High performance transducers utilizing piezoceramic, electrostrictive, magnetostrictive or shape memory elements offer novel control capabilities in applications ranging from flow control to precision placement for nanoconstruction. To achieve the full potential of these materials, however, models, numerical methods and control designs which accommodate the constitutive nonlinearities and hysteresis inherent to the compounds must be employed. Furthermore, it is advantageous to consider material characterization, model development, numerical approximation, and control design in concert to fully exploit the novel sensor and actuator capabilities of these materials in coupled systems.

In this presentation, the speaker will discuss recent advances in the development of model-based control strategies for high performance smart material systems. The presentation will focus on the development of unified nonlinear hysteresis models, inverse compensators, reduced-order approximation techniques, and nonlinear control strategies for high precision or high drive regimes. The range for which linear models and control methods are applicable will also be outlined. Examples will be drawn from problems arising in structural acoustics, high speed milling, deformable mirror design, artificial muscle development, tendon design to minimize earthquake damage, and atomic force microscopy.

In this talk I will discuss the Keller-Segel equation, which is a nonlocal PDE modeling the collective motion of cells attracted by a self-emitted chemical substance. When this equation is set up in 2D with a degenerate diffusion term, it is known that solutions exist globally in time, but their long-time behavior remain unclear. To answer this question, we investigate a general aggregation equation with degenerate diffusion, and prove that all stationary solutions must be radially symmetric up to a translation. As a consequence, this enables us to obtain a convergence result for solutions to 2D Keller-Segel equation with degenerate diffusion as the time goes to infinity. This is a joint work with J. Carrillo, S. Hittmeir and B. Volzone.

The Vicsek model, describing alignment and self-organisation in large systems of self-propelled particles, such as fish schools or flocks of birds, has attracted a lot of attention with respect to its simplicity and its ability to reproduce complex phenomena. We consider here a time-continuous version of this model, in the spirit of the one proposed by P. Degond and S. Motsch, but where the rate of alignment is proportional to the mean speed of the neighboring particles. In the hydrodynamic limit, this model undergoes a phase transition phenomenon between a disordered and an ordered phase, when the local density crosses a threshold value. We present the two different macroscopic limits we can obtain under and over this threshold, namely a nonlinear diffusion equation for the density, and a first-order non-conservative hydrodynamic system of evolution equations for the local density and orientation. (joint work with Pierre Degond and Jian-Guo Liu).

There has been high scientific interest to understand the behavior of the surface quasi-geostrophic (SQG) equation because it is a possible model to explain the formation of fronts of hot and cold air and because it also exhibits analogies with the 3D incompressible Euler equations. It is not known at this moment if this equation can produce singularities or if solutions exist globally. In this talk I will discuss some recent works on the existence of global solutions.

In this work, we combine the adaptive mesh refinement (AMR) framework with high order finite difference weighted essentially non-oscillatory (WENO) method in space and TVD Runge-Kutta (RK) method in time (WENO-RK) for hyperbolic conservation laws. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. To maintain high order accuracy, we use high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from the coarse to find grid, and at ghost points. The resulting scheme is high order accuracy, robust and efficient, due to the mesh adaptivity and has high order accuracy in both space and time. We have experimented the third and fifth order AMR-finite difference WENO-RK schemes. The accuracy of the scheme is demonstrated by applying the method to several smooth test problems, and the quality and efficiency are demonstrated by applying the method to the shallow water and Euler equations with different challenging initial conditions. From our numerical experiment, we conclude a significant improvement of the fifth order AMR - WENO scheme over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, which we think is due to the very low numerical diffusion of high order schemes. This work is in collaboration with Dr. Chaopeng Shen and Professor Jing-Mei Qiu.

I review some works on modeling and the mathematical theory for dilute suspensions of rigid rods. Such problems appear in modeling sedimentation of suspensions of particles. Similar in spirit models are also used for modeling swimming micro-organisms. Here, we focus on a class of models introduced by Doi and describing suspensions of rod¨Clike molecules in a solvent fluid. They couple a microscopic Fokker-Planck type equation for the probability distribution of rod orientations to a macroscopic Stokes flow. One objective is to compare such models with traditional models used in macoscopic viscoelasticity as the well known Oldroyd model. In particular: For the problem of sedimenting rods under the influence of gravity we discuss the instability of the quiescent flow and the derivation of effective equations describing the collective response. We derive two such effective theories: (i) One ammounts to a classical diffusive limit and produces a Keller-Segel type of model. (ii) A second approach involves the derivation of a moment closure theory and the approximation of moments via a quasi-dynamic approximation. This produces a model that belongs to the class of flux-limited Keller-Segel systems. The two theories are compared numerically with the kinetic equation. (joint work with Christiane Helzel, Univ. Duesseldorf).

