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.

Analyzing and inferring the underlying global intrinsic structures of data from its local information are critical in many fields. In practice, coherent structures of data allow us to model data as low dimensional manifolds, represented as point clouds, in a possible high dimensional space. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data sets are far more challenging due to their complicated geometry. For example, the same geometric object can take very different coordinate representations due to the variety of embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). These ambiguities form an infinite dimensional isometric group and make higher-level tasks in manifold-structured data analysis and understanding even more challenging. To overcome these ambiguities, I will first discuss modeling based methods. This approach uses geometric PDEs to adapt the intrinsic manifolds structure of data and extracts various invariant descriptors to characterize and understand data through solutions of differential equations on manifolds. Inspired by recent developments of deep learning, I will also discuss our recent work of a new way of defining convolution on manifolds and demonstrate its potential to conduct geometric deep learning on manifolds. This geometric way of defining convolution provides a natural combination of modeling and learning on manifolds. It enables further applications of comparing, classifying and understanding manifold-structured data by combing with recent advances in deep learning.

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.

Electrophysiological recordings of neurons in the cortex of mammals reveal a ubiquitous high degree of irregularity of single neuron activity. The mechanisms and functional role of this irregular activity remain the subject of debate. Here, I will describe simplified models of networks of neurons, and analytical tools that can be used to understand their dynamics. Under some conditions, such networks can be described using a system of coupled Fokker-Planck equations (one for each class of neurons composing the network), in which the drift and diffusion terms depend on the probability flux at firing threshold. Provided specific conditions on network connectivity are satisfied, these models reproduce some of the landmark features observed in experiments (highly irregular firing at low rates, weak correlations between neurons, wide distributions of firing rates). Interestingly, these networks show a rich diversity of irregular states (chaotic or not, asynchronous or synchronous).

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.

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.

Chemotaxis plays a crucial role in a variety of processes in biology and ecology. One of the most studied PDE models of chemotaxis is given by Keller-Segel equation, which describes a population density of bacteria or mold which attract chemically to substance they secrete. However, solution of Keller-Segel equation can exhibit dramatic collapsing behavior. In other words, there exist initial data leading to finite time blow up. In this talk, we will discuss the possible effects resulting from interaction of chemotactic and fluid transport processes, namely we will consider the Keller-Segel equation with additional advection term modeling ambient fluid flow. We will prove that the presence of fluid can prevent the singularity formation. We will discuss two classes of flows that have the explosion arresting property. Both classes are known as very efficient mixers.

I will present various results on the Langevin dynamics, both from theoretical and numerical perspectives. This dynamics is quite popular for sampling purposes in computational statistical physics. It can be seen as a Hamiltonian dynamics perturbed by an Ornstein-Uhlenbeck process on the momenta. I will start on the theoretical side with an account of the hypocoercive approach by Dolbeault, Mouhot and Schmeiser, which is a key technique to prove that the asymptotic variance of time averages is well defined, and also to obtain quantitative bounds on it. I will then discuss various extensions/modifications of the standard Langevin dynamics, such as replacing the standard quadratic kinetic energy by a more general one, constructing control variates relying on a simplified Poisson equation, proving the convergence of nonequilibrium versions such as the one encountered in the Temperature Accelerated Molecular Dynamics method, etc.

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).

Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such models leads to the question of representing general L^2-data as the divergence of uniformly bounded vector fields.
We use a multi-scale approach to construct uniformly bounded solutions of div(U)=f for general f’s in the critical regularity space L^2(T^2). The study of this equation and related problems was motivated by results of Bourgain & Brezis. The intriguing critical aspect here is that although the problems are linear, construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations $U = \sum_j u_j$ which we introduced earlier in the context of image processing, yielding a multi-scale decomposition of “images” U.

Many problems arising from scientific and engineering applications can be naturally formulated as optimization problems, most of which are nonconvex. For nonconvex problems, obtaining a local minimizer is computationally hard in theory, never mind the global minimizer. In practice, however, simple numerical methods often work surprisingly well in finding high-quality solutions for specific problems at hand.

