Recently there has been a large class of research that utilizes nonlinear mappings into low dimensional spaces in order to organize potentially high dimensional data. Examples include, but are not limited to, locally linear embedding (LLE), ISOMAP, Hessian LLE, Laplacian eigenmaps, and diffusion maps. In this talk we will focus on the latter, and in particular consider how to generalize diffusion maps to the setting in which we are given a data set that evolves over time or changes depending on some set of parameters. Along with describing the current theory, various synthetic and real world examples will be presented to illustrate these ideas in practice.

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.

Sparsity plays a central role in recent developments of many fields such as signal and image processing, compressed sensing, statistics, and optimization. In practice, sparsity is promoted through the additional of an L1 norm (or related quantity) as a constraint or penalty in a variational model. Motivated by the success of sparsity-inducing methods in imaging and information sciences, there is a growing interest in exploiting sparsity in dynamical systems and partial differential equations. In this talk, we will investigate the connections between compressed sensing, sparse optimization, and numerical methods for nonlinear differential equations. In particular, we will discuss about sparse modeling as well as the advantage of sparse optimization in solving various differential equations arising from physical and data sciences.

When iterative methods are used to solve a discretized linear system for partial differential equations, the key issue is how to make the convergence fast. For different type of problems convergence mechanism can be quite different. In this talk, I will present an efficient iterative method, the fast sweeping method, for a class of nonlinear hyperbolic partial differential equation, Hamilton-Jacobi equation, which is widely used in optimal control, geometric optics, geophysics, classical mechanics, image processing, etc. We show that the fast sweeping method can converge in a finite number of iterations when monotone upwind scheme,  Gauss-Seidel iterations with causality enforcement and proper orderings are used. We  analyze its convergence, which is very different from that for iterative method for elliptic problems. If time permit I will present a new formulation to compute effective Hamiltonians for homogenization of a class of Hamilton-Jacobi equations. Both error estimate and stability analysis will be shown.

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.

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.

I will discuss variational problems arising in machine learning and their limits as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying "ground-truth" measure. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. Many machine learning tasks, such as clustering and classification, can be posed as minimizing functionals on such graphs. We consider functionals involving graph cuts and graph laplacians and their limits as the number of data points goes to infinity. In particular we establish for what graph constructions the minimizers of discrete problems converge to a minimizer of a functional defined in the continuum setting. The talk is primarily based on joint work with Nicolas Garcia Trillos, as well as on works with Xavier Bresson, Moritz Gerlach, Matthias Hein, Thomas Laurent, James von Brecht and Matt Thorpe.

There are two proofs of Vinogradov's Mean Value Theorem (VMVT), the harmonic analysis decoupling proof by Bourgain, Demeter, and Guth from 2015 and the number theoretic efficient congruencing proof by Wooley from 2017. While there has been some work illustrating the relation between these two methods, VMVT has been around since 1935. It is then natural to ask: What does previous partial progress on VMVT look like in harmonic analysis language? How similar or different does it look from current decoupling proofs? We talk about a classical argument due to Karatsuba that shows VMVT "asymptotically" and interpret this in decoupling language. This is joint work with Brian Cook, Kevin Hughes, Olivier Robert, Akshat Mudgal, and Po-Lam Yung.

This talk is about representation learning with a nontrivial geometry of variables. A convolutional neural network can be viewed as a statistical machine to detect and count features in an image progressively through a multi-scale system. The constructed features are insensitive to nuance variations in the input, while sufficiently discriminative to predict labels. We introduce the Haar scattering transform as a model of such a system for unsupervised learning. Employing Haar wavelets makes it applicable to data lying on graphs that are not necessarily pixel grids. When the underlying graph is unknown, an adaptive version of the algorithm infers the geometry of variables by optimizing the construction of the Haar basis so as to minimize data variation. Given time, I will also mention an undergoing project of flow cytometry data analysis, where histogram-like features are used for comparing empirical distributions. After "binning" samples on a mesh in space, the problem can be closely related to feature learning when a variable geometry is present.

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.

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).

Topological insulators (TIs) are materials characterized by topological invariants. One of their remarkable features is the asymmetric transport observed at the interface between materials in different topological phases. Such transport is itself described by a topological invariant, and therefore ``protected" against random perturbations. This immunity makes TIs extremely promising for many engineering applications and actively researched.

In this talk, we present a PDE model for such TIs, introduce a topology based on indices of Fredholm operators, and analyze the influence of random perturbations. We confirm that topology is an obstruction to Anderson localization, a hallmark of wave propagation in strongly heterogeneous media in the topologically trivial case and to some extent quantify what is or is not protected topologically. For instance, a quantized amount of transmission is protected while back-scattering, a practical nuisance, is not.

In the first part of the talk, we will discuss a numerical method for wave propagation in inhomogeneous media. The Trefftz method relies on basis functions that are solution of the homogeneous equation. In the case of variable coefficients, basis functions are designed to solve an approximation of the homogeneous equation. The design process yields high order interpolation properties for solutions of the homogeneous equation. This introduces a consistency error, requiring a specific analysis.
In the second part of the talk, we will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudo-spectral algorithms were developed for inverting the Laplace-Beltrami operator and computing the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, well-conditioned solvers for the Maxwell equations will rely on these algorithms.

