Benedikt Wirth : Optimal fine-scale structures in elastic shape optimization
- Applied Math and Analysis ( 113 Views )A very classical shape optimization problem consists in optimizing the topology and geometry of an elastic structure subjected to fixed boundary loads. One typically aims to minimize a weighted sum of material volume, structure perimeter, and structure compliance (a measure of the inverse structure stiffness). This task is not only of interest for optimal designs in engineering, but e.g. also helps to better understand biological structures. The high nonconvexity of the problem makes it impossible to find the globally optimal design; if in addition the weight of the perimeter is chosen small, very fine material structures are optimal that cannot even be resolved numerically. However, one can prove an energy scaling law that describes how the minimum of the objective functional scales with the model parameters. Part of such a proof involves the construction of a near-optimal design, which typically exhibits fine-scale structure in the form of branching and which gives an idea of how optimal geometries look like. (Joint with Robert Kohn)
Shi Jin : An Eulerian surface hopping method for the Schr\{o}dinger equation with conical crossings
- Applied Math and Analysis ( 104 Views )In a nucleonic propagation through conical crossings of electronic energy levels, the codimension two conical crossings are the simplest energy level crossings, which affect the Born-Oppenheimer approximation in the zeroth order term. The purpose of this paper is to develop the surface hopping method for the Schr\"{o}dinger equation with conical crossings in the Eulerian formulation. The approach is based on the semiclassical approximation governed by the Liouville equations, which are valid away from the conical crossing manifold. At the crossing manifold, electrons hop to another energy level with the probability determined by the Landau-Zener formula. This hopping mechanics is formulated as an interface condition, which is then built into the numerical flux for solving the underlying Liouville equation for each energy level. While a Lagrangian particle method requires the increase in time of the particle numbers, or a large number of statistical samples in a Monte Carlo setting, the advantage of an Eulerian method is that it relies on fixed number of partial differential equations with a uniform in time computational accuracy. We prove the positivity and $l^{1}$-stability and illustrate by several numerical examples the validity and accuracy of the proposed method.
Jie Shen : Phase-field models for multiphase complex fluids: modeling, numerical analysis and simulations
- Applied Math and Analysis ( 98 Views )I shall present some recent work on phase-field model for multiphase incompressible flows. We shall pay particular attention to situations with large density ratios as they lead to formidable challenges in both analysis and simulation. I shall present efficient and accurate numerical schemes for solving this coupled nonlinear system, in many case prove that they are energy stable, and show ample numerical results which not only demonstrate the effectiveness of the numerical schemes, but also validate the flexibility and robustness of the phase-field model.
Bob Pego : Merging-splitting group dynamics via Bernstein function theory (or: How to count fish using mathematics)
- Applied Math and Analysis ( 96 Views )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.
Robert Pego : Scaling dynamics of solvable models of coagulation
- Applied Math and Analysis ( 140 Views )We study limiting behavior of rescaled size distributions in several models of clustering or coagulation dynamics, `solvable' in the sense that the Laplace transform converts them into nonlinear PDE. The scaling analysis that emerges has many connections with the classical limit theorems of probability theory, and a surprising application to the study of shock clustering in the inviscid Burgers equation with random-walk initial data. This is joint work with Govind Menon.
Hau-tieng Wu : Medical challenges meet time series analysis and manifold learning
- Applied Math and Analysis ( 96 Views )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.
Benjamin Stamm : Efficient numerical methods for polarization effects in molecular systems
- Applied Math and Analysis ( 120 Views )In this talk we provide two examples of models and numerical methods involving N-body polarization effects. One characteristic feature of simulations involving molecular systems is that the scaling in the number of atoms or particles is important and traditional computational methods, like domain decomposition methods for example, may behave differently than problems with a fixed computational domain.
We will first see an example of a domain decomposition method in the context of the Poisson-Boltzmann continuum solvation model and present a numerical method that relies on an integral equation coupled with a domain decomposition strategy. Numerical examples illustrate the behaviour of the proposed method.
