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.

We will discuss recent approximations to boundary value problems to non-linear systems of Boltzmann or Landau (for Coulombic interactions) equations coupled to the Poisson equation. The proposed approximation methods involve hybrid schemes of spectral and Galerkin type, were conservation of flow invariants are achieved by a constrain minimization problem. We will discuss some analytical and computational issues related to these approximations.

The first reaction-diffusion equation developed and studied is the Fisher-KPP equation. Introduced in 1937, this population-dynamics model accounts for the spatial spreading and growth of a species. Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al. I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions. Afterwards I will describe the new model for the cane toads equations and give new results regarding this model. In particular, I will show how the model may be viewed as a perturbation of a local equation using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads. The talk is based on a joint work with Bouin and Ryzhik.

The emergence of massive data sets, over the past twenty or so years, has lead to the development of Randomized Numerical Linear Algebra. Fast and accurate randomized matrix algorithms are being designed for applications like machine learning, population genomics, astronomy, nuclear engineering, and optimal experimental design. We give a flavour of randomized algorithms for the solution of least squares/regression problems. Along the way we illustrate important concepts from numerical analysis (conditioning and pre-conditioning), probability (concentration inequalities), and statistics (sampling and leverage scores).

We report on an experimental and theoretical study of a molecularly thin polymer Langmuir layers on the surface of a Stokesian subfluid. Langmuir layers can have multiple phases (fluid, gas, liquid crystal, isotropic or anisotropic solid); at phase boundaries a line tension force is observed. By comparing theory and experiment we can estimate this line tension. We first consider two co-existing fluid phases; specifically a localized phase embedded in an infinite secondary phase. When the localized phase is stretched (by a transient stagnation flow), it takes the form of a bola consisting of two roughly circular reservoirs connected by a thin tether. This shape relaxes to the minimum energy configuration of a circular domain. The tether is never observed to rupture, even when it is more than a hundred times as long as it is thin. We model these experiments by taking previous descriptions of the full hydrodynamics (primarily those of Stone & McConnell and Lubensky & Goldstein), identifying the dominant effects via dimensional analysis, and reducing the system to a more tractable form. The result is a free boundary problem where motion is driven by the line tension of the domain and damped by the viscosity of the subfluid. The problem has a boundary integral formulation which allows us to numerically simulate the tether relaxation; comparison with the experiments allows us to estimate the line tension in the system. We also report on incorporating dipolar repulsion into the force balance and simulating the formation of "labyrinth" patterns.

Sleep-wake cycling is an example of switching between discrete states in mammalian brain. Based on the experimental data on the activity of populations of neurons, we develop a mathematical model. The model incorporates several different time scales: firing of action potentials (milliseconds), sleep and wake bout times (seconds), developmental time (days). Bifurcation diagrams in a deterministic dynamical system gives the occupancy time distributions in the corresponding stochastic system. The model correctly predicts that forebrain regions help to stabilize wake state and thus modifies the wake bout distribution.

The nascent technique of 4D printing has the potential to revolutionize manufacturing in fields ranging from organs-on-a-chip to architecture to soft robotics. By expanding the pallet of 3D printable materials to include the use stimuli responsive inks, 4D printing promises precise control over patterned shape transformations. With the goal of creating a new manufacturing technique, we have recently introduced a biomimetic printing platform that enables the direct control of local anisotropy into both the elastic moduli and the swelling response of the ink.

We have drawn inspiration from nastic plant movements to design a phytomimetic ink and printing process that enables patterned dynamic shape change upon exposure to water, and possibly other external stimuli. Our novel fiber-reinforced hydrogel ink enables local control over anisotropies not only in the elastic moduli, but more importantly in the swelling. Upon hydration, the hydrogel changes shape accord- ing the arbitrarily complex microstructure imparted during the printing process.

To use this process as a design tool, we must solve the inverse problem of prescribing the pattern of anisotropies required to generate a given curved target structure. We show how to do this by constructing a theory of anisotropic plates and shells that can respond to local metric changes induced by anisotropic swelling. A series of experiments corroborate our model by producing a range of target shapes inspired by the morphological diversity of flower petals.

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.

The Patlak-Keller-Segel equations (PKS) are widely applied to model the chemotaxis phenomena in biology. It is well-known that if the total mass of the initial cell density is large enough, the PKS equations exhibit finite time blow-up. In this talk, I present some recent results on applying additional fluid flows to suppress chemotactic blow-up in the PKS equations. These are joint works with Jacob Bedrossian and Eitan Tadmor.