In seismic and SAR imaging, fitting cross-correlations of wavefields rather than the wavefields themselves can result in much improved robustness vis-a-vis model uncertainties. This approach however raises two challenges: (i) new spurious local minima may complicate the inversion, and (ii) one must find a good subset of cross-correlations to make the problem well-posed. I will explain how to address these two problems with lifting, semidefinite relaxation, and expander graphs. This mix of ideas has recently proved to be the right approach in other contexts as well, such as angular synchronization (Singer et al.) and phase retrieval (Candes et al.). Joint work with Vincent Jugnon.

The Explicit Jump Immersed Interface Method (EJIIM) is a finite difference method for elliptic partial differential equations that, like all Immersed Interface Methods, works on a regular grid in spite of non-grid aligned discontinuities in equation parameters and solution. The specific idea is to introduce jumps in function and its derivatives explicitely as additional variables. We present a finite difference based EJIIM for the stationary Stokes flow in saddle point formulation. Challenges related to staggered grid, fast Stokes solver and non-simply connected domains will be discussed.

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, 382–386 (2018).

Multi-segment reconstruction (MSR) problem consists of recovering a signal from noisy segments with unknown positions of the observation windows. One example arises in DNA sequence assembly, which is typically solved by matching short reads to form longer sequences. Instead of trying to locate the segment within the sequence through pair-wise matching, we propose a new approach that uses shift-invariant features to estimate both the underlying signal and the distribution of the positions of the segments. Using the invariant features, we formulate the problem as a constrained nonlinear least-squares. The non-convexity of the problem leads to its sensitivity to the initialization. However, with clean data, we show empirically that for longer segment lengths, random initialization achieves exact recovery. Furthermore, we compare the performance of our approach to the results of expectation maximization and demonstrate that the new approach is robust to noise and computationally more efficient.

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.

The theory of homogenization for equations with random coefficients is now quite well-developed. What is less studied is the theory for the correctors to homogenization, which asymptotically characterize the randomness in the solution of the equation and as such are important to quantify in many areas of applied sciences. I will present recent results in the theory of correctors for elliptic and parabolic problems and briefly mention how such correctors may be used to improve reconstructions in inverse problems. Homogenized (deterministic effective medium) solutions are not the only possible limits for solutions of equations with highly oscillatory random coefficients as the correlation length in the medium converges to zero. When fluctuations are sufficiently large, the limit may take the form of a stochastic equation and stochastic partial differential equations (SPDE) are routinely used to model small scale random forcing. In the very specific setting of a parabolic equation with large, Gaussian, random potential, I will show the following result: in low spatial dimensions, the solution to the parabolic equation indeed converges to the solution of a SPDE, which however needs to be written in a (somewhat unconventional) Stratonovich form; in high spatial dimension, the solution to the parabolic equation converges to a homogenized (hence deterministic) equation and randomness appears as a central limit-type corrector. One of the possible corollaries for this result is that SPDE models may indeed be appropriate in low spatial dimensions but not necessarily in higher spatial dimensions.

The simplest example of a boundary value problem is the Dirichlet Poisson problem: we seek to recover a function, defined on a smooth domain, its values at the boundary of the domain and the divergence of its gradient for all points inside the domain. This problem has been studied for more than 200 years, and has many applications in science and engineering. Analytic solutions are available only for a limited number of cases. Therefore one has to use a numerical method. The basic goals in designing a numerical method is guaranteed quality of the solution, in reasonable time, in a black-box fashion. Surprisingly, a robust, black-box, algorithmically scalable method for the Poisson problem does not exist. The main difficulties are related to robust mesh generation in complex geometries in three dimensions. I will review different approaches in solving the Poisson problem and present a new method based on classical Fredholm integral equation formulation. The main components of the new method are a kernel-independent fast summation method, manifold surface representations, and superalgebraically accurate quadrature methods. The method directly extends to problems with non-oscillatory known Green's functions. In addition to the Poisson problem I will present results for the Navier, modified Poisson, and Stokes operators.

We will discuss some recent results on singularity formation for finite-energy strong solutions to the 3D Euler system based on the analysis of scale-invariant solutions. The work consists of three parts: local well-posedness in critical spaces, the proof of blow-up for scale-invariant solutions, and then a cut-off argument to ensure finite energy. This is joint work with I. Jeong.

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.

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.

We survey recent advances on the complexity and methods for solving Markov decision problems (MDP) and Reinforcement Learning (RL) with finitely many states and actions - a basic mathematical model for reinforcement learning.

For model reduction of large scale MDP in reinforcement learning, we propose a bilinear primal-dual pi learning method that utilizes given state and action features. The method is motivated from a saddle point formulation of the Bellman equation. The sample complexity of bilinear pi learning depends only on the number of parameters and is variant with respect to the dimension of the problem.

In the second part we study the statistical state compression of general Markov processes. We propose a spectral state compression method for learning the state features from data. The state compression method is able to “ sketch” a black-box Markov process from its empirical data and output state features, for which we provide both minimax statistical guarantees and scalable computational tools.

We discuss some recent results on stability of channel flows and boundary layers expansions in the inviscid limit.

A neuron receives thousands of synaptic inputs from other neurons and integrates them to process information. Many experimental results demonstrate this integration could be highly nonlinear, yet few theoretical analyses have been performed to obtain a precise quantitative characterization. Based on asymptotic analysis of an idealized cable model, we derive a bilinear spatiotemporal integration rule for a pair of time-dependent synaptic inputs. Note that the above rule is obtained from idealized models. However, we have confirmed this rule both in simulations of a realistic pyramidal neuron model and in electrophysiological experiments of rat hippocampal CA1 neurons. Our results demonstrate that the integration of multiple synaptic inputs can be decomposed into the sum of all possible pairwise integration with each paired integration obeying a bilinear rule.