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talk021302: Galactic Central Regions: Wavelet Methods and Numerical Simulations Chien-Chang Yen University of Minnesota Most of the nearby galaxies are found to have a central gas-dust disk. Their structures, however, are often obscured by the behind luminous star lights . We probe these structures of the galactic central regions by observation(wavelet method) and numerical simulations(relaxed method). Wavelet method decomposes a signal into various information at various levels. They are extremely useful in extracting those hidden structures of the galactic central regions. We have analyzed the NICMOS and WFPC (WFPC2) data from HST for more than 20 nearby disk galaxies. In general, the central regions are characterized by spiral or/and bar structures, and we have the following conclusions: For galaxies with a major bar, there are two possible scenarios; one is that the two-arm spirals can be traced all the way to the center; the other is a nuclear bar (bar within a bar). On the other hand, most of the galaxies without a major bar have a central or nuclear bar coupled with two-arm spirals. It is well known that spiral density waves can be generated by a rotating bar through a resonance excitation mechanism. Associated with these waves is the angular momentum transport between the bar and the disk. As waves attenuated by viscosity, the angular momentum will be deposited into the disk. This will cause the disk matter moving inward or outward, depending respectively on whether the angular momentum carried by the waves is negative or positive. Numerical simulations confirm the spiral density theory that the disk matter would gain angular momentum and move outward to form a tightly wound spiral-ring in the case of a fast bar resonance, and it would lose angular momentum and move inward to form an open-spiral and oval-ring structure in the case of a low bar. These works are supported by NSC Grant 90-2112-M-001-052.
talk022102: Date: Thursday, 2/21 Place: Math Common Room Time: Tea begin at 4:30pm, "Fermat's Last Tango" 5:00-7:00pm Fermat's Last Tango is a musical comedy, performed off broadway, and taped. We have a dvd disk of this. Play is very clever. Good to bring spouses, and significant others. Good for all levels: undergraduate, graduate, postdocs, staff and professors. All will enjoy!
talk022702: WHAT LANDSCAPE THEORY HAS TO TEACH US ABOUT SIMULATED ANNEALING Edward Weinberger Polytechnic University and Blumenthal Associates The success of simulated annealing depends critically on how configurations of high and low energy are distributed in the space of all possible solutions to the problem being considered. Evolutionary biologists, having come to the same conclusion about the importance of the locations of high and low fitness "solutions" to the problem of "optimal design" for an organism, have, by now, some useful results on how to characterize such "fitness landscapes" via a variant of Fourier analysis. A parallel development is a class of relatively simple landscapes, known collectively as "Kauffman's N-K Model", that have the useful feature that their ruggedness can be "tuned" by varying a single parameter. The goal of this talk is to explain these conceptual tools and to sketch how they might be used to improve cooling schedules, design parallel annealing algorithms, etc.
talk041702: =========================================================================== Adaptive and Parallel Discontinuous Galerkin Methods for Hyperbolic Systems Joseph E. Flaherty Scientific Computation Research Center Rensselaer Polytechnic Institute Troy, NY 12180 USA Abstract The discontinuous Galerkin method (DGM) provides an appealing approach to address problems having discontinuities, such as those that arise in hyperbolic conservation laws. Originally developed for neutron transport problems, the DGM has been used to solve both ordinary and partial differential equations. The DGM may be regarded as a way of extending finite volume methods to arbitrarily high orders of accuracy. The solution space is a piecewise continuous (polynomial) function relative to a structured or unstructured mesh. As such, it can sharply capture solution discontinuities relative to the computational mesh. It maintains local conservation on an elemental basis. Regardless of order, the DGM has a simple communication pattern to elements with a common face that makes it useful for parallel computation. It can handle problems in complex geometries to high order. And, it is useful with adaptivity since interelement continuity is neither required for h-refinement (mesh refinement and coarsening) nor p-refinement (method order variation). We describe several aspect of the method including basis construction, data structures, flux evaluation, solution limiting, local time stepping, and a posteriori error estimation. We further describe a framework for controlling parallel adaptive computation. The parallel data management system can handle high-order techniques and maintain a dynamic load balance in homogeneous and heterogeneous computing environments. Results of serial and parallel computations are are presented for unsteady compressible flow problems involving instabilities and other complex two- and three-dimensional phenomena.
talk041002: New Developments in Numerical Reservoir Simulation Zhangxin Chen Department of Mathematics Southern Methodist University This talk will address some new developments of scanning, gridding, discretizing, and visualizing technologies in numerical reservoir simulation. The scanning technology scans and extracts various geometrical data such as depth, thickness, porosity, permeability, and the location of wells, fractures, and faults. From scanning, the gridding technology generates corresponding 2D or 3D unstructured meshes. New discretization methods over these meshes have been developed. These methods are based on control volume finite elements and are capable to handle faults, horizontal wells, and unstructured meshes. The visualizing technology possesses real-time calculation and real-time display capabilities and provides streamline computations. As model examples in reservoirs, black-oil and compositional flow models will be discussed.
talk042402: Title: Designer Gene Networks: De novo constructs-in numero descriptions. Jeff Hasty Dept. of Biomedical Engineering Boston University Uncovering the structure and function of gene regulatory networks has become one of the central challenges of the post-genomic era. Theoretical models of protein-DNA feedback loops and gene regulatory networks have long been proposed, and recently, certain qualitative features of such models have been experimentally corroborated. This talk will focus on model and experimental results that demonstrate how a naturally occurring gene network can be used as a "parts List" for synthetic network design. The model formulation leads to computational and analytical approaches relevant to nonlinear dynamics and statistical physics, and the utility of such a formulation will be demonstrated through the consideration of specific design criteria for several novel genetic devices. Fluctuations originating from small molecule-number effects will be discussed in the context of model predictions, and the experimental validation of these stochastic effects underscores the importance of internal noise in gene expression. Potential biotech applications will be highlighted within the framework of cellular control schemes. Specifically, the coupling of an oscillating cellular process to a synthetic oscillator will be considered, and the resulting model behavior will be analyzed in the context of synchronization. The underlying methodology highlights the utility of engineering-based methods in the design of synthetic gene regulatory networks.
talk050102: Shock/Vortex/Entropy Interactions Gordon Erlebacher School of Computational Science & Information Technology and Department of Mathematics Florida State University I will present a series of high order numerical experiments that describe the interaction of a planar shocks with vortical and entropic structures. I will discuss the problem setup, numerical method, various types of upstream disturbances, and the structure of the shock and the downstream flow.
talk050802: The Vacuum in Isentropic Gas Dynamics Robin Young University of Massachusetts We are interested in global solutions to the equations of isentropic gas dynamics. We consider solutions having arbitrarily large data, so that the celebrated Glimm-Lax theory does not apply. One of the central difficulties in this program is the possible appearance of a vacuum. Liu and Smoller have shown that Glimm's interaction estimates do not apply near the vacuum, in that wave interactions cannot be approximated linearly. By considering interactions exactly rather than asymptotically, we analyze the vacuum in detail. It is well-known that certain Riemann problems give rise to a vacuum; we show that this is essentially the only way a vacuum can develop. We describe interactions of waves with the vacuum, and the annihilation of the vacuum. In particular, when a vacuum is annihilated, two shocks are emitted, and these form a cusp at the point of annihilation. I will describe progress on the problem of existence if time permits.