We consider a droplet spreading on a thin liquid film where both fluids are Newtonian, incompressible, and immiscible. Rather than following the typical asymptotic derivation for a thin film, we formulate the model through a gradient flow approach. The sign of the spreading parameter indicates the spreading behavior (complete or partial spreading) and is a relation between the three interfacial tensions: fluid/air, fluid/drop, and drop/air. We are particularly interested in the case where the spreading parameter is negative. In this case, the drop is expected to spread to a static lens and we find the corresponding equilibrium solution. Finally, we make a comparison between the theoretical model and experimental results.

Multiscale problems are often treated via asymptotic of homogenization techniques: one first determines the asymptotic limit and then finds an appropriate numerical methods to solve it. Variable scale problems which exhibit a continuous variation of the perturbation parameter from a finite to an infinitesimal value cannot be solved by this method alone. They require the coupling of the asymptotic problem to the original one across the region of scale variation. This coupling is often quite complex and lacks robustness. Asymptotic-Preserving methods represent an alternative to the coupling strategy and provide a way to resolve the original problem without resorting to its asymptotic limit. They provide a systematic methodology to resolve multiscale problems even in situations where the asymptotic limit is quite complex. We will provide examples of this methodology for the treatment of the low-Mach number regime, of quasineutrality in plasmas, large magnetic fields or strong anisotropy in diffusion equations.

The potential impact of blood flow simulations on the diagnosis and treatment of patients suffering from vascular disease is tremendous. Empowering models of the full arterial tree can provide insight into diseases such as arterial hypertension and enables the study of the influence of local factors on global hemodynamics. We present a new, highly scalable implementation of the Lattice Boltzmann method which addresses key challenges such as multiscale coupling, limited memory capacity and bandwidth, and robust load balancing in complex geometries. We demonstrate the strong scaling of a three-dimensional, high-resolution simulation of hemodynamics in the systemic arterial tree on 1,572,864 cores of Blue Gene/Q. Faster calculation of flow in full arterial networks enables unprecedented risk stratification on a per-patient basis. In pursuit of this goal, we have introduced computational advances that significantly reduce time-to-solution for biofluidic simulations. In this talk, I will discuss the development of HARVEY, a parallel fluid dynamics application designed to model hemodynamics in patient-specific geometries. I will cover the methods introduced to reduce the overall time-to-solution and enable near-linear strong scaling on the IBM Blue Gene/Q supercomputer. Finally, I will present the expansion of the scope of projects to address not only vascular diseases, but also treatment planning and the movement of circulating tumor cells in the bloodstream.

With Jon Weare (Chicago), we study the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. With computational experiments and theoretical arguments we are able to derive several partial differential equation limits identified (or nearly identified) in previous studies and to clarify the correct choice of surface tension appearing in the PDE and the correct scaling regime giving rise to each PDE. We also provide preliminary computational and analytic investigations of a number of interesting qualitative features of the large scale behavior of the models. The PDE models involved are fully non-linear Fourth order diffusion type equations with many interesting geometric features. We will given time discuss recent progress analyzing properties of solutions to such PDE.

Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids along with the closely related class of polynomials called strongly log-concave polynomials. Strong log-concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients. Discrete probability distributions defined by these coefficients inherit several of these nice properties. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and describe applications to random walks on the faces of simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets of a matroid is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

We consider the vanishing viscosity limit of the Navier-Stokes equations in a half space, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the Navier-Stokes solutions remain bounded in $L^2_t L^\infty_x$ independently of the kinematic viscosity, and if they are equicontinuous at $x_2 = 0$. These conditions imply that there is no boundary layer separation: the Lagrangian paths originating in a boundary layer, stay in a proportional boundary layer during the time interval considered. We then give a proof of the (numerical) conjecture of vanDommelen and Shen (1980) which predicts the finite time blowup of the displacement thickness in the Prandtl boundary layer equations. This shows that the Prandtl layer exhibits separation in finite time.

The efficient parallelization of numerical methods for ordinary or partial differential equations in the temporal direction is an intriguing possibility that has of yet not been fully realized despite decades of investigation. For partial differential equations, virtually all large scale computations now employ parallelization across space, and there are freely available computational tools and libraries to aid in the development of spatially parallelized codes. Conversely, parallelization in the temporal direction is rarely even considered. I will discuss a relatively recent parallel strategy called the parareal algorithm that has generated a renewed wave of interest in time parallelization. I will show how the iterative structure of the parareal algorithm can be interpreted as a particular form of deferred corrections and then present a modified parareal strategy based on spectral deferred corrections that can significantly reduce the computational cost of the method. Finally I will make some observations as to why parallel in time methods may be attractive in the future.

