Manoj Gopalkrishnan : On catalysis in biochemical networks
- Applied Math and Analysis ( 121 Views )It is a common expectation in chemistry that a chemical transformation which takes place in the presence of a catalyst must also take place in its absence, though perhaps at a much slower rate. A reaction network will be called ``saturated'' if it satisfies such an expectation. I propose a mathematical definition for saturated networks and show that the associated dynamical systems have no boundary equilibria in positive stoichiometric classes, and are therefore permanent. This result is independent of the specific rates, and generalizes previous results for complete networks by Gnacadja, atomic event-systems by Adleman et al. and constructive networks by Shinar et al. I require no assumption of complex balance or deficiency restrictions. The question of permanence for weakly-reversible reaction networks remains a long-standing open problem.
Christian Mendl : Matrix-valued Boltzmann equation for the Hubbard model
- Applied Math and Analysis ( 118 Views )The talk is concerned with a matrix-valued Boltzmann equation derived from the Fermi-Hubbard or Bose-Hubbard model for weak interactions. The quantum analogue of the classical distribution function is the Wigner function, which is matrix-valued to accommodate the spin degree of freedom. Conservation laws and the H-theorem can be proven analytically, and numerical simulations illustrate the time dynamics.
Costas Pozrikidis : Biofluid-dynamics of blood cells
- Applied Math and Analysis ( 157 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.
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.
Lei Li : Some algorithms and analysis for first order interacting particle systems
- Applied Math and Analysis ( 113 Views )We focus on first order interacting particle systems, which can be viewed as overdamped Langevin equations. In the first part, we will look at the so-called random batch methods (RBM) for simulating the interacting particle systems. The algorithms are motivated by the mini-batch idea in machine learning. For some special cases, we show the convergence of RBMs for the first marginal under Wasserstein distance. In the second part, we look at the Coulomb interaction in 3D space. We show that as the number of particles go to infinity, almost surely, the empirical measure converges in law to weak solutions of the limiting nonlinear Fokker-Planck equation. This talk is based on joint works with Shi Jin (Shanghai Jiao Tong), Jian-Guo Liu (Duke University) and Pu Yu (Peking University).
Anil N. Hirani : Applied Topology and Numerical PDEs
- Applied Math and Analysis ( 94 Views )Exterior calculus generalizes vector calculus to manifolds. For numerical solutions of PDEs on meshes this language has been discretized as finite element exterior calculus and discrete exterior calculus. I'll first give a very brief introduction to these discretizations. Tools from geometry and topology, such as Hodge theory, and basic ideas from cohomology and homology will be seen to be an integral part of these discretizations. A specific example I'll describe will be the computation of harmonic forms. This is a crucial first step in a finite element solution of even the most basic elliptic PDE like Poisson's equation. I'll show how the availability of a homology basis allows one to find a basis for discrete harmonic forms using least squares. When viewed appropriately, the concepts, language, and software for these PDE discretizations can be easily used to solve some interesting problems in data analysis. A slight generalization also leads to some problems in computational topology. Specifically, this involves moving from 2-norms to 1-norms. In some sense, this is an example of how work in numerical PDEs can lead to a very combinatorial and classical problem in computational topology.
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.
Anna Mazzucato : Optimal mixing and irregular transport by incompressible flows
- Applied Math and Analysis ( 99 Views )I will discuss transport of passive scalars by incompressible flows (such as a die in a fluid) and measures of optimal mixing and stirring under physical constraint on the flow. In particular, I will present recent results concerning examples of flows that achieve the optimal theoretical rate in the case of flows with a prescribed bound on certain Sobolev norms of the associated velocity, such as under an energy or an enstrophy budget. These examples are related to examples of (instantaneous) loss of Sobolev regularity for solutions to linear transport equations with non-Lipschitz velocity.
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.
Zongming Ma : Community detection in stochastic block models and beyond
- Applied Math and Analysis ( 91 Views )In this talk, I will start with a simple algorithm for community detection in stochastic block models and discuss its statistical optimality. After that, we will discuss two related issues. One is model selection for stochastic block models. The other is the extension to community detection in degree-corrected block models. We shall pay close attention to the achievability of statistical optimality by computationally feasible procedures throughout the talk.
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.
Anna Gilbert : Fast Algorithms for Sparse Analysis
- Applied Math and Analysis ( 139 Views )I will present several extremely fast algorithms for recovering a compressible signal from a few linear measurements. These examples span a variety of orthonormal bases, including one large redundant dictionary. As part of the presentation of these algorithms, I will give an explanation of the crucial role of group testing in each algorithm.