talk091602: Talk Title: Self-Similar Solutions to 2-D Riemann Problems Speaker: Prof. Suncica Canic Department of Mathematics University of Houston Abstract: In this talk a brief overview of the problems and methods used to study the structure of solutions for a class of two-dimensional Riemann problems will be presented. The speaker will focus on the analysis of models arising in gas dynamics (the steady and the unsteady transonic small disturbance equations, the nonlinear wave system) and pay a special attention on the treatment of nonlinear waves and their interaction with a nontrivial subsonic region. Since the interaction between the supersonic and subsonic flow occurs either through a transonic shock, through a rarefaction wave or via a sonic curve, different techniques need to be used to analyze the solution in each case. An overview of the techniques and a comparison between the methods used by several authors, will be given. In the end the speaker will suggest how one method can be used in the analysis of self-similar nonlinear wave structures arising in compressible Euler equations (isentropic and adiabatic case) where linearly degenerate modes are present. The corresponding reduced (self-similar) system is of mixed (elliptic-hyperbolic) type. More precisely, the density satisfied a degenerate elliptic equation, whereas vorticity satisfies a transport equation. In the low-velocity regime, the mixed system decouples (giving rise to the nonlinear wave system) and the structure of both the nonlinear and the linearly degenerate waves can be analyzed. A similarity between the structure of the decoupled systems and the fully coupled equations (corresponding to the compressible Euler equations) will be emphasized thereby hinting how the techniques presented in the first half of the lecture could be employed in the analysis of the structure of self-similar solutions of the full set of compressible Euler equations. Collaborators: Barbara Lee Keyfitz, University of Houston, Eun Heui Kim, CalState Long Beach, Gary Lieberman, Iowa State University, Dragan Mirkovic, University of Houston.
talk091902: Coupling the Sierra FEA code to smooth faceted surface evaluations in the Common Geometry Module (CGM) Timothy J. Tautges Sandia National Laboratories Albuquerque, NM, USA e-mail: tjtautg@sandia.gov Recent advances in the speed and capability of computational simulation are driving the incor- poration of geometric modeling methods in computational simulation codes. Several examples of analysis methods making use of geometric modeling include adaptive mesh refinement on curved boundaries and modeling of free surface flows over curvilinear bodies. This trend is also reflected on the pre-processing side, where mesh generation tools are forging ever-closer links to CAD tools and other sources of continuous domain representations. These efforts can all be thought of as restoring associative links between the various representations of the spatial com- putational domain. The Common Geometry Module, or CGM, is a set of libraries providing a consistent interface to geometric models in a variety of representation formats. CGM includes links to geometry in the ACIS modeling format as well as facet-based and virtual geometry representations. CGM can be linked into analysis codes to provide the same geometry functionality used in mesh gen- eration codes; in fact, the CUBIT mesh generation code accesses all its geometry functionality through CGM. We have developed a smooth facet-based surface representation in the CGM framework, where facet-based surfaces support C2-continuous differential geometry evaluations. In this presenta- tion we describe the use of facet-based surfaces to support adaptive mesh refinement in the SIERRA finite element code. Techniques used for minimizing data duplication and for associating the triangle-based facets needed by CGM to the (possibly non-conformal and h- refined) quadrilateral and triangle elements in SIERRA will be described. A general discussion of coupling physics codes to the CGM geometry component will conclude this talk. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.
talk092502: Accurate Computation of Tidal Bores in Estuaries Professor Grafton W. H. Hui Hong Kong University of Science and Technology Tidal waves and bores belong to shallow-water flow, which is traditionally formulated in terms of water depth and fluid velocity. This formulation enjoys great success for flow with horizontal bottom and zero friction when the governing equations reduce to conservation laws. It, however, encounters difficulties in the presence of uneven bottom topography; in particular, it fails to replicate stationary flow and fails to compute tidal bores when the tide is receeding. To overcome these difficulties, we formulate the problem of shallow-water flow in terms of water level and fluid velocity. The non-homogeneous equations are solved using the fractional step method together with: (1) a Godunov-type scheme for the homogeneous conservation law equations and (2) a balanced discretisation for the source terms arising from bottom topography. The Riemann problem in this formulation is solved with an approximation equivalent to coarsening the grid for bottom topography by doubling its size locally. Our method exactly replicates the stationary flow, and accurately computes steady and unsteady flow. When applied to compute the famous tidal bores on the Qiantang River on the East Coast of China, it produces excellent agreement with field observations.
talk100202: A Nonconventional Eulerian-Lagrangian Single-Node Collocation Method for Unsteady-State Advection-Diffusion Equations Li Wu Department of Mathematics University of Rhode Island We developed a nonconventional Eulerian-Lagrangian single-node collocation method (ELSCM) with piecewise-cubic Hermite polynomials as basis functions for the numerical simulation to unsteady-state advection-diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The method is relatively easy to formulate. Numerical experiments in one, two, and three-dimensional spaces are presented to show the strong potential of this method.
talk020503: Experimental and Computational Study of Fuel Injection Jet Constantine Tzanos Argonne National Laboratory Monochromatic synchrotron x-rays from the Advanced Photon Source (APS) at Argonne National Laboratory have been used to make time-resolved absorption measurements in the spray generated by a high-pressure diesel fuel injector. From these measurements, diesel fuel mass distributions, density and volume fraction have been determined as a function of time and position from the tip of the injector nozzle. The speed of the leading and trailing edges of the spray were also calculated. The measurements show that the fuel volume fraction drops off quickly as we move away from the tip of the nozzle. The front-tracking code FronTier has been used to analyze these experiments. The experimental measurements provide a basis for the validation of the code, and the validated code can be used to provide an understanding of the spray dynamics, and a quantitative description of spray breakup for the simulation of combustion in an internal combustion engine. Experimental measurements and analyses and the application of FronTier at ANL to analyze one of the APS experiment will be discussed. The potential application of FronTier for the design of an injector-based lithium thin-film-stripper generator will also be discussed.