In this talk, I will describe our recent effort in bridging the mysterious theory-practice gap for nonconvex optimization. I will highlight a family of nonconvex problems that can be solved to global optimality using simple numerical methods, independent of initialization. This family has the characteristic global structure that (1) all local minimizers are global, and (2) all saddle points have directional negative curvatures. Problems lying in this family cover various applications across machine learning, signal processing, scientific imaging, and more. I will focus on two examples we worked out: learning sparsifying bases for massive data and recovery of complex signals from phaseless measurements. In both examples, the benign global structure allows us to derive geometric insights and computational results that are inaccessible from previous methods. In contrast, alternative approaches to solving nonconvex problems often entail either expensive convex relaxation (e.g., solving large-scale semidefinite programs) or delicate problem-specific initializations.

Completing and enriching this framework is an active research endeavor that is being undertaken by several research communities. At the end of the talk, I will discuss open problems to be tackled to move forward.

Single-cell RNA sequencing (scRNA-Seq) has emerged as a powerful tool to sample the complexity of large populations of cells and observe biological processes at unprecedented molecular resolution. This offers the exciting prospect of understanding the molecular programs that guide cellular differentiation during development. Here, we introduce Waddington-OT: a mathematical framework for understanding the temporal dynamics of development based on snapshots of expression profiles. The central challenge in analyzing these data arises from the fact that scRNA-Seq is destructive, which means that one cannot directly measure the trajectory of any given cell over time. We model the population of developing cells mathematically with a time-varying probability distribution (i.e. stochastic process) on a high-dimensional gene expression space, and we propose to recover the temporal coupling of the process with optimal transport. We demonstrate the power of Waddington-OT by applying the approach to study 315,000 scRNA-seq profiles collected at 40 time points over 16 days during reprogramming of fibroblasts to induced pluripotent stem cells. We construct a high-resolution map of reprogramming that rediscovers known features; uncovers new alternative cell fates including neural- and placental-like cells; predicts the origin and fate of any cell class; and implicates regulatory models in particular trajectories. Of these findings, we highlight Obox6, which we experimentally show enhances reprogramming efficiency. Our approach provides a general framework for investigating cellular differentiation.

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.

In this talk I will discuss the application of techniques initially developed to study singularities of Yang Mill's fields and harmonic maps to obtain Betti number bounds, especially for negatively curved manifolds.

For initial datum of finite kinetic energy Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this talk, I will discuss recent joint work with Vlad Vicol in which we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy.

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.

Granular materials are common in everyday life but are historically difficult to model. This has direct ramifications owing to the prominent role granular media play in multiple industries and terrain dynamics. One can attempt to track every grain with discrete particle methods, but realistic systems are often too large for this approach and a continuum model is desired. However, granular media display unusual behaviors that complicate the continuum treatment: they can behave like solid, flow like liquid, or separate into a "gas", and the rheology of the flowing state displays remarkable subtleties that have been historically difficult to model. To address these challenges, in this talk we develop a family of continuum models and solvers, permitting quantitative modeling capabilities for a variety of applications, ranging from general problems to specific techniques for problems of intrusion, impact, driving, and locomotion in grains.

To calculate flows in general cases, a rather significant nonlocal effect is evident, which is well-described with our recent nonlocal model accounting for grain cooperativity within the flow rule. This model enables us to capture a number of seemingly disparate manifestations of particle size-effects in granular flows including: (i) the wide shear-band widths observed in many inhomogeneous flows, (ii) the apparent strengthening exhibited in thin layers of grains, and (iii) the fluidization observed due to far-away motion of a boundary. On the other hand, to model only intrusion forces on submerged objects, we will show, and explain why, many of the experimentally observed results can be captured from a much simpler tension-free frictional plasticity model. This approach gives way to some surprisingly simple general tools, including the granular Resistive Force Theory, and a broad set of scaling laws inherent to the problem of granular locomotion. These scalings are validated experimentally and in discrete particle simulations suggesting a new down-scaled paradigm for granular locomotive design, on earth and beyond, to be used much like scaling laws in fluid mechanics.

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.

For 2D Euler flows, it is known that the L^\infty norm of the gradient of vorticity can grow with at most double exponential rate in time. In recent years, this bound has been proven to be sharp by Kiselev and Sverak on the unit disc. We will examine the possibility of growth in the 3D axisymmetric setting for flows without swirl component.