In a second case, we consider a N-body problem of interacting dielectric charged spheres whose solution satisfies an integral equation of the second kind. We present results from an a priori analysis with error bounds that are independent of the number particles N allowing for, in combination with the Fast Multipole Method (FMM), a linear scaling method. Towards the end, we finish the talk with applications to dynamic processes and enhanced stabilization of binary superlattices through polarization effects.
Dmytro Bilyk : Discrepancy Theory and Analysis
- Applied Math and Analysis ( 101 Views )In this talk, we shall look at discrepancy theory through the prism of harmonic and functional analysis. Discrepancy theory deals with finding optimal approximations of continuous objects by discrete sets of points and quantifying the inevitably arising errors (irregularities of distribution). This field lies at the interface of several areas of mathematics: approximation, probability, discrete geometry, number theory. Historically, methods of analysis (Fourier techniques, Riesz product, wavelet expansions etc) played a pivotal role in the development of the subject.
A number of exciting new connections of discrepancy theory to other fields were discovered recently and are not yet fully understood. These include approximation theory (metric entropy of spaces with mixed smoothness, hyperbolic approximations), probability (small deviations of Gaussian processes, empirical processes), harmonic analysis (small ball inequality, Sidon theorem), compressed sensing etc.
We shall describe some of the recent results in the field, the main ideas and methods, and numerous relations to other areas of mathematics.
Joceline Lega : Molecular dynamics simulations of live particles
- Applied Math and Analysis ( 138 Views )I will show results of molecular dynamics simulations of hard disks with non-classical collision rules. In particular, I will focus on how local interactions at the microscopic level between these particles can lead to large-scale coherent dynamics at the mesoscopic level.
This work is inspired by collective behaviors, in the form of vortices and jets, recently observed in bacterial colonies. I will start with a brief summary of basic experimental facts and review a hydrodynamic model developed in collaboration with Thierry Passot (Observatoire de la Cote d'Azur, Nice, France). I will then motivate the need for a complementary approach that includes microscopic considerations, and describe the principal computational issues that arise in molecular dynamics simulations, as well as the standard ways to address them. Finally, I will discuss how classical collision rules that conserve energy and momentum may be modified to describe ensembles of live particles, and will show results of numerical simulations in which such rules have been implemented. Randomness, included in the form of random reorientation of the direction of motion of the particles, plays an important role in the type of collective behaviors that are observed.
Svitlana Mayboroda : Partial differential equations, analysis, and potential theory in irregular media
- Applied Math and Analysis ( 108 Views )Elliptic boundary value problems are well-understood in the case when the boundary, the data, and the coefficients exhibit smoothness. However, it has been long recognized in physics and engineering that irregularities (non-smooth boundary, abrupt change of media, noise or disorder) can decisively influence the properties of the solutions and give rise to completely new phenomena.
The analysis of general non-smooth elliptic PDEs gives rise to decisively new challenges: possible failure of maximal principle and positivity, breakdown of boundary regularity, lack of the classical L^2 estimates, to mention just a few. Further progress builds on an involved blend of harmonic analysis, potential theory and geometric measure theory techniques. In this talk we are going to discuss some highlights of the history, conjectures, paradoxes, and recent discoveries such as the higher-order Wiener criterion and maximum principle for higher order PDEs, solvability of rough elliptic boundary problems, harmonic measure, as well as an intriguing phenomenon of localization of eigenfunctions -- within and beyond the limits of the famous Anderson localization.