talk042303: Dr. Folkert Tangerman Principal Scientist Photon Research Associates In Image Analysis Linear Analysis is your friend The statistical analysis of even a single large image, leads to a translation invariant image correlation function. Correlation functions tend to arise in two but usually disparate ways: 1. as Green's functions of suitable operators 2. resulting from convolution with uncorrelated random variables. These ways are associated with two different square root operations from the symmetric positive definite operator (Toeplitz) C, given by convolution with the correlation function: 1. (Cholesky) find a lower triangular matrix for which ACA'=Id, A=inverse Cholesky factorization of C. 2. (Principal Component Analysis) C=EDE', with E orthogonal, D diagonal While the second factorization is 'normal' factor analysis, the first is not only equally useful, but also more intriguing as the operator A tends to be a differential operator, appproximately translation invariant, with its coefficients fast computed. Example: In the one dimensional case: if C is N by N matrix for which C(i,j)=exp-|i-j|, A is banded (diagonal and one sub-band of opposite sign). Check this! In general A is banded dominated, as approximately explained by Szego's theory of Toeplitz matrices. We show how to extend this theory to higher dimensions (1 to 2 illustrative of the induction step), The result is a 'superfast' inverse Cholesky decomposition of C for 2d (and n-d) correlation functions.
talk091003: Speaker: Chun Liu Associate Professor Math Department Penn State University Title: Variational Approach in Studying the Mixture of the Fluids: Transport and Induced Elastic Stress Abstract: From the energetic point of view, most complicated hydrodynamical and rheological properties of the non-Newtonian complex fluids arise from the coupling and competing between the kinetic energy and different types of internal "elastic" energy. The examples include liquid crystal materials where the alignment of the molecule director contributes to the elastic energy; the Magneto-hydrodynamics (MHD) and Electro- hydrodynamics (EHD) where the magnetical and electrical fields are the source of the elasticity; different polymerical fluids; viscoelastical fluids; mixtures of different materials (where the the elasticity is due to the heterogeneity) and fluids involving different surfactant materials. The coupling between the transport of these elastic effects by the flow field and the induced elastic stresses in the momentum equations assure the Hamiltonian (or dissipative) nature of the whole system. On the other hand, such coupling also reflect the influence of the micro-structure of the material to the hydrodynamical properties of the fluid and the vice versa. The hydrodynamic theory of mixtures is a good example for these theories. In this talk, I will discuss a energetic variational approach involving phase field methods to model the dynamics of mixtures with free interfaces. The method can be generalized to the cases of more complicated cases, such as Marangoni effects, surface viscosity or more general surfactant situations. When the mixture involves viscoelastic materials, we employed a formulation of the system in Eulerian coordinates. Some analytical, numerical results as well as open problems will be presented.
talk101703: Accurate, Stable and Efficient Navier-Stokes Solvers Based on Explicit Treatment of the Pressure Term Jianguo Liu Professor Mathematics Department University of Maryland, College Park We present numerical schemes for the incompressible Navier-Stokes equations based on primitive variables formulations in which pressure is treated explicitly in time and the incompressibility constraint has been replaced by a pressure Poisson equation. The crucial point for these schemes is the proper enforcement of a Neumann boundary condition for the pressure Poisson equation, which in turn ensures satisfaction of the divergence-free condition. The computation of the momentum and kinematic equations are fully decoupled, resulting is a class of extremely efficient Navier-Stokes solvers. Moreover, the schemes are not projection-type methods, which are plagued by numerical boundary layers which result from time consistency issues inherent in such splitting methods. A finite difference version of the current approach was discussed by the authors in Johnston and Liu, and also by Henshaw. In that setting the decoupling is realized via local pressure boundary conditions, in analogy with local vorticity boundary conditions. The focus of the current work is the extension of the local pressure boundary condition framework to collocation and Galerkin spectral methods. In the Galerkin approach the decoupling of the momentum and pressure equations is realized via a variational formulation, while in the collocation scheme direct spatial discretization of the pressure Poisson equation formulation by differentiation matrices is used. Additionally, the Galerkin formulation requires only C^0 elements, highly desirable for implementation using the more general finite element method. Various numerical examples are presented, including both implicit and explicit time discretizations, demonstrating the high accuracy, robustness, and efficiency of this class of schemes. This is a joint work with Hans Johnston, University of Michigan.
talk110603: On the use of level sets for solving some inverse problems Oliver Dorn Universidad Carlos III de Madrid Department of Mathematics ABSTRACT: We will discuss the use of level sets for object detection and specification in two important inverse problems arising in geophysical and medical imaging. The geophysical application which we address is 'Low Frequency electromagnetic Induction Tomography' (in 3D), which is governed by the 3D system of Maxwell's equations. The medical application which we address is 'Diffuse Optical Tomography' (in 2D) which is governed by a linear Boltzmann or radiative transfer equation. The inverse problem will be formulated as a shape reconstruction problem. For solving this problem, the physical data are used for deforming some initial guess for the unknown shape according to a suitably designed velocity function. The final shape of this evolution is taken as final solution of the inverse problem. Since the evolving shapes usually change topology many times during this evolution, a powerful numerical tool for describing the evolving interfaces is needed. We choose to use a level set representation for this purpose. We will present and discuss various numerical examples of reconstructions with level sets for both applications.
talk111003: Bill Henshaw Centre for Applied Scientific Computing Lawrence Livermore National Laboratory Livermore CA The Overture framework enables the accurate and efficient solution of PDEs in complex geometry using the method of composite overlapping grids. In this talk I will consider the following topics: (1) solving the high-speed reacting euler equations with adaptive mesh refinement, (2) solving the incompressible Navier-Stokes equations with a fourth-order accurate split-step scheme, (3) solving elliptic equations with the multigrid algorithm, (4) solving wave equations with higher order accurate methods.
talk120303: Direct Numerical Simulations of Multiphase Flow Gretar Tryggvason Department of Mechanical Engineering Worcester Polytechnic Institute Worcester, MA 01609-2280 Direct numerical simulations have recently emerged as a viable tool to understand finite Reynolds number multiphase flows. The approach parallels direct numerical simulations of turbulent flows, but the unsteady motion of a deforming phase boundary adds considerable complexity. Recent progress for relatively simple flows containing many bubbles and drops is discussed. The Navier-Stokes equations are solved by a front tracking technique that allows the inclusion of fully deformable interfaces and surface tension, in addition to inertial and viscous effects. A parallel version of the method makes it possible to use large grids and resolve flows containing a few hundred bubbles, making it possible to examine statistical properties of the flow. The development of numerical methods for more complex multiphase flows, where it is necessary to account for thermal and/or electric fields and phase changes is also underway. A few examples of the influence of electric fields on the dispersion of drops in a channel flow, the effect of flow on the growth of microstructures during solidification, and boiling flows are presented.