The heart is a coupled electro-fluid-mechanical system. The contractions of the cardiac muscle are stimulated and coordinated by the electrophysiology of the heart; these contractions in turn affect the electrical function of the heart by altering the macroscopic conductivity of the tissue and by influencing stretch-activated transmembrane ion channels. In this talk, I will present mathematical models and adaptive numerical methods for describing cardiac mechanics, fluid dynamics, and electrophysiology, as well as applications of these models and methods to cardiac fluid-structure and electro-mechanical interaction. I will also describe novel models of cardiac electrophysiology that go beyond traditional macroscopic (tissue-scale) descriptions of cardiac electrical impulse propagation by explicitly incorporating details of the cellular microstructure into the model equations. Standard models of cardiac electrophysiology, such as the monodomain or bidomain equations, account for this cellular microstructure in only a homogenized or averaged sense, and we have demonstrated that such homogenized models yield incorrect results in certain pathophysiological parameter regimes. To obtain accurate model predictions in these parameter regimes without resorting to a fully cellular model, we have developed an adaptive multiscale model of cardiac conduction that uses detailed cellular models only where needed, while resorting to the more efficient macroscale equations where those equations suffice. Applications of these methods will be presented to simulating cardiac and cardiovascular dynamics in whole heart models, as well as in detailed models of cardiac valves and novel models of aortic dissection. Necessary physiological details will be introduced as needed.

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.

Functionalized polymer membranes have a strong affinity for solvent, imbibing it to form charge-lined networks which serve as charge-selective ion conductions in a host of energy conversion applications. We present a continuum model, based upon a reformulation of the Cahn-Hilliard free energy, which incorporates solvation energy and counter-ion entropy to stabilize a host of network morphologies. We derive geometric evolution for co-dimension 1 bilayers and co-dimension two pore morphologies and show that the system possesses a simple mechanism for competitive evolution of co-existing networks through the common far-field chemical potential.

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.

In order to understand the nonlinear stability of many types of time-periodic traveling waves on unbounded domains, one must overcome two main difficulties: the presence of zero eigenvalues that are embedded in the continuous spectrum and the time-periodicity of the associated linear operator. I will outline these issues and show how they can be overcome in the context of time-periodic Lax shocks in systems of viscous conservation laws. The method involves the development of a contour integral representation of the linear evolution, similar to that of a strongly continuous semigroup, and detailed pointwise estimates on the resultant Greens function, which are sufficient for proving nonlinear stability under the necessary assumption of spectral stability.

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).

The use of PDE models to describe complex systems in the social sciences, such as socio-economic segregation and crime, has been popularized during the past decade. In this talk I will introduce some PDE models which can be seen as basic models for a variety of social phenomena. I will then discuss how these models can be used to explore and gain understanding of the real-world systems they describe. For example, we learn that a populations innate views toward criminal activity can play a significant role in the prevention of crime-wave propagation.

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.

In this talk, I will present a kernel-free boundary integral (KFBI) method for the variable coefficient elliptic partial differential equation on complex domains. The KFBI method is a generalization of the standard boundary integral method. But, unlike the standard boundary integral method, the KFBI method does not need to know an analytical expression for the kernel of the boundary integral operator or the Green's function associated with the elliptic PDE. So it is not limited to the constant-coefficient PDEs. The KFBI method solves the discrete integral equations by an iterative method, in which only part of the matrix vector multiplication involves the discretization of the boundary integral. With the KFBI method, the evaluation of the boundary integral is replaced by interpolation from a structured grid based solution to an equivalent interface problem, which is solved quickly by a Fourier transform or geometric multigrid based fast elliptic solver. Numerical examples for Dirichlet and Neumann BVPs, interface problems with different conductivity constants and the Poisson-Boltzmann equations will be presented.

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).

Finite difference methods are awkward for solving boundary value problems, such as the Dirichlet problem, with general boundaries, but they are well suited for interface problems, which have prescribed jumps in an unknown across a general interface or boundary. The two problems can be connected through potential theory: The Dirichlet boundary value problem is converted to an integral equation on the boundary, and the integrals can be thought of as solutions to interface problems. Wenjun Ying et al. have developed a practical method for solving the Dirichlet problem, and more general ones, by solving interface problems with finite difference methods and iterating to mimic the solution of the integral equation. We will describe some analysis which proves that a simplified version of Ying's method works. A recent view of classical potential theory leads to a finite difference version of the theory in which, remarkably, the crude discrete operators have much of the structure of the exact operators. This simplified method produces the Shortley-Weller solution of the Dirichlet problem. Details can be found at arxiv.org/abs/1803.08532 .

In this talk, we settle a long-standing problem on the connectivity of spaces of finite unit norm tight frames (FUNTFs), essentially affirming a conjecture first appearing in Dykema and Strawn (2003). Our central technique involves continuous liftings of paths from the polytope of eigensteps (see Cahill et al. (2012)) to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense, and that generic FUNTFs are full spark.

In phase retrieval, one aims to recover a signal from magnitude measurements. In the literature, an effective SDP algorithm, referred to as PhaseLift, was proposed with numerical success as well as strong theoretical guarantees. In this talk, I will first introduce some recent theoretical developments for PhaseLift, which demonstrate the applicability and adaptivity of this convex method. Although convex methods are provably effective and robust, the computational complexity may be relatively high. Moreover, there is often an issue of storage to solve the lifted problem. To address these issues, we introduce a nonconvex optimization algorithm, named Wirtinger flow, with theoretically guaranteed performance. It is much more efficient than convex methods in terms of computation and memory. Finally, I will introduce how to modify Wirtinger flow when the signal is known to be sparse, in order to improve the accuracy of the recovery.