The detailed behavior of solutions to the biharmonic equation on regions with corners has been historically difficult to characterize. It is conjectured by Osher (and proven in certain special cases) that the Green’s function for the biharmonic equation on regions with corners has infinitely many oscillations in the vicinity of each corner. In this talk, we show that, when the biharmonic equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of elementary functions which oscillate with a frequency proportional to the logarithm of the distance from the corner. These representations are used to construct highly accurate and efficient Nyström discretizations, significantly reducing the number of degrees of freedom required for solving the corresponding integral equations. We illustrate the performance of our method with several numerical examples.

In this talk, we will explore stochastic motion of cellular cycles inside CW complexes. This serves as a generalization of random walks on graphs, and a discretization of stochastic flows on smooth manifolds. We will define a notion of stochastic current, connect it to classical electric current, and show it satisfies a quantization result. Along the way, we will define the main combinatorial objects of study, namely spanning trees and spanning co-trees in higher dimensions. We will relate these to stochastic current, as well as discrete Hodge theory.

We study coagulation-fragmentation equations inspired by a simple model derived in fisheries science to explain data on the size distribution of schools of pelagic fish. The equations lack detailed balance and admit no H-theorem, but we are anyway able to develop a rather complete description of equilibrium profiles and large-time behavior, based on complex function theory for Bernstein and Pick (Herglotz) functions. The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers that count all ternary trees with $n$ nodes. The structure of equilibrium profiles and other related sequences is explained through a new and elegant characterization of the generating functions of completely monotone sequences, as those Pick functions analytic and nonnegative on a half line. This is joint work with Jian-Guo Liu and Pierre Degond.

In this talk, I will start with a simple algorithm for community detection in stochastic block models and discuss its statistical optimality. After that, we will discuss two related issues. One is model selection for stochastic block models. The other is the extension to community detection in degree-corrected block models. We shall pay close attention to the achievability of statistical optimality by computationally feasible procedures throughout the talk.

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 need to deduce reduced computational models from discrete observations of complex systems arises in many climate and engineering applications. The challenges come mainly from memory effects due to the unresolved scales and nonlinear interactions between resolved and unresolved scales, and from the difficulty in inference from discrete data.

We address these challenges by introducing a discrete-time stochastic parametrization framework, through which we construct discrete-time stochastic models that can take memory into account. We show by examples that the resulting stochastic reduced models that can capture the long-time statistics and can make accurate short-term predictions. The examples include the Lorenz 96 system (which is a simplified model of the atmosphere) and the Kuramoto-Sivashinsky equation of spatiotemporally chaotic dynamics.

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.

Transportation networks are ubiquitous as they are possibly the most important building blocks of nature. They cover microscopic and macroscopic length scales and evolve on fast to slow times scales. Examples are networks of blood vessels in mammals, genetic regulatory networks and signaling pathways in biological cells, neural networks in mammalian brains, venation networks in plant leafs and fracture networks in rocks. We present and analyze a PDE (Continuum) framework to model transportation networks in nature, consisting of a reaction-diffusion gradient-flow system for the network conductivity constrained by an elliptic equation for the transported commodity (fluid).

We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales char- acterized by the magnitude of a dimensionless parameter—the Knudsen number (Kn). In the diffusive scaling (Kn ≪ 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well- posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn → 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Knp (p = 1 or 2), and as a result lead to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.

The Hilbert space dimension of quantum-many body systems grows exponentially with the system size. Fortunately, nature does usually not explore this monstrous number of degrees of freedom and we have a chance to describe quantum systems with much smaller sets of effective degrees of freedom. A very precise description for systems with one spatial dimension is based on so-called matrix product states (MPS). With such a reduced parametrization, the computation cost, needed to achieve a certain accuracy, is determined by entanglement properties (quantum non-locality) in the system.
I will give a short introduction to the notion of entanglement entropies and their scaling behavior in typical many-body systems. I will then employ entanglement entropies to bound the required computation costs in MPS simulations. This will lead us to the amazing conclusion that 1D quantum many-body systems can usually be simulated efficiently on classical computers, both for zero and finite temperatures, and for both gapless and critical systems.
In these considerations, we will encounter a number of mathematical concepts such as the theorem of typical sequences (central limit theorem), concentration of measure (Levy's lemma), singular value decomposition, path integrals, and conformal invariance.

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.