talk121003: Scalable Solvers and Software for PDE Applications David E. Keyes Fu Foundation Professor of Applied Mathematics Department of Applied Physics & Applied Mathematics Columbia University Like the theoretical peak performance of a computer system, theoretical efficiency for algorithms is rarely closely approached for real applications. While the quest for the "textbook efficiency" continues on many fronts, real users need to have their solver capabilities upgraded today to exploit the platform potential to run more highly resolved computations. The Terascale Optimal PDE Simulations (TOPS) project of the Scientific Discovery through Advanced Computing (SciDAC) initiative is working on both fronts --- attempting to make fundamental advances in numerical algorithms that will be integrated into tomorrow's scalable solver software while achieving gains for SciDAC application developers at the outset of the initiative. In this talk, we dwell on some practical aspects of migrating from a legacy (usually operator-split) nonlinear solver for evolutionary or equilibrium systems of PDEs to a Jacobian-free Newton-Krylov framework that provides strong controls on splitting error while still incorporating physically-based operator-split methodology (and even legacy subroutines) where possible. It is emphasized that to support even a single application from development through production use on various platforms, contemporary solver libraries must offer a menu of flexibly combinable and tunable components to allow application-specific and architecture-specific trade-offs (e.g., memory versus flops, synchronization frequency versus stability, robustness versus efficiency). We also discuss some experiences with the M3D extended magnetohydrodynamics code of our PPPL-based SciDAC partners, which is designed to underscore the desirability of being able to draw from a broad family of solvers within a single application. This talk is partially based on a 2003 review article for J. Comp. Phys. on Jacobian-Free Newton-Krylov methods co-authored with Dana Knoll of Los Alamos. Speaker URL: http://www.columbia.edu/~kd2112/ Project URL: http://www.tops-scidac.org
talk121703: The effect of contact-line conditions on the evolution of solid-liquid and liquid-vapor interfaces Vladimir S. Ajaev Department of Mathematics Southern Methodist University Numerical simulations of phase change at moving solid-liquid or liquid-vapor interfaces are important for many industrial applications including crystal growth, boiling in microchannels, and laser-induced melting. In experiments, the interface is often in contact with the constraining walls of the apparatus or, for solid-liquid transitions, with the gas phase surrounding the system. The boundary conditions at the line of contact can have a significant influence on the evolution of the interface. Standard numerical methods for simulations of moving interfaces often fail to capture the proper physical contact-line conditions and may even turn out to be numerically unstable. Recent progress in dealing with the issue of imposing the contact-line conditions for moving interfaces with phase change will be discussed in the talk. It will be shown that common difficulties in the numerical simulations are related to the local behaviour of solutions of the governing equations, such as equations for fluid flow and heat transfer, in the vicinity of the contact line. It will be explained how these difficulties can be avoided by including all relevant physical effects near the contact line in the framework of an efficient boundary integral type approach. Examples of physical problems used to illustrate the numerical method include solidification of non-uniformly cooled liquid droplets surrounded by air and spreading of volatile liquid droplets on heated surfaces.
talk012804: The QCDOC Supercomputer: Hardware, Software, and Performance Dr. Chulwoo Jung Physics Department Columbia University and Brookhaven National Laboratory An overview is given of the QCDOC architecture, a massively parallel and highly scalable computer optimized for lattice QCD using system-on-a-chip technology. The heart of a single node is the PowerPC-based QCDOC ASIC, developed in collaboration with IBM Research, with a peak speed of 1 GFlop/s. The nodes communicate via high-speed serial links in a 6-dimensional mesh with nearest-neighbor connections. Highly optimized four-dimensional QCD code obtains over 50% efficiency, even for problems of fixed computational difficulty run on tens of thousands of nodes. We also provide an overview of the QCDOC operating system, which manages and runs QCDOC applications on partitions of variable dimensionality.
talk021804: Imaging the Function of the Brain with Single Synapse Resolution Prof. Karel Svoboda Cold Spring Harbor Lab and Howard Hughes Medical Institute Abstract What neural substrate underlies our stable perception of the world? What changes in neural circuits when we learn? Biophysical methods based on optics hold the promise of aswering these fundamental question. I will discuss examples of such methods, with applications drawn from ourt work on plasticity in the somatosensory cortex.
talk030304: Dong Chen BlueGene/L: project status and early results Dr. Dong Chen IBM T J Watson Research Center Abstract: BlueGene/L is a partnership between IBM, ASIC-Trilab and Universities to develop and build a 180/360 Tflops computer. A 512 node prototype has been running in the IBM T.J. Watson Research Lab for a few months now. In this talk, we describe the architecture of the BlueGene/L computer, and show some early performance results obtained on the prototype.
talk031004: Computer modeling of the interaction of proteins with membrane surfaces: Insights into subcellular localization Diana Murray Cornell Medical School The reversible binding of proteins to membranes is crucial to many biological processes, such as signal transduction, vesicle trafficking and viral assembly. Many of these "peripheral " proteins contain lipid-interacting domains that recruit the proteins to specific intracellular membranes in response to signals, such as an increase in cellular calcium or the production of a phosphoinositide lipid. Our computational research and complementary experimental studies suggest that the binding of lipid-interacting domains to ligands, such as calcium ions or phosphoinositide head groups, dramatically alters the biophysical properties of the domains and that these changes are responsible for regulating membrane association. Further, it appears that various combinations of two physical factors electrostatics and hydrophobicity are major determinants of membrane binding. The finite difference Poisson-Boltzmann (FDPB) method has proved extremely accurate in its ability to account for many of the experimentally determined electrostatic properties of protein/membrane systems. This talk will focus on recent applications of the FDPB method to model the subcellular targeting of proteins to membrane surfaces. Our calculations of the physical forces between atomic-level models of proteins and phospholipid membranes provide insight, at the molecular level, into how different proteins are recruited to specific membranes and how proteins and lipids may be organized at membrane surfaces to facilitate the formation of macromolecular complexes. The overall computational approach we are developing provides a comprehensive framework with which to examine how proteins are designed to effect the wide range of membrane binding behaviors observed
talk051204: The Heterogeneous Multiscale Method Weinan E Princeton University The heterogeneous multiscale method (HMM) is a general framework for designing multiscale methods that might involve multi-levels of physics. It has been applied to a variety of interesting and challenging problems including contact line dynamics, complex fluids, fluid flow over chemically heterogeneous surface, dynamics of twinn boundaries in solids, and transport in strongly heterogeneous medium. In this talk, we will discuss the basic principles of HMM, as well as some examples of applications and analysis.