In radio signal encoding and decoding, frequency modulation (FM) has several advantages over amplitude modulation (AM) - we all enjoy the high fidelity and nearly static free reception of FM radio. Sensing and imaging can also be approached with AM or FM modalities. All imaging methods practiced today are AM implementations, and FM for imaging has never been explored or its advantages exploited. In this talk I'll introduce the FM approach to sensing and imaging in its infant form. I'll show that the FM approach is closely related to design of Gaussian quadratures for bandlimited functions. I'll demonstrate the superiorities of the FM approach over AM by proposing three FM methods to deal with Gibbs phenomenon encountered in imaging.

For a more detailed abstract, see http://www.math.duke.edu/~jonm/yuChen.html

The speaker will talk about several projects that are taking place in an interdisciplinary endeavor between the researchers in the Mathematics Department at the University of Houston, the Texas Heart Institute, Baylor College of Medicine, the Mathematics Department at the University of Zagreb, and the Mathematics Department of the University of Lyon 1. The projects are related to non-surgical treatment of aortic abdominal aneurysm and coronary artery disease using endovascular prostheses called stents and stent-grafts. Through a collaboration between mathematicians, cardiovascular specialists and engineers we have developed a novel mathematical model to study blood flow in compliant (viscoelastic) arteries treated with stents and stent-grafts. The mathematical tools used in the derivation of the effective, reduced equations utilize asymptotic analysis and homogenization methods for porous media flows. The existence of a unique solution to the resulting fluid-structure interaction model is obtained by using novel techniques to study systems of mixed, hyperbolic-parabolic type. A numerical method, based on the finite element approach, was developed, and numerical solutions were compared with the experimental measurements. Experimental measurements based on ultrasound and Doppler methods were performed at the Cardiovascular Research Laboratory located at the Texas Heart Institute. Excellent agreement between the experiment and the numerical solution was obtained. This year marks a giant step forward in the development of medical devices and in the development of the partnership between mathematics and medicine: the FDA (the United States Food and Drug Administration) is getting ready to, for the first time, require mathematical modeling and numerical simulations to be used in the development of peripheral vascular devices. The speaker acknowledges research support from the NSF, NIH, and Texas Higher Education Board, and donations from Medtronic Inc. and Kent Elastomer Inc.

We consider composite media with a broad range of scales, whose effective properties are important in materials science, biophysics, and climate modeling. Examples include random resistor networks, polycrystalline media, porous bone, the brine microstructure of sea ice, ocean eddies, melt ponds on the surface of Arctic sea ice, and the polar ice packs themselves. The analytic continuation method provides Stieltjes integral representations for the bulk transport coefficients of such systems, involving spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices or discretizations of continuum systems, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. In this lecture we will discuss various implications and applications of these integral representations. We will also discuss computations of the spectral measures of the operators, as well as statistical measures of their eigenvalues. For example, the effective behavior of composite materials often exhibits large changes associated with transitions in the connectedness or percolation properties of a particular phase. We demonstrate that an onset of connectedness gives rise to striking transitional behavior in the short and long range correlations in the eigenvalues of the associated random matrix. This, in turn, gives rise to transitional behavior in the spectral measures, leading to observed critical behavior in the effective transport properties of the media.

From a mathematical point of view self-organization can be described as the formation of patterns, where certain dynamical systems modeling social dynamics tend autonomously to converge. The fascinating mechanism to be revealed by such a modeling is how to connect the microscopical and usually binary rules or social forces of interaction between individuals with the eventual global behavior or group pattern, forming as a superposition in time of the different microscopical effects. In this talk we explore mechanisms to go beyond self-organization, in particular how to externally control such dynamical systems in order to eventually enforce pattern formation also in those situations where this wished phenomenon does not result from spontaneous and autonomous convergence. Our focus is on dynamical systems of Cucker-Smale type, modeling consensus emergence, and we question the existence of stabilization and optimal control strategies which require the minimal amount of external intervention for nevertheless inducing consensus in a group of interacting agents. On the one hand and formally, our main result realizes the connection between certain variational problems involving L1-norm terms and optimal sparse controls. On the other hand, our findings can be informally stated in terms of the general principle for which "A policy maker should always consider more favorable to intervene with stronger actions on the fewest possible instantaneous optimal leaders than trying to control more agents, with minor strength".

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.

Geometric problems - such as finding corresponding points over a collection of shapes, or computing shape deformation under geometric constraints - pose various computational challenges. I will show that despite the very different nature of these two highly non-convex problems, Semidefinite Programming (SDP) can be leveraged to provide a tight convex approximation in both cases. A different approach is used for each problem, demonstrating the versatility of SDP: (i) For establishing point correspondences between shapes, we devise an SDP relaxation. I will show it is a hybrid of the popular spectral and doubly-stochastic relaxations, and is in fact tighter than both. (ii) For the computation of piecewise-linear mappings, we introduce a family of maximal SDP restrictions. Solving a sequence of such SDPs enables the optimization of functionals and constraints expressed in terms of singular values, which naturally model various geometry processing problems.