Lek-Heng Lim : Multilinear Algebra and Its Applications
- Applied Math and Analysis ( 109 Views )In mathematics, the study of multilinear algebra is largely limited to properties of a whole space of tensors --- tensor products of k vector spaces, modules, vector bundles, Hilbert spaces, operator algebras, etc. There is also a tendency to take an abstract coordinate-free approach. In most applications, instead of a whole space of tensors, we are often given just a single tensor from that space; and it usually takes the form of a hypermatrix, i.e.\ a k-dimensional array of numerical values that represents the tensor with respect to some coordinates/bases determined by the units and nature of measurements. How could one analyze this one single tensor then? If the order of the tensor k = 2, then the hypermatrix is just a matrix and we have access to a rich collection of tools: rank, determinant, norms, singular values, eigenvalues, condition number, etc. This talk is about the case when k > 2. We will see that one may often define higher-order analogues of common matrix notions rather naturally: tensor ranks, hyperdeterminants, tensor norms (Hilbert-Schmidt, spectral, Schatten, Ky Fan, etc), tensor eigenvalues and singular values, etc. We will discuss the utility as well as difficulties of various tensorial analogues of matrix problems. In particular we shall look at how tensors arise in a variety of applications including: computational complexity, control engineering, mathematical biology, neuroimaging, quantum computing, signal processing, spectroscopy, and statistics.
Pierre Degond : Collective dynamics and self-organization
- Applied Math and Analysis ( 100 Views )We are interested in large systems of agents collectively looking for a consensus (about e.g. their direction of motion, like in bird flocks). In spite of the local character of the interactions (only a few neighbours are involved), these systems often exhibit large scale coordinated structures. The understanding of how this self-organization emerges at the large scale is still poorly understood and offer fascinating challenges to the modelling science. We will discuss a few of these issues among (time permitting) phase transitions, propagation of chaos and the derivation of macroscopic models.
Paolo Aluffi : Chern class identities from string theory
- Applied Math and Analysis ( 143 Views )(joint with Mboyo Esole) String theory considerations lead to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. After giving a very sketchy idea of the physics arguments leading to this identity, I will present a Chern class identity which confirms it, generalizing it to arbitrary dimension and to varieties that are not necessarily Calabi-Yaus. The relevant loci are singular, and this plays a key role in the identity.
Alexander Litvak : Order statistics and Mallat--Zeitouni problem
- Applied Math and Analysis ( 123 Views )Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\mathbb{R}^n$. We show that the random vector $Y=T(X)$ satisfies $$ \mathbb{E} \sum_{j=1}^k j\mbox{-}\min_{i \leq n} {X_{i}}^2 \leq C \mathbb{E} \sum_{j=1}^k j\mbox{-}\min_{i\leq n}{Y_{i}}^2 $$ for all $k \leq n$, where $ j\mbox{-}\min$ denotes the $j$-th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the Karhunen--Lo`eve basis for the nonlinear reconstruction. We also show some relations for order statistics of random vectors (not only Gaussian), which are of independent interest. This is a joint work with Konstantin Tikhomirov.
Alexandr Labovschii : High accuracy numerical methods for fluid flow problems and turbulence modeling
- Applied Math and Analysis ( 98 Views )We present several high accuracy numerical methods for fluid flow problems and turbulence modeling.
First we consider a stabilized finite element method for the Navier-Stokes equations which has second order temporal accuracy. The method requires only the solution of one linear system (arising from an Oseen problem) per time step.
We proceed by introducing a family of defect correction methods for the time dependent Navier-Stokes equations, aiming at higher Reynolds' number. The method presented is unconditionally stable, computationally cheap and gives an accurate approximation to the quantities sought.
Next, we present a defect correction method with increased time accuracy. The method is applied to the evolutionary transport problem, it is proven to be unconditionally stable, and the desired time accuracy is attained with no extra computational cost.
We then turn to the turbulence modeling in coupled Navier-Stokes systems - namely, MagnetoHydroDynamics. We consider the mathematical properties of a model for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence, uniqueness and convergence of solutions for the simplest closed MHD model. Furthermore, we show that the model preserves the properties of the 3D MHD equations.
Lastly, we consider the family of approximate deconvolution models (ADM) for turbulent MHD flows. We prove existence, uniqueness and convergence of solutions, and derive a bound on the modeling error. We verify the physical properties of the models and provide the results of the computational tests.