talk101504: Challenge of CIP as a Universal Solver from Atom to Space T.Yabe Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152-8552, JAPAN e-mail: yabe@mech.titech.ac.jp Abstract We present a review of the CIP(Cubic-Interpolated Propagation/Constrained Interpolation Profile) method[1,2] that is known as a general numerical solver for solid, liquid , gas and plasmas. This method is a kind of semi-Lagrangian scheme and has been extended to treat incompressible flow in the framework of compressible fluid. Since it uses primitive Euler representation, it is suitable for multi-phase analysis. The recent version of this method guarantees the exact mass conservation even in the framework of semi-Lagrangian scheme[3]. Comprehensive review is given for the strategy of the CIP method that has a compact support and subcell resolution including front capturing algorithm with functional transformation, pressure-based algorithm and other miscellaneous physics such as elastic-plastic effect and surface tension. Some practical applications are also reviewed such as skimmer, killifish, laser-induced melting, and so on. Recently we found that the CIP method can be extended to all kinds of equations in differential forms and one of the example is the high accurate solution of Shroedinger equation[4]. In addition to this, we have recently proposed a new class of body-fitted grid system that can keep the third-order accuracy in time and space with the help of the CIP. The grid system consists of the straight lines and grid points moving along these lines like abacus - Soroban in Japanese[5]. The length of each line and the number of grid points in each line can be different. The CIP scheme is suitable to this mesh system and the calculation of large CFL(>10) at locally refined mesh is easily performed. Mesh generation and searching of upstream departure point are very simple and almost mesh-free treatment is possible. Adaptive grid movement and local mesh refinement are demonstrated. We show some applications of this scheme to three-dimensional propagation of electromagnetic wave and hydrodynamics in complex surface. REFERENCES [1] H.Takewaki,A.Nishiguchi and T.Yabe , J.Comput.Phys., 61 (1985) 261-268. [2] T.Yabe, and T. Aoki, Comput. Phys. Commun.66 (1991), 219-232 [3] T.Yabe, F.Xiao and T.Utsumi, J. Comput. Phys. 169 (2001), 556-593 [4] T.Utsumi, T.Yabe, J.Koga, T.Aoki and M.Sekine, Comput.Phys.Commun. 157 (2004)., 121-138. [5] T.Yabe et.al., J. Comput. Phys. 194 (2004), 57-77
talk110304: Danny Z. Chen Department of Computer Science and Engineering University of Notre Dame Notre Dame, IN 46556, USA Computer-assisted surgery is a newly emerging interdisciplinary area that applies the state-of-the-art computer technologies to help the diagnosis, design, operation, and optimization of modern medical surgery. In this talk, we discuss some new developments in this important and exciting area. Intensity-modulated radiation therapy (IMRT) uses radiation beams to surgically eradicate tumors while sparing surrounding critical structures and healthy tissues in human bodies. We present new computer algorithms for several surgical planning problems in IMRT, called leaf sequencing problems in the medical community, and prove that the leaf sequencing problems are in general NP-hard. The previously best leaf sequencing algorithms which are currently used in clinical treatments are all heuristics that do not guarantee any good quality of the output surgical solutions and may take a long computational time. Our new algorithms, based on a novel unified approach and geometric optimization techniques, are very efficient and generate surgical plans of much better quality. Our algorithmic ideas include formulating the leaf sequencing problems as computing generalized shortest paths and maximum flows in a directed graph and building the graph by computing optimal bipartite matchings on various geometric objects. Experimental and comparison results have shown that our new IMRT algorithms run very fast (within a minute) on real medical data, and the quality of our output treatment plans is significantly better than the plans produced by the current commercial surgical planning systems. Our software has been incorporated into real medical systems for clinical treatments of cancer patients in several hospitals.
talk041305: Prof. Michael Mascagni Department of Computer Science and School of Computational Science Florida State University Tallahassee, FL 32306-4530 Title: Stochastic Methods in Electrostatics: Applications to Biological and Physical Science Abstract: We present an overview of stochastic methods for the solution of elliptic partial differential equations (PDEs). In particular, we consider the solution of linear and nonlinear problems that arise in electrostatics computations in various applications. We discuss the "Walk On Spheres" (WOS), "Greens Function First-Passage" (GFFP), and "Simulation-Tabulation" (S-T) Monte Carlo methods for the computation of capacitance, charge density, and related problems in materials science and biophysics. In addition, we introduce the "Walk on the Boundary" method for rapid calculation of capacitance. We then present generalizations that permit direct computation of charge density. Finally, we consider the problem of the electrostatics of large molecules in aqueous solution. An implicit model of the solvent leads to consideration of the Poisson-Boltzmann equation (PBE) as a continuum electrostatic model. We present new Monte Carlo methods for the solution of the linear PBE based on WOS, GFFP, and other methods. In particular, we solve an elliptic PDE system with the Poisson equation inside the molecule of interest, the linear PBE outside, and matching Neumann boundary conditions on the molecular surface. We demonstrate the intrinsic advantages of these methods on an electrostatic internal energy computation with the use of new, efficient Monte Carlo approaches to the boundary conditions. This work is joint with Dr. Nikolai Simonov of FSU and the Siberian Branch of the Russian Academy of Sciences
talk042005: Face-Offsetting Framework for Dynamic Surface Meshes Xiangmin (Jim) Jiao, UIUC Dynamic surface meshes are ubiquitous nowadays in scientific and engineering applications, such as heart or blood-cell simulations and other fluid-solid interaction problems in computational biology, and burning propellant in solid rocket motors. In this talk, we introduce a new class of methods, called face-offsetting, based on an entropy-satisfying Lagrangian formulation for moving interfaces. Unlike level set methods, face-offsetting operates directly on Lagrangian surface meshes, without requiring Eulerian volume meshes. Unlike traditional Lagrangian methods, face-offsetting propagates faces and then reconstructs vertices by solving a constrained minimization problem at each vertex, instead of moving vertices along approximate normal directions. We present the theoretical foundation and some experimental results for face-offsetting methods. In addition, we outline some recent progress and on-going efforts on numerical treatments for constraints, dissipation, and conservation, as well as combinatorial treatments for adaptivity and topological changes within our face-offsetting framework.
talk050405: Joint Seminar Computational Applied Math Computational Geometry Wed. May 4, 10:30 AM Room 1-122A, Math Towe Prof. Brent Lindquist Applied Math and Statistics Stony Brook University The (Computational) Geometry of Primary Drainage Abstract: We show that the movement of liquid-liquid menisci under primary drainage and imbibition in capillary tube cross sections, and, by extension, in throats in rock and soil pore structure, can be understood in terms of the computational geometry theory of medial axis analysis of polygon interiors. As a result we extend the existing capability to predict menisci configuration from cases in which cross sections are either triangular or regular convex polygons to that of arbitrary polygonal cross section. This prediction of the shape of the cross sectional region occupied by the two (or more) fluids is a problem of interest to chemical, fiber, and petroleum industries. For two fluid flow, in which one fluid is perfectly wetting, the centers of curvature of the menisci must be located along the medial axis of the polygon interior. If neither fluid is perfectly wetting, the centers of curvature of the arc menisci are located on an offset to the medial axis which we refer to as the drainage axis. This offset is obtained by retaining hyperbolic edge- vertex bisectors rather than the parabolic edge-vertex bisectors used in construction of a medial axis. The drainage axis has not been investigated in computational geometry. Like the medial axis, the drainage axis has a tree structure, but, unlike the medial axis, its tree structure is not spatially connected. Open questions include whether the drainage axis can theoretically be computed in linear time and whether the drainage axis is linked to a Voronoi-like tesselation of the polygon.
