Since the underlying dynamics frequently involve a range of temporal and spatial scales, specialized methods that can handle numerical challenges such as stiffness and high dimensionality are required. In this talk I will present several novel approaches for constructing custom-designed time integration techniques that achieve improved efficiency on specific application problems and computational hardware. In particular, I will introduce a framework based on interpolating polynomials that can be used to construct time integrators with desirable properties such as parallelism, high-order of accuracy, and varying degrees of implicitness. I will show how the polynomial framework naturally incorporates various paradigms such as exponential and additive integration. In addition I will also discuss the construction of parallel-in-time integrators for non-diffusive systems. Specifically, I will show how convergence and stability analysis can be used to select stable Parareal configurations that allow for massively parallel simulations of hyperbolic and dispersive equations.

Abstract: Iteratively Reweighted Least Squares (IRLS), also known as half-quadratic minimization, is a simple algorithmic framework for non-smooth optimization that has been studied since the 1930’s and has been widely used in approximation theory, statistics, image processing and beyond. Despite its popularity, a thorough understanding of the framework had been elusive. 

In this talk, we provide several advances in both the theory and formulation of IRLS for well-known problems in data science. We provide the first global linear convergence results for IRLS applied to l1-norm minimization. Furthermore, we formulate an optimal formulation for IRLS optimizing non-convex spectral functions, for which we show fast local convergence guarantees, and illustrate its data-efficiency and scalability compared to the state-of-the-art. Finally, we show that IRLS can be used to infer the topology of a graph from a near-optimal number of spatio-temporal samples.

Abstract: Thromboembolism, one of the leading causes of morbidity and mortality worldwide, is characterized by formation of obstructive intravascular clots (thrombi) and their mechanical breakage (embolization). While blood clot formation has been relatively well studied, little is known about the mechanisms underlying the subsequent structural and mechanical clot remodeling called contraction or retraction. Impairment of the clot contraction process is associated with both life-threatening bleeding and thrombotic conditions, such as ischemic stroke, venous thromboembolism, and others. Recently, blood clot contraction was observed to be hindered in patients with COVID-19. A novel multi-phase computational model will be described that simulates active interactions between the main components of the clot, including platelets and fibrin. Model simulations provided new insights into mechanisms underlying clot stability and embolization that cannot be studied experimentally at this time. In particular, model simulations, calibrated using experimental intravital imaging of an established arterial clot, show that flow-induced changes in size, shape and internal structure of the clot are largely determined by two shear-dependent mechanisms: reversible attachment of platelets to the exterior of the clot and removal of large clot pieces [1]. Model simulations also predict that blood clots with higher permeability are more prone to embolization with enhanced disintegration under increasing shear rate. In contrast, less permeable clots are more resistant to rupture due to shear rate dependent clot stiffening originating from enhanced platelet adhesion and aggregation. Role of platelets-fibrin network mechanical interactions in determining shape of a clot and clot contraction will be also discussed and quantified using model simulations calibrated using experimental data [2,3]. These results can be used in future to predict risk of thromboembolism based on the patient specific data about composition, permeability and deformability of a clot formed from patient’s blood in a microfluidic device under specific local hemodynamic conditions.

[1] Xu S, Xu Z, Kim OV, Litvinov RI, Weisel JW, Alber M. Model predictions of deformation, embolization and permeability of partially obstructive blood clots under variable shear flow. J. R. Soc. Interface 14 (2017) 20170441.

[2] Oleg V. Kim, Rustem I. Litvinov, Mark S. Alber & John W. Weisel, Quantitative structural mechanobiology of platelet driven blood clot contraction, Nature Communications 8 (2017) 1274. 

[3] Samuel Britton, Oleg Kim, Francesco Pancaldi, Zhiliang Xu, Rustem I. Litvinov, John W.Weisel, Mark Alber [2019], Contribution of nascent cohesive fiber-fiber interactions to the non-linear elasticity of fibrin networks under tensile load, Acta Biomaterialia 94 (2019) 514–523.

Abstract: In modern data analysis, the datasets are often represented by large matrices or tensors(the generalization of matrices to higher dimensions). To have a better understanding of the data, an important step is to construct a low-dimensional/compressed representation of the data that may be better to analyze and interpret in light of a corpus of field-specific information. To implement the goal, a primary tool is the matrix/tensor decomposition. In this talk, I will talk about novel matrix/tensor decompositions, CUR decompositions, which are memory efficient and computationally cheap. Besides, I will also discuss how skeleton decompositions (also termed CUR decompositions) are applied to develop efficient robust decomposition algorithms and show the efficiency of the algorithms on some real and synthetic datasets.

Abstract: This talk is concerned with the inverse problem of recovering a discrete measure on the torus given a finite number of its noisy Fourier coefficients. We focus on the diffraction limited regime where at least two atoms are closer together than the Rayleigh length. We show that the fundamental limits of this problem and the stability of subspace (algebraic) methods, such as ESPRIT and MUSIC, are closely connected to the minimum singular value of non-harmonic Fourier matrices. We provide novel bounds for the latter in the case where the atoms are located in clumps. We also provide an analogous theory for a statistical model, where the measure is time-dependent and Fourier measurements are collected over at various times. Joint work with Wenjing Liao, Albert Fannjiang, Zengying Zhu, and Weiguo Gao.