PART 1: I will construct a mathematical model of a defect embedded in an infinite homogeneous crystal. I will then establish a regularity result for minimisers, which given the crucial information on which approximation schemes are based. As an elementary application of this framework I will prove convergence rates for two computational schemes: (1) clamped far-field and (2) coupling to harmonic far-field model.
PART 2: The conditions under which the theory of Part 1 holds are separability and locality of the total energy. In Part 2 I will show how for a tight-binding model (a minimalistic electronic structure model) these two condition arise. This analysis raises some interesting (open) questions.
PART 3: Finally, I will use the theory developed in PART 1 and PART 2 to construct and analyse a new family of QM/MM embedding schemes with rigorous error estimates.

PART 1: I will construct a mathematical model of a defect embedded in an infinite homogeneous crystal. I will then establish a regularity result for minimisers, which given the crucial information on which approximation schemes are based. As an elementary application of this framework I will prove convergence rates for two computational schemes: (1) clamped far-field and (2) coupling to harmonic far-field model.
PART 2: The conditions under which the theory of Part 1 holds are separability and locality of the total energy. In Part 2 I will show how for a tight-binding model (a minimalistic electronic structure model) these two condition arise. This analysis raises some interesting (open) questions.
PART 3: Finally, I will use the theory developed in PART 1 and PART 2 to construct and analyse a new family of QM/MM embedding schemes with rigorous error estimates.

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.

PART 1: I will construct a mathematical model of a defect embedded in an infinite homogeneous crystal. I will then establish a regularity result for minimisers, which given the crucial information on which approximation schemes are based. As an elementary application of this framework I will prove convergence rates for two computational schemes: (1) clamped far-field and (2) coupling to harmonic far-field model.
PART 2: The conditions under which the theory of Part 1 holds are separability and locality of the total energy. In Part 2 I will show how for a tight-binding model (a minimalistic electronic structure model) these two condition arise. This analysis raises some interesting (open) questions.
PART 3: Finally, I will use the theory developed in PART 1 and PART 2 to construct and analyse a new family of QM/MM embedding schemes with rigorous error estimates.

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 this talk, I will survey the recent understandings on the motion of water waves obtained via rigorous mathematical tools, this includes the evolution of smooth initial data and some typical singular behaviors. In particular, I will present our recently results on gravity water waves with angled crests.

The idea of using nonlinear approximations as a tool for solving equations is as natural as that of using bases and, in fact, was proposed in 1960 in the context of quantum chemistry. The usual approach to solving partial differential and integral equations is to select a basis (possibly a multiresolution basis) or a grid, project equations onto such basis and solve the resulting discrete equations. The nonlinear alternative is to look for the solution within a large lass of functions (larger than any basis) by constructing optimal or near optimal approximations at every step of an algorithm for solving the equations. While this approach can theoretically be very efficient, the difficulties of constructing optimal approximations prevented any significant use of it in practice. However, during the last 15 years, nonlinear approximations have been successfully used to approximate operator kernels via exponentials or Gaussians to any user-specified accuracy, thus enabling a number of multidimensional multiresolution algorithms. In a new development several years ago, we constructed a fast and accurate reduction algorithm for optimal approximation of functions via exponentials or Gaussians (or, in a dual form, by rational functions) than can be used for solving partial differential and integral equations equations. We present two examples of the resulting solvers: one for the viscous Burgers' equation and another for solving the Hartree-Fock equations of quantum chemistry. Burgers' equation is often used as a testbed for numerical methods: if the viscosity \vu; is small, its solutions develop sharp (moving) transition regions of width O (\vu) presenting significant challenges for numerical methods. Using nonlinear approximations for solving the Hartree-Fock equations is the first step to a wider use of the approach in quantum chemistry. We maintain a functional form for the spatial orbitals as a linear combinations of products of decaying exponentials and spherical harmonics entered at the nuclear cusps. While such representations are similar to the classial Slater-type orbitals, in the course of computation we optimize both the exponents and the coefficients in order to achieve an efficient representation of solutions and to obtain guaranteed error bounds.

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.

In computational mathematics and science, it is now essential to consider computer hardware issues if a new algorithm is to be deployed. One such issue is the prevalence of parallelism in almost all levels of computer hardware. We discuss some of the challenges presented by computer hardware and some potential solutions in the context on quantum chemistry algorithms. Important considerations include reducing data movement, load balance across processors, and use of SIMD (single instruction, multiple data) features in modern processors. Specific results we have obtained include efficient computations using Hartree--Fock approximations on more than 1.5 million processor cores, and a new library for computing electron repulsion integrals that is designed for SIMD operation. These results are joint work with Ben Pritchard, Xing Liu, and the Intel Parallel Computing Lab.