Ioannis Kevrekidis : No Equations, No Variables, No Parameters, No Space, No Time -- Data, and the Crystal Ball Modeling of Complex/Multiscale Systems
- Applied Math and Analysis ( 176 Views )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.
Mark Levi : Arnold diffusion in physical examples
- Applied Math and Analysis ( 150 Views )Arnold diffusion is the phenomenon of loss of stability of a completely integrable Hamiltonian system: an arbitrarily small perturbation can cause action to change along some orbit by a finite amount. Arnold produced the first example of diffusion and gave an outline of the proof. After a brief overview of related results I will describe the simplest example of Arnold diffusion which we found recently with Vadim Kaloshin. We consider geodesics on the 3-torus, or equivalently rays in a periodic optical medium in $ {\mathbb R} ^3 $ (or equivalently a point mass in a periodic potential in $ {\mathbb R} ^3 $.) Arnold diffusion has a transparent intuitive explanation and a simple proof. Resonances and the so-called ``whiskered tori" acquire a clear geometrical interpretation as well. I will conclude with a sketch of a different but related manifestation of Arnold diffusion as acceleration of a particle by a pulsating potential. This is joint work with Vadim Kaloshin.
Peter Smereka : The Gaussian Wave Packet Transform: Efficient Computation of the Semi-Classical Limit of the Schroedinger Equation
- Applied Math and Analysis ( 158 Views )An efficient method for simulating the propagation of a localized solution of the Schroedinger equation near the semiclassical limit is presented. The method is based on a time dependent transformation closely related to Gaussian wave packets and yields a Schroedinger type equation that is very ammenable to numerical solution in the semi-classical limit. The wavefunction can be reconstructed from the transformed wavefunction whereas expectation values can easily be evaluated directly from the transformed wavefunction. The number of grid points needed per degree of freedom is small enough that computations in dimensions of up to 4 or 5 are feasible without the use of any basis thinning procedures. This is joint work with Giovanni Russo.
Shahed Sharif : Who wants to be a millionaire?
- Applied Math and Analysis ( 93 Views )The Mordell-Weil theorem shows that the rational points on an elliptic curve defined over the field of rational numbers is a finitely generated abelian group. The Birch and Swinnerton-Dyer conjecture relates the rank of this group to a number of analytic and algebraic invariants of the curve. (More generally it considers an elliptic curve defined over a number field.) The conjecture is one of the Millennium Prize problems and the Clay Institute is offering a reward of 1 million dollars for a solution. This talk will be an introduction to the conjecture. In following weeks we will have lectures explaining each of the terms in the formula.
Rong Ge : Learning Two-Layer Neural Networks with Symmetric Inputs
- Applied Math and Analysis ( 110 Views )Deep learning has been extremely successful in practice. However, existing guarantees for learning neural networks are limited even when the network has only two layers - they require strong assumptions either on the input distribution or on the norm of the weight vectors. In this talk we give a new algorithm that is guaranteed to learn a two-layer neural network under much milder assumptions on the input distribution. Our algorithms works whenever the input distribution is symmetric - which means two inputs $x$ and $-x$ have the same probability.
Based on joint work with Rohith Kuditipudi, Zhize Li and Xiang Wang
Almut Burchard : Geometry in Wasserstein Space: Geodesics, Gradients, and Curvature, from an Eulerian Point of View
- Applied Math and Analysis ( 95 Views )The optimal transportation problem defines a notion of distance in the space of probability measures over a manifold, the *Wasserstein space*. In his 1994 Ph.D. thesis, McCann discovered that this space is a length space: the distance between probability measures is given by the length of minimizing geodesics called *displacement interpolants*. A surprising number of important functionals in physics and geometry turned out to be geodesically convex. In contrast with classical function spaces, the Wasserstein space is not a linear space, but rather an infinite-dimensional analogue of a Riemannian manifold. This analogy has motivated new functional inequalities and new methods for studying evolution equations; however, it has rarely been used in rigorous proofs. I will describe recent work with Benjamin Schachter on differentiating functionals (such as the entropy or the Dirichlet integral) along displacement interpolants. Starting from an Eulerian formulation for the underlying optimal transportation problem, we take advantage of the system of transport equations to compute derivatives of arbitrary order, for probability densities that need not be smooth.