talk052505: Cartesian Grid Based Conservative Front-Tracking Methods. Professor Mao De-kang Department of Mathematics, Shanghai University No. 99, Shangda Rd. Shanghai, 200444, P.R. China In this talk I am going to present a new approach for developing conservative front-tracking methods. The main idea is as the follows: A discontinuity in a solution to hyperbolic conservation laws is actually a lower dimensional moving manifold and its evolutions can be described by a conservative differential equation in the lower dimensional space. To do the front-tracking, we discretize this differential equation on the underlying Cartesian grid in a conservative fashion and embed its discretization into the computation in the smooth region. In doing this way, the developed front-tracking method becomes a combination of two capturing schemes, one is for the solution in the smooth region and the other is for the tracked discontinuity in the lower dimensional space. The method runs on Cartesian grid, no irregular grid cells are used, and it is conservative.
talk061705: A robust and practical numerical model for multi-fluid simulations Professor Feng Xiao Tokyo Institute of Technology Abstract This talk presents a novel numerical formulation for direct simulations of multi-fluid flows including free boundaries. The numerical framework, namely VSIAM3 (Volume/Surface Integrated Average based Multi-Moment Method)[1,2], is constructed by combining the fundamental concept of the CIP method [3] and the finite volume formulation (FVM). Two integrated moments of each physical quantity are defined as the volume integrated average (VIA) and the surface integrated average (SIA), and separately used as the model variables. Different from conventional finite volume method (FVM), the VSIAM3 provides a general framework for simulating various flows. Numerical model for multi-fluid simulations can be straightforwardly built based on the VSIAM3 formulation as well. The formulation based on VIA and SIA gives a great convenience to present solid obstacles or boundaries of arbitrary geometrical complexities. Similar to the FAVOR method of Hirt [4], modifications to the effective fluxes due to the presence of the solid bodies can be straightforwardly obtained based on VIA and SIA as well as fluid fractions. In terms of VIA and SIA, the present model does not meet the 'zero volume' problem. The projection based on the VIA of pressure and the SIAs of velocity field guarantees the incompressibility even obstacles of complex geometry are involved. The resulting algorithm is quite simple and appears robust in various simulations. To get around of the complexity in the surface reconstruction in the presence of complex geometry, we developed three interface capturing methods, namely C3VOF scheme [2], STAA scheme [2] and THINC[5] scheme for practical use. These schemes don't require the geometrical reconstruction for the moving interface, still the numerical diffusion across the interface can be effectively eliminated and the thickness of the interface remains compact. The presented numerical model has been used to simulate various interfacial multi-fluid flows and appears to be computationally efficient and robust even with solid obstacles of much complex geometry. References [1] F.Xiao: Unified formulation for compressible and incompressible flows by using multi integrated moments I: One-dimensional inviscid compressible flow. J. Comput. Phys., 195, 629-654 (2004). [2] F.Xiao, A.Ikebata and T.Hasegawa: Numerical simulations of free-interface fluids by a multi integrated moment method. Computers & Structures, 83, 409-423 (2005). [3] T.Yabe, F.Xiao and T. Utsumi: The constrained interpolation profile method for multiphase analysis. J. Comput. Phys., 169, 556-593 (2001). [4] C.W.Hirt: Volume-fraction techniques: powerful tools for wind engineering. J. Wind Engrg. & Industr. Aerodyn., 46-47, 327-338, (1993),. [5] F.Xiao, Y.Honma and T.Kono: A simple algebraic interface capturing scheme using hyperbolic tangent function, Int. J. Numer. Method in Fluids, in press(2005).
talk101205: From Climate Models to Earth System Models Robert Jacob Argonne National Laboratory The DOE/NSF developed Community Climate System Model is a state-of-the-art, fully coupled global climate model which provides simulations of the Earth's past present and future climate. CCSM consists of models of the atmosphere, ocean, land surface and sea ice united into one system by a coupler. An overview of CCSM will be presented with an emphasis on its basic design and software structure as well as results from recent integrations. The evolution of climate models to earth system models, which include representations of chemistry in the land, ocean and atmosphere, will also be described.
talk102605: SMALL IS DIFFERENT: Highly Confined Fluids -- Nanotribology and Nanojets Uzi Landman School of Physics, Georgia Institute of Technology, Atlanta, GA 30332 Computationally-based theoretical modeling and simulations play an increasingly important role in modern condensed matter physics, chemistry, materials science, and biology. In particular, such studies, that may be called computational microscopies”, allow explorations of complex phenomena with refined resolution in space and time [1]. The use of atomistic simulations as tools of discovery will be discussed and demonstrated through a discussion of two simulation-based studies: 1. Molecular dynamics simulations of the formation and breakup of liquid jets of nanoscale dimensions, lead to a stochastic formulation of the Navier-Stokes equations, thus extending continuum hydrodynamics to the nanoscale domain. The emergence of new classes of break-up solutions for nano-scale liquid structures, differing from those found for the corresponding macroscopic ones, will be analyzed [2]. 2. Amontons’ law, which was already known to Leonardo da Vinci, states that the friction force is directly proportional to the (normal) applied load, with a constant of proportionality - the friction coefficient - that is constant and independent of the contact area, the surface roughness and the sliding velocity. No theory has yet satisfacorily explained this surprisingly general law, all attempts being model or system dependent. On the basis of large-scale molecular dynamics simulations pertaining to lubricated adhesive and non-adhesive junctions, with morphologically rough (as well as crystallographically flat) confining solid surfaces, and in conjunction with recent experiments, we show that the local energy-dissipation mechanisms are not 'mechanical', as assumed in most models, but thermodynamic” in nature. We show that a local analysis of the simulation results, based on division of the system into small cells, leads to a natural description in terms of the Weibull distribution. For the dynamic. non-equilibrium, energy-dissipating process that we study, this long-tail distribution serves a similar purpose as the Boltzmann distribution for classical systems at equilibrium. While Amontons law does not hold on the local scale, it is recovered on the global scale, with the spatio-temporal averaging utilizing the Weibull distribution of the local friction forces. Interestingly, the concept of "area of contact", often used in frictional studies, does not enter into our analysis [3]. 1. U. Landman, Materials by Numbers: Computations as Tools of Discovery”, Perspective article in Proc. Nat. Acad. Sci. (USA) 102, 6671 (2005). 2. (a) M. Moseler and U. Landman, Science, 289, 1165 (2000); (b) W. Kang and U. Landman, to be published. 3. J. Gao, W. D. Luedtke and U. Landman, Feature article in J. Phys. Chem. B 108, 3480 (2004).
talk110205: Lawrence Sirovich Director, Laboratory of Applied Mathematics Chairman, Department of Biomathematics Mt. Sinai School of Medicine Dynamics of Neural Populations: Stability & Synchrony Abstract A population formulation of neuronal activity is employed to study an excitatory network of (spiking) neurons receiving external input as well as recurrent feedback. At relatively low levels of feedback, the network exhibits time stationary asynchronous behavior. A stability analysis of this time stationary state leads to an analytical criterion for the critical gain at which time asynchronous behavior becomes unstable. At instability the dynamics can undergo a supercritical Hopf bifurcation and the population passes to a synchronous state. Under different conditions it can pass to synchrony through a subcritical Hopf bifurcation. And at high gain a network can reach a runaway state, in finite time, after which the network no longer supports bounded solutions. The introduction of time delayed feedback leads to a rich range of phenomena. For example, for a given external input, increasing gain produces transition from asynchrony, to synchrony, to asynchrony and finally can lead to divergence. Time delay is also shown to strongly mollify the amplitude of synchronous oscillations. Perhaps, of general importance, is the result that synchronous behavior can exist only for a narrow range of time delays, which range is an order of magnitude smaller than periods of oscillation.
