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.

Adaptive acquisition of correct features from massive datasets is at the core of modern data analysis. One particular interest in medicine is the extraction of hidden dynamics from an observed time series composed of multiple oscillatory signals. The mathematical and statistical problems are made challenging by the structure of the signal which consists of non-sinusoidal oscillations with time varying amplitude and time varying frequency, and by the heteroscedastic nature of the noise. In this talk, I will discuss recent progress in solving this kind of problem. Based on the cepstrum-based nonlinear time-frequency analysis and manifold learning technique, a particular solution will be given along with its theoretical properties. I will also discuss the application of this method to two medical problems – (1) the extraction of a fetal ECG signal from a single lead maternal abdominal ECG signal; (2) the simultaneous extraction of the instantaneous heart rate and instantaneous respiratory rate from a PPG signal during exercise. If time permits, an extension to multiple-time series will be discussed.

For a range of physical and biological processes—from dynamics of granular media to biological swarming—the evolution of a large number of interacting agents is modeled according to the competing effects of pairwise attraction and (possibly degenerate) diffusion. In the slow diffusion limit, the degenerate diffusion formally becomes a hard height constraint on the density of the population, as arises in models of pedestrian crown motion. Motivated by these applications, we bring together new results on the Wasserstein gradient flow of nonconvex energies with the theory of free boundaries to study a model of Coulomb interaction with a hard height constraint. Our analysis demonstrates the utility of Wasserstein gradient flow as a tool to construct and approximate solutions, alongside the strength of viscosity solution theory in examining their precise dynamics. By combining these two perspectives, we are able to prove quantitative estimates on convergence to equilibrium, which relates to recent work on asymptotic behavior of the Keller-Segel equation. This is joint work with Inwon Kim and Yao Yao.

Contemporary and ancient demographic structure in human populations has shaped the genomic variation observed in modern humans, and severely affected the distribution of functional and disease related genetic variation. Using next-generation sequencing technologies, researchers gather increasing amounts of genomic sequencing data for large samples in many different human population groups. These datasets present unprecedented opportunities to study genomic variation in complex demographic scenarios, and this area has received a lot of attention in recent years. In this talk, I will present a method for the inference of demographic histories from full-genome sequencing data of multiple individuals developed by me and my collaborators. I will apply this method to a genomic dataset of Native American individuals to unravel the ancient demographic events underlying the peopling of the Americas. Moreover, I will discuss a novel method for demographic inference that has the potential to improve inference especially in the recent past, which is of particular importance in the context of complex genetic diseases in humans.

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.

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.

This talk focuses on two instances in which scientific fields outside mathematics benefit from incorporating the geometry of the data. In each instance, the application area motivates the need for new mathematical approaches and algorithms and leads to interesting new questions. (1) A method to determine and predict drug treatment effectiveness for patients based off their baseline information. This motivates building a function adapted diffusion operator on high-dimensional data X when the function F can only be evaluated on large subsets of X, and defining a localized filtration of F and estimation values of F at a finer scale than it is reliable naively. (2) The current empirical success of deep learning in imaging and medical applications, in which theory and understanding is lagging far behind. By assuming the data lie near low-dimensional manifolds and building local wavelet frames, we improve on existing theory that breaks down when the ambient dimension is large (the regime in which deep learning has seen the most success).

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.

At the beginning of this talk, We will introduce the background of tensor network states (TNS) in various areas such as quantum physics, quantum chemistry and numerical partial differential equations. Famous TNS including tensor trains (TT), matrix product states (MPS), projected entangled pair states (PEPS) and multi-scale entanglement renormalization ansatz (MERA). Then We will explain how to define TNS by graphs and we will define tensor network ranks which can be used to measure the complexity of TNS. We will see that the notion of tensor network ranks is an analogue of tensor rank and multilinear rank. We will discuss basic properties of tensor network ranks and the comparison among tensor network ranks, tensors rank and multilinear rank. If time permits, we will also discuss the dimension of tensor networks and the geometry of TNS. This talk is based on papers joined with Lek-Heng Lim.