Wencai Liu : Spectral transitions for Schr\odinger operators with decaying potentials and Laplacians on asymptotically flat (hyperbolic) manifolds
- Applied Math and Analysis ( 109 Views )We apply piecewise constructions and gluing technics to construct asymptotically flat (hyperbolic) manifolds such that associated Laplacians have dense embedded eigenvalues or singular continuous spectra. The method also allows us to provide various examples of operators with embedded singular spectra, including perturbed periodic operators, periodic Jacobi operators, and Stark operators. We establish the asymptotic behavior (WKB for example) of eigensolutions under small perturbations, which implies certain rules for the absence of singular spectra. As a result, several sharp spectral transitions (even criteria) for a single (finitely many or countably many) embedded eigenvalues, singular continuous spectra and essential supports of spectral measures are obtained. The talk is based on several papers, some joint with Jitomirskaya and Ong.
Andrea Bertozzi : Geometry based image processing - a survey of recent results
- Applied Math and Analysis ( 101 Views )I will present a survey of recent results on geometry-based image processing. The topics will include wavelet-based diffuse interface methods, pan sharpening and hyperspectral sharpening, and sparse image representation.
Dionisios Margetis : A tale of three scales in modeling crystal surface relaxation
- Applied Math and Analysis ( 113 Views )Crystals play a critical role in the design of novel devices. The surface of a crystal can evolve with time and give rise to a variety of interesting structures used in applications. In this evolution, several length and time scales, from the atomistic to the continuum, are implicated. The description of their linkages poses challenging questions. How can surface evolution at large scales, possibly described by PDEs, emerge from the motion of mesoscale crystal defects? And how does the description of such defects arise from atomistic motion? In this talk, I will address recent progress and open challenges in answering these questions. In particular, I will discuss how facets, macroscopically flat surface regions, lead to PDE free boundary problems with nontrivial microstructures.
Matt Holzer : Invasion fronts and wavespeed selection in systems of reaction-diffusion equations
- Applied Math and Analysis ( 135 Views )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.
Nathan Totz : A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem
- Applied Math and Analysis ( 133 Views )In this joint work with Sijue Wu (U. Mich.), we consider solutions to the 2D
inviscid infinite depth water wave problem neglecting surface tension which
are to leading order wave packets of the form $\alpha + \epsilon B(\epsilon
\alpha, \epsilon t, \epsilon^2 t)e^{i(k\alpha + \omega t)}$ for $k > 0$.
Multiscale calculations formally suggest that such solutions have
modulations $B$ that evolve on slow time scales according to a focusing
cubic NLS equation. Justifying this rigorously is a real problem, since
standard existence results do not yield solutions which exist for long
enough to see the NLS dynamics. Nonetheless, given initial data within
$O(\epsilon^{3/2})$ of such wave packets in $L^2$ Sobolev space, we show
that there exists a unique solution to the water wave problem which remains
within $O(\epsilon^{3/2})$ to the approximate solution for times of order
$O(\epsilon^{-2})$. This is done by using a version of the evolution
equations for the water wave problem developed by Sijue Wu with no quadratic nonlinearity.
See arXiv:1101.0545
Xiaochun Tian : Interface problems with nonlocal diffusion
- Applied Math and Analysis ( 176 Views )Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, they also come with increased difficulty in numerical analysis with nonlocality involved. In the first part of this talk, I will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. This also motivates the development of new mathematical results -- for instance, a new trace theorem that extends the classical results. In the second part of this talk, I will describe our recent effort in computing a nonlocal interface problem arising from segregation of two species with high competition. One species moves according to the classical diffusion and the other adopts a nonlocal strategy. A novel iterative scheme will be presented that constructs a sequence of supersolutions shown to be convergent to the viscosity solution of the interface problem.
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.
Fengyan Li : High order asymptotic preserving methods for some kinetic models
- Applied Math and Analysis ( 132 Views )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 BurgersÂ’ 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.
Rayan Saab : Quantization of compressed sensing measurements and frame expansions
- Applied Math and Analysis ( 101 Views )Compressed sensing, as a signal acquisition technique, has been shown to be highly effective for dimensionality reduction. On the other hand, reconstruction from compressed sensing measurements is highly non-linear and requires digital computers. Thus, quantizing (i.e., digitizing) compressed sensing measurements is an important, albeit under-addressed topic. In this talk, we show that by using $\Sigma\Delta$ quantization instead of the most commonly assumed approach (uniform quantization), a significant reduction in the reconstruction error is possible. In particular, we prove error decay rates of $\lambda^{-c r}$ where $\lambda$ is the ratio of the number of measurements to the sparsity of the underlying signal, and $r$ is the order of the $\Sigma\Delta$ scheme. In addition to the compressed sensing scenario we also consider the quantization of frame expansions, where one collects more measurements than the ambient dimension. We show state of the art results for certain frames (including random frames) and $\Sigma\Delta$ schemes. In particular, we prove error rates of $e^{-c\sqrt{\lambda}}$, where $\lambda$ is the oversampling ratio.