talk121905: Yuan-nan Young NJIT We study the effect of surface tension on the incompressible Rayleigh-Taylor instability. We modify Goncharov's local analysis \cite{Go02} to consider the surface tension effect on the Rayleigh-Taylor bubble velocity. The surface tension damps the linear instability and reduces the nonlinear terminal bubble velocity. We summarize the development of a finite-volume, particle-level-set, two-phase flow solver with an adaptive Cartesian mesh, and results from convergence and validation studies of this two-phase flow solver are provided. We use this code to simulate the single-mode, viscous Rayleigh-Taylor instability with surface tension, and good agreement in terminal bubble velocity is found when compared with analytic results. We also simulate the immiscible Rayleigh-Taylor instability with random initial perturbations. The ensuing mixing flow is characterized by the effective mixing rate and the flow anisotropy. Surface tension tends to reduce the effective mixing rate and homogenizes the Rayleigh-Taylor mixing flow. Finally we provide a scaling argument for detecting the onset of the self-similar Rayleigh-Taylor growth.
talk020806: Title: A new implementation of the elliptic systems method in time dependent diffusion tomography applied to back reflected and transmitted data. YinTzer Jerry Shih Department of Applied Mathematics National Chung HsingUniversity Taichung, Taiwan 4002 It is common in applied work in engineering such as the search for buried land mines, or in medical imaging for diagnosis of possible breast tumors, to have only limited boundary measurement data, back reflected in the first case or transmitted in the second. Here we formulate the problem as one of coefficient recovery from incomplete boundary data in inverse problems. We have completed a new implementation of the Elliptic Systems Method (ESM) in time dependent diffusion tomography. The basic formulation of the ESM involves solving a system of (typically 4) coupled 4th-order PDE's, with the time variable integrated out using Legendre polynomials. Here, unlike the previous implementation that creates a larger (typically of size 8) mixed system of 2nd-order problems with quadratic elements over triangles, we use Bogner-Fox-Schmit bi-cubic elements over rectangles, with a new treatment of boundary conditions in the common case of incomplete boundary data. This new method is 4th-order accurate for sufficiently smooth functions. The new BC approach allows the use of homogeneous natural boundary conditions on parts of the boundary where no measured data is available. This combined effort is being reported elsewhere, but without extensive comparisons of difficult applications against the literature. Here we will focus on three previously published examples using back reflected or transmitted data with one or two inclusions. The new implementation in comparison gives markedly improved results for inclusion recovery, all of which are achieved without use of additional aids such as weight functions which have previously been thought to be essential. In addition the new implementation is shown to be surprisingly robust with respect to noise. We conclude with two examples illustrating the effect of increasing levels of noise.
talk041206: Robustness of Morphogen Gradients Qing Nie Department of Mathematics Center for Mathematical and Computational Biology Department of Biomedical Engineering Center for Complex Biological Systems University of California, Irvine Many patterns of cell and tissue organization are specified during development by gradients of morphogens, substances that assign different cell fates at different concentrations. One of the central questions in cell and developmental biology is to identify mechanisms by which the morphogen gradient systems might achieve robustness to ensure reproducible embryonic patterns despite genetic or environmental fluctuations. Recently, through computations and analysis of various bio-chemical models and examination of old and new experimental data, we found a set of of new mechanisms for enhancing robustness of cell-cell signaling through non-signaling cell surface molecules (e.g., HSPG). In addition, we examined the roles of diffusive ligands (e.g., Sog) on the formation and robustness of BMP (Bone Morphogenetic Protein) gradients in the Drosophila embryo. In this talk, I shall also discuss some mathematical and computational challenges associated with such study, and present a new class of numerical algorithms for reaction-diffusion equations arising from biological models.
talk081606: Non-selfsimilar global solutions and new structures in multi-dimensional conservation laws Professor Xiaozhou Yang Department of Mathematics, Shantou University, Guangdong 515063, China Abstract In this talk, we will discuss the multi-dimensional (M-D) conservation laws whose Riemann data just contain two different constant states which are separated by a smooth curve or surface. Non-selfsimilar M-D elementary waves, their new structures and properties are disclosed.
talk092706: Phase field modeling and simulation of vesicle membranes Professor Qiang Du Department of Mathematics Penn State University We report some recent works on the phase field modeling and simulations of the vesicle membrane deformation under elastic bending energy and the interaction with background fluid flows. We illustrate the effectiveness of the phase field modeling through simulations of recent biological experiments on two-component membranes. We also discuss how to effectively retrieve topological information within the phase field framework.
talk030707: Fast sweeping method for static convex Hamilton-Jacobi equation Professor Hongkai Zhao Department of Mathematics UC Irvine Abstract: Hamilton-Jacobi (HJ) equation is a class of nonlinear hyperbolic partial differential equations that have wide applications in optimal control, geometric optics, image processing and computer graphics, etc. In this talk I will present an efficient iterative method, the fast sweeping method, for computing the numerical solution of static convex HJ equations on both structured and unstructured meshs. Convergence, error estimate and optimal complexity will be shown. Every iterative method converges for a reason. The fast sweeping method can converge in a finite number of iterations that is independent of mesh size. I will explain the two most crucial mechanisms, ordering and causality enforcement during Gauss-Seidel iterations, for the fast convergence of fast sweeping method. Finally, applications to image processing and computer vision will be shown.
talk030907: New numerical solvers for Hydo- and Magnetohydrodynamics Christian Klingenberg Mathematisches Institut Universitaet Wuerzburg Am Hubland 97074 Wuerzburg GERMANY Abstract We present a relaxation system for the Euler equations and for ideal MHD, from which one may derive approximate Riemann solvers. The solvers satisfy a discrete entropy inequality, and preserve positivity of density and pressure under a subcharacteristic condition. We present applications to realistic flows. This is joint work with F. Bouchut.
talk042507: Mesh Construction for Adaptive High-order and Multiscale Simulations Xiaojuan (Sarah) Luo Scientific Computation Research Center (SCOREC) Rensselaer Polytechnic Institute xluo@scorec.rpi.edu High-order (p-version) finite element methods are capable of achieving exponential rate of convergence. However, the full realization of this high convergence rate requires optimal mesh construction, which is particularly challenging in general 3D curved domains. This talk presents a curvilinear mesh generation procedure for high-order curved meshes that applied Bezier polynomial as mesh entity shape representation. An algorithm to determine the validity of curved Bezier simplex elements has been developed. For general 3D curved domains, the mesh generation procedure includes the construction of geometrically graded meshes from singular model features, construction of prismatic elements in thin sections, and local mesh modifications to ensure meshes can be properly curved to the domain boundary. In case of a given curved mesh without any model information, local mesh modifications are incrementally applied to correct the invalid curved elements and the surface approximation of the original mesh is constraint. Recently, a curved mesh correction tool has been developed. This tool has been used to correct curved meshes used by the SLAC T3P solver and the results demonstrated that 40% computational efficiency could be achieved. In the average-theory based multiscale simulation for engineered tissues, mixed topology meshes with layered prisms are constructed for the thin model geometries. The meshes are applied in a component-based parallel multiscale simulation toolkit (MCTK) that makes the computation of soft tissues with complex geometries and hundreds of millions of degree of freedom practical.
talk050807: Place: Math 1-122A, AMS Colloquium Room Time: Wednesday, 5/8, 11:00AM Title: Monte Carlo Methods for Partial Differential Equations Name: Prof. Michael Mascagni Address: Department of Computer Science and School of Computational Science Florida State University Tallahassee, FL 32306-4530 USA Abstract: We begin with a quick review of the Feynman-Kac equations. These allow one to represent the solution of linear elliptic and parabolic partial differential equations (PDEs) as expected values over stochastic processes. The particular stochastic process for a given PDE is the solution to a stochastic differential equation defined via the elliptic operator in the PDE. We then briefly discuss methods for nonlinear parabolic equations known as reaction-diffusion equations. We then return to elliptic PDEs and discuss, in detail, several acceleration techniques that are widely applicable Monte Carlo methods. We begin with the "walk on spheres" algorithm, followed by the the "Greens function first-passage" method, the "simulation-tabulation" method, "last passage" methods, the "walk on the boundary" method, and finally the "walk on subdomains" method. These various Monte Carlo methods are presented within the context of various problems that arise in flow through porous media, electrostatics, and continuum biochemistry. We also present the example of the "telegrapher's" equation, an hyperbolic equation, as solved stochastically, and some nonlinear PDE examples.
talk052307: MAGNETOHYDRODYNAMIC COMPUTATIONS IN A ROTATING SPHERE David Montgomery, Dept. of Physics and Astronomy Dartmouth College, Hanover, NH 03755 An old but currently active problem is that of the magnetic dynamo, defined as a mechanical process by means of which a moving electrically conducting fluid, probably turbulent, can excite and sustain a macroscopic magnetic field against resistive decay. The geomagnetic field is a classic example, but current laboratory experiments in liquid sodium have also become able to achieve persistent dynamos. The strong nonlinearities in the magnetohydrodynamic equations require numerical computation. The large range of length and time scales involved puts the computational problem out of reach without some compromise with reality in dealing with the smaller spatial scales. Here, a new spectral method will be described, using orthonormal Chandrasekhar-Kendall vector eigenfunctions of the curl as an expansion basis. The method lacks the resolution necessary to compute at geomagnetically large Reynolds numbers or realistically small Rossby and Ekman numbers, but nonetheless has resolution enough to recover a wide variety of dynamo action, inside a rotating spherical boundary, that is thought to be of some physical interest. (This work has been performed with P.D. Mininni and L. Turner; see: Phys. Fluids 18, 116602 (2006) and arXiv:physics/0702082.)
talk091207: Data-Based Analysis of Winner-Loser Models of Hierarchy Formation in Animals W. Brent Lindquist, Department of Applied Mathematics and Statistics Stony Brook University Ivan D. Chase, Department of Sociology, Stony Brook University
talk091907: Towards Breaking Temperature Equilibrium in Multi-Component Eulerian Schemes John Grove Los Alamos National Laboratory
talk092607: Graduate Research Opportunities in Energy Related Projects Roman Samulyak Computational Science Center Brookhaven National Laboratory A reliable energy supply is the cornerstone of sustained economic growth and prosperity. World energy demand is expected to more than double by 2050. Energy research is the highest priority for the Department of Energy (DOE). Research in the area of thermonuclear energy and an expansion of nuclear energy is viewed by DOE as main paths towards meeting the future energy demand while reducing air pollution and carbon. Stony Brook University and Brookhaven National Laboratory are involved in DOE energy related projects that present significant challenges in applied mathematics and computational science. I'll describe two projects of potential interest to AMS graduate students. The main goal of the first project is the development of new mathematical models, numerical algorithms, and computational software for the study of magnetohydrodynamics (MHD) of 3D multiphase flows in the presence of phase transitions and external energy sources. Our computational models and software are currently being used to study the fueling of thermonuclear reactors. The second project aims at the deployment of a suite of high performance computational tools for multiscale physics simulations of generation-IV nuclear reactors. I'll describe applied mathematics and computational aspects of these projects, current progress and future work.
talk100307: Data-Based Analysis of Winner-Loser Models of Hierarchy Formation in Animals W. Brent Lindquist, Department of Applied Mathematics and Statistics Stony Brook University Ivan D. Chase, Department of Sociology, Stony Brook University Linear dominance hierarchies occur in small groups across a broad range of species: insects, crustaceans, fish, birds, and mammals, including humans. Hierarchical rank mediates many aspects of individuals’ lives including physiology, reproduction, susceptibility to diseases, and access to scarce resources. In spite of their biological importance and their unique form as social structures, it is still not clear what accounts for the linear form of dominance hierarchies. An earlier view assumed that these linear structures were simply reflections of linear rankings on attributes associated with dominance ability (e.g., combination of traits such as weight, aggressiveness, genotype, and hormonal profiles) that animals had prior to joining a group. Theoretical work indicates that stringent, but not always recognized, mathematical requirements must be met in order for this view to be correct, and recent experimental work demonstrates that differences in prior attributes cannot generally account for linear structures. The popular current view suggests that linear hierarchy structures arise from series of pair-wise interactions involving winner and loser effects. In a winner effect, an individual winning a contest increases its ability to win a subsequent contest; in a loser effect, an individual losing a contest decreases its ability to win a subsequent dominance encounter. A number of recent models incorporating these effects can indeed produce highly linear structures. These models reflect four assumptions. 1) Animals in groups forming hierarchies experience winner and loser effects in response to the outcomes of their contests. 2) Each individual has a dominance “score” which reflects its past history of winning and losing contests; the dominance score of the winner is incremented and that of the loser decremented in a specified manner. 3) When two animals interact in a dominance contest, a specific mathematical probability formulation, which is based upon the difference in their scores, determines the probability that one or the other will win. 4) Animals do not identify one another as individuals; consequently, in successive meetings they are not influenced by memories of their previous encounters. Methodology/Principal Findings We evaluate winner-loser models as explanations for the occurrence of linear dominance hierarchies. Our methodology is divided into three parts. Part one examines three of the most widely-cited winner-loser