Obtaining predictive dynamical equations from data lies at the heart of science and engineering modeling, and is the linchpin of our technology. In mathematical modeling one typically progresses from observations of the world (and some serious thinking!) first to selection of variables, then to equations for a model, and finally to the analysis of the model to make predictions. Good mathematical models give good predictions (and inaccurate ones do not) --- but the computational tools for analyzing them are the same: algorithms that are typically operating on closed form equations.
While the skeleton of the process remains the same, today we witness the development of mathematical techniques that operate directly on observations --- data, and appear to circumvent the serious thinking that goes into selecting variables and parameters and deriving accurate equations. The process then may appear to the user a little like making predictions by "looking into a crystal ball". Yet the "serious thinking" is still there and uses the same --- and some new --- mathematics: it goes into building algorithms that "jump directly" from data to the analysis of the model (which is now not available in closed form) so as to make predictions. Our work here presents a couple of efforts that illustrate this "new" path from data to predictions. It really is the same old path, but it is traveled by new means.

Guided waves arise in a variety of applications like underwater acoustics, optics, the design of musical instruments, and so on. We present an analysis of wave propagation and reflection in an acoustic waveguide with random sound soft boundary and a turning point. The waveguide has slowly bending axis and variable cross section. The variation consists of a slow and monotone change of the width of the waveguide and small and rapid fluctuations of the boundary, on the scale of the wavelength. These fluctuations are modeled as random. The turning point is many wavelengths away from the source, which emits a pulse that propagates toward the turning point, where it is reflected. We consider a regime where scattering at the random boundary has a significant effect on the reflected pulse. We determine from first principles when this effects amounts to a deterministic pulse deformation. This is known as a pulse stabilization result. The reflected pulse shape is not the same as the emitted one. It is damped, due to scattering at the boundary, and is deformed by dispersion in the waveguide. An example of an application of this result is in inverse problems, where the travel time of reflected pulses at the turning points can be used to determine the geometry of the waveguide.

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.

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).)

A common question arising in the study of graphs is how to partition nodes into well-connected clusters. One standard methodology is known as spectral clustering and utilizes an eigenvector embedding as a staring point for clustering the nodes. Given that embedding, we present a new algorithm for spectral clustering based on a column-pivoted QR factorization. Our method is simple to implement, direct, scalable, and requires no initial guess. We also provide theoretical justification for our algorithm and experimentally demonstrate that its performance tracks recent information theoretic bounds for exact recovery in the stochastic block model. Algorithmically, the origins of this work are in methods for building a localized basis for Kohn-Sham orbitals, and we briefly discuss those connections.

The incompressible Euler equation of fluid mechanics has been derived in 1755. It is one of the central equations of applied analysis, yet due to its nonlinearity and non-locality many fundamental properties of solutions remain poorly understood. In particular, the global regularity vs finite time blow up question for incompressible three dimensional Euler equation remains open. In two dimensions, it has been known since 1930s that solutions to Euler equation with smooth initial data are globally regular. The best available upper bound on the growth of derivatives of the solution has been double exponential in time. I will describe a construction showing that such fast generation of small scales can actually happen, so that the double exponential bound is qualitatively sharp. This work has been motivated by numerical experiments due to Hou and Luo who propose a new scenario for singularity formation in solutions of 3D Euler equation. The scenario is axi-symmetric. The geometry of the scenario is related to the geometry of 2D Euler double exponential growth example and involves hyperbolic points of the flow located at the boundary of the domain. If time permits, I will discuss some recent attempts to gain insight into the three-dimensional fluid behavior in this scenario.

Exploring data structures (e.g, periodicity, sparsity, low-rankness) is a universal method in designing fast algorithms in scientific computing. In the first part of this talk, I will show how this idea is applied to the analysis of oscillatory data in applied harmonic analysis. These fast algorithms have been applied to data analysis ranging from materials science, medicine, and art. In the second part, I will discuss how this idea works in some basic numerical linear algebra routines like matrix multiplications and decompositions, with an emphasis in electronic structure calculation.