Xiu Yang : Enhancing Sparsity of Hermite Polynomial Expansions by Iterative Rotations
- Applied Math and Analysis ( 90 Views )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.
Brittany Froese : Meshfree finite difference methods for fully nonlinear elliptic equations
- Applied Math and Analysis ( 106 Views )The relatively recent introduction of viscosity solutions and the Barles-Souganidis convergence framework have allowed for considerable progress in the numerical solution of fully nonlinear elliptic equations. Convergent, wide-stencil finite difference methods now exist for a variety of problems. However, these schemes are defined only on uniform Cartesian meshes over a rectangular domain. We describe a framework for constructing convergent meshfree finite difference approximations for a class of nonlinear elliptic operators. These approximations are defined on unstructured point clouds, which allows for computation on non-uniform meshes and complicated geometries. Because the schemes are monotone, they fit within the Barles-Souganidis convergence framework and can serve as a foundation for higher-order filtered methods. We present computational results for several examples including problems posed on random point clouds, computation of convex envelopes, obstacle problems, non-continuous surfaces of prescribed Gaussian curvature, and Monge-Ampere equations arising in optimal transportation.
Chi-Wang Shu : Discontinuous Galerkin Finite Element Method for Multiscale Problems
- Applied Math and Analysis ( 98 Views )In this talk, we first give a brief introduction to the discontinuous Galerkin method, which is a finite element method using completely discontinuous basis functions, for solving hyperbolic conservation laws and parabolic and elliptic equations. We will then survey the progress in developing discontinuous Galerkin methods for multiscale problems, in three different approaches, namely using the heterogeneous multiscale method (HMM) framework, using domain decompositions, and using multiscale basis in the discontinuous Galerkin method. Numerical results will be shown to demonstrate the effectiveness of the multiscale discontinuous Galerkin methods.
Edmond Chow : Parallel Computing Issues in Computational Chemistry
- Applied Math and Analysis ( 105 Views )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.
Zhouping Xin : On Gases Expanding into Vacuum with or without Self-Gravitations
- Applied Math and Analysis ( 96 Views )In this talk I will discuss several issues concerning the motions of gases expanding into vacuum with or without self-gravitations which are governed by a free-boundary value problem for the 3-dimnesional compressible Euler system with/or without Poisson equation. A general uniqueness theorem for classical solutions to such a free boundary-value problem is presented for physical vacuums. A typical physical vacuum solution includes the famous Lane-Emdan solution in astrophysics. The uniqueness is proved by a relative entropy argument. Then a local well-posedness theory for spherically symmetric motions is established in a less regular space by a deliberate choice of weighted functional to overcome difficulties arising both at the free surface and the symmetry center. Finally, the uniqueness of the spherically symmetric motions is discussed for general equation of state without self-gravitations. This is a joint work with Professor Tao Luo and Professor Huihui Zeng.
Costas Pozrikidis : Biofluid-dynamics of blood cells
- Applied Math and Analysis ( 153 Views )Blood is a concentrated suspension of red cells, white cells, and platelets, each having a unique constitution and serving a different function. Red cells are highly deformable liquid capsules enclosed by a thin incompressible membrane whose resting shape is a biconcave disk. White cells are viscoelastic spherical particles enclosed by a cortical shell. In the unactivated state, platelets are oblate spheroids with an average aspect ratio approximately equal to 0.25. Like red cells, platelets lack a nucleus; like white cells, platelets exhibit a low degree of flow-induced deformation. In this talk, the biomechanics and biofluid-dynamics of these three types of cells will be discussed, recent progress in modeling and simulation methods will be reviewed, and open problems will be outlined.