The Boltzmann equation governs the dynamics of a dilute gas. In this talk, I will address the L^p-stability problem of the Boltzmann equation near vacuum and a global Maxwellian. In a close-to-vacuum regime, I will explain the nonlinear functional approach motivated by Glimm's theory in hyperbolic conservation laws. This functional approach yields the uniform L^1-stability estimate. In contrast, in a close-to-global maxwellian regime, I will present the L^2-stability theory which establishes the uniform L^2-stability of several classical solutions.

The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic microswimmers.

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.

This talk introduces a framework that uses ordinary differential equations to model, analyze, and interpret gradient-based optimization methods. In the first part of the talk, we derive a second-order ODE that is the limit of Nesterov’s accelerated gradient method for non-strongly objectives (NAG-C). The continuous-time ODE is shown to allow for a better understanding of NAG-C and, as a byproduct, we obtain a family of accelerated methods with similar convergence rates. In the second part, we begin by recognizing that existing ODEs in the literature are inadequate to distinguish between two fundamentally different methods, Nesterov’s accelerated gradient method for strongly convex functions (NAG-SC) and Polyak’s heavy-ball method. In response, we derive high-resolution ODEs as more accurate surrogates for the three aforementioned methods. These novel ODEs can be integrated into a general framework that allows for a fine-grained analysis of the discrete optimization algorithms through translating properties of the amenable ODEs into those of their discrete counterparts. As the first application of this framework, we identify the effect of a term referred to as ‘gradient correction’ in NAG-SC but not in the heavy-ball method, shedding insight into why the former achieves acceleration while the latter does not. Moreover, in this high-resolution ODE framework, NAG-C is shown to boost the squared gradient norm minimization at the inverse cubic rate, which is the sharpest known rate concerning NAG-C itself. Finally, by modifying the high-resolution ODE of NAG-C, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates as NAG-C for smooth convex functions. This is based on joint work with Stephen Boyd, Emmanuel Candes, Simon Du, Michael Jordan, and Bin Shi.

We will review the primitive equations of the atmoshere and the ocean and their coupling. We will descibe some mathematical poblems that they raise, some recent results and some less recent ones.

In this talk, we attempt to derive some experimentally verifiable physical laws of nature based only on a few fundamental first principles. First, we present two basic principles, the principle of interaction dynamics (PID) and the principle of representation invariance (PRI). Intuitively, PID takes the variation of the action under energy-momentum conservation constraint. PID offers a completely different and natural way of introducing Higgs fields. For gravity, we show that PID is the direct consequence of Einstein’s principle of general relativity and the presence of dark matter and dark energy. PRI requires that the SU(N) gauge theory be independent of representations of SU(N). PRI has remarkably rich physical consequences. Second, we show that the physical laws of the four fundamental forces—gravity, electromagnetic force, weak and strong forces—are dictated by 1) the Einstein principle of general relativity, 2) the principle of gauge symmetry, 3) PID, and 4) PRI. The new theory will lead to solutions to a number of longstanding problems in particle physics and cosmology. The talk is based on recent joint work with Tian Ma.

Whenever we wish to determine the characteristics of an object based on data of how it scatters incoming radiation we must solve an inverse scattering problem. This is a frequent situation in many fields, such as geophysical imaging or biomedical imaging. To reconstruct objects from the measured data, we can design optimization problems in which the boundary value problems governing the incident radiation act as constraints. Then we implement descent techniques to approach a global minimum. However, the process may stagnate without converging, either due to lack of convexity or to small gradients. We propose a method to overcome this difficulty combining topological derivative based optimization to generate first approximations with iteratively regularized Gauss-Newton techniques to ensure convergence. Numerical simulations illustrate fast reconstruction of objects formed by multiple non convex components in extreme situations such as holographic microscopy set-ups, in which the only data available are intensity measurements for one incident light beam on a limited screen.

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.

This talk introduces new approximation theories for deep learning in parallel computing and high dimensional problems. We will explain the power of function composition in deep neural networks and characterize the approximation capacity of shallow and deep neural networks for various functions on a high-dimensional compact domain. Combining parallel computing, our analysis leads to an important point of view, which was not paid attention to in the literature of approximation theory, for choosing network architectures, especially for large-scale deep learning training in parallel computing: deep is good but too deep might be less attractive. Our analysis also inspires a new regularization method that achieves state-of-the-art performance in most kinds of network architectures.