In this talk, we study blow-up mechanism of solutions to an integrable equation with cubic nonlinearities and nonlinear dispersion. We will show that singularities of the solutions can occur only in the form of wave-breaking. Some wave-breaking conditons on the initial data are provided. In addition, this equation is known to admit single and multi-peaked solitons, of a different character than those of the Camassa-Holm equation. We will prove that the shapes of these waves are stable under small perturbations in the energy space.

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.

Many problems in science and engineering involve parameters in their mathematical models. Depending on the values of the parameters, the equations can differ greatly in nature. Asymptotic preserving (AP) methods are one type of methods which are designed to work uniformly with respect to different scales or regimes of the equations when the parameters vary.

In this talk, I will present our work in developing high order AP methods for some kinetic models, including discrete-velocity models in a diffusive scaling and the BGK model in a hyperbolic scaling. When the Knudson number approaches zero, the limiting equations of the former model can be heat equation, viscous Burgers’ equation, or porous medium equation, while the limiting equations for the latter are the compressible Euler equations. When the Knudson number is very small, the BGK model also leads to compressible Navier-Stokes equations. The proposed methods are built upon a micro-macro decomposition of the equations, high order discontinuous Galerkin (DG) spatial discretizations, and the globally stiffly accurate implicit-explicit Runge-Kutta (IMEX-RK) temporal discretizations. Theoretical results are partially established for uniform stability, error estimates, and rigorous asymptotic analysis. Numerical experiments will further demonstrate the performance of the methods.

Out-of-equilibrium behavior is the characteristic feature of the long-time dynamics of nonlinear dispersive equations on compact domain. This means that solutions typically do not exhibit any form of long-time stability near equilibrium solutions or configurations. We shall survey several aspects of this behavior both from a dynamical systems and statistical mechanics point of view.

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.

When we want to understand a geometric picture, finding the zero set of a polynomial hiding in it can be very helpful: it can reveal structure and allow computations. Polynomial partitioning, developed by Guth and Katz, is a technique to find such a nice algebraic hypersurface. Polynomial partitioning has revolutionised discrete incidence geometry in the recent years, thanks to the fact that interaction of lines with algebraic hypersurfaces is well-understood. Recently, however, Guth discovered agreeable interaction between tubes and algebraic hypersurfaces, and thus used polynomial partitioning to improve on the 3-dim restriction problem. In this talk, we will present polynomial partitioning via a discrete analogue of the Kakeya problem, and discuss its potential to be extensively used in harmonic analysis.

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 Boltzmann equation models the evolution of the probability density of gas particles that interact through predominantly binary collisions. The equation consists of a transport operator and a collision operator. The latter is a bilinear integral with a non-integrable angular kernel. For a long time the equation was simplified by assuming that the kernel is integrable (so called Grad's cutoff), with a belief that such an assumption does not affect the equation significantly. Recently, however, it has been observed that a non-integrable singularity carries regularizing properties, which motivates further analysis of the equation in this setting. We study the behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime, by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. We show how the singularity rate of the angular kernel affects the order of tails that can be propagated. The result uses Mittag-Leffler functions, which are a generalization of exponential functions. This is based on joint works with Alonso, Gamba, Pavlovic and with Gamba, Pavlovic.

Compressive sensing has become a powerful addition to uncertainty quantification in recent years. This paper identifies "new" bases for random variables through linear mappings such that the representation of the quantity of interest is more sparse with new basis functions associated with the new random variables. This sparsity increases both the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. Specifically, we consider rotation-based linear mappings which are determined iteratively for Hermite polynomial expansions. We demonstrate the effectiveness of the new method with applications in solving stochastic partial differential equations and high-dimensional problems.

The forces generated by moving interfaces usually include the parts due to tangential stretching. We derive the evolution equation for the tangential stretching, which then forms the basis for an Eulerian formulation based on level set functions. The jump conditions are then derived using the level set and stretch functions. The derived jump conditions under this Eulerian formulation are clean. This makes possible a local level set method for immersed interface method to simulate membranes or vesicles where the tangential forces are present. This is a continuation of one piece of my work during my Ph.D. study.

Wavespeed selection refers to the problem of determining the long time asymptotic speed of invasion of an unstable homogeneous state by some other secondary state. This talk will review wavespeed selection mechanisms in the context of reaction-diffusion equations. Particular emphasis will be placed on the qualitative differences between wavespeed selection in systems of reaction-diffusion equations and scalar problems as well as some surprising consequences. The primary example will be a system of coupled Fisher-KPP equations that exhibit anomalous spreading wherein the coupling of two equations leads to faster spreading speeds.

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.

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 kernel-free boundary integral method is a Cartesian grid based method for solving elliptic partial differential equations (PDEs). It solves elliptic PDEs in the framework of boundary integral equations (BIEs). The method evaluates boundary and volume integrals by solving equivalent simple interface problems on Cartesian grids. It takes advantages of the well-conditioning properties of the BIE formulation, the convenience of grid generation with Cartesian grids and the availability of fast and efficient elliptic solvers for the simple interface problems. In this talk, I will present recent developments of the method for the reaction-diffusion equations in computational cardiology, the nonlinear Poisson-Boltzmann equation in biophysics, the Stokes equation in fluid dynamics as well as some free boundary and moving interface problems.

We discuss one approach to determining the hazard threat to a locale due to a large volcanic avalanche. The methodology employed includes large-scale numerical simulations, field data reporting the volume and runout of flow events, and a detailed statistical analysis of uncertainties in the modeling and data. The probability of a catastrophic event impacting a locale is calculated, together with a estimate of the uncertainty in that calculation. By a careful use of simulations, a hazard map for an entire region can be determined. The calculation can be turned around quickly, and the methodology can be applied to other hazard scenarios.

We propose a general framework to design asymptotic preserving schemes for the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We will show how this idea can be applied to other collision operators, such as the Landau-Fokker-Planck operator, Ullenbeck-Urling model, and in the kinetic-fluid model of disperse multiphase flows, and can be implemented in the Monte-Carlo framework in which is time step is not limited by the possibly small mean free time.

It is well-known that the asymptotic complexity of matrix-matrix product and matrix inversion is given by the rank of a 3-tensor, recently shown to be at most O(n^2.3728639) by Le Gall. This approach is attractive as a rank decomposition of that 3-tensor gives an explicit algorithm that is guaranteed to be fastest possible and its tensor nuclear norm quantifies the optimal numerical stability. There is also an alternative approach due to Cohn and Umans that relies on embedding matrices into group algebras. We will see that the tensor decomposition and group algebra approaches, when combined, allow one to systematically discover fast(est) algorithms. We will determine the exact (as opposed to asymptotic) tensor ranks, and correspondingly the fastest algorithms, for products of Circulant, Toeplitz, Hankel, and other structured matrices. This is joint work with Ke Ye (Chicago).

We consider a coupled system of Schroedinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. In the first case, the classical limit is only taken in one of the two equations, leading to a mixed quantum-classical model which is closely connected to the well-known Ehrenfest method in molecular dynamics. In the second case, the classical limit of the full system is rigorously established, resulting in a system of coupled Vlasov-type equations. In the second part of our work, we provide a numerical study of the coupled semiclassically scaled Schroedinger equations and of the mixed quantum-classical model obtained via Ehrenfest's method. A second order (in time) method is introduced for each case. We show that the proposed methods allow time steps independent of the semi-classical parameter(s) while still capturing the correct behavior of physical observables. It also becomes clear that the order of accuracy of our methods can be improved in a straightforward way.

Recently there has been extensive development of numerical methods for fluid flow interacting with moving boundaries or interfaces, using regular finite difference grids which do not conform to the boundaries. Simulations at low Reynolds number have demonstrated that, with certain choices in the design of the method, the velocity can be accurate to about O(h^2) while discretizing near the interface with truncation error as large as O(h). We will describe error estimates which verify that such accuracy can be achieved in a simple prototype problem, even near the interface, using corrections to difference operators as in the immersed interface method. We neglect errors in the interface location and derive uniform estimates for the fluid velocity and pressure. We will first discuss maximum norm estimates for finite difference versions of the Poisson equation and diffusion equation with a gain of regularity. We will then describe the application to the Navier-Stokes equations.

The geometric notion of curvature is closely related to the analytic concept of Fourier decay. This will be our starting point to explore some restriction inequalities on spheres sitting in d-dimensional Euclidean space. The case d=2 is of special interest as it can be approached with a variety of tools ranging from elementary combinatorics and planar geometry to Fourier analysis and special function theory. Among other things, we shall see how new convexity estimates for certain integrals involving six-fold products of Bessel functions allow for partial progress in this tantalizing problem.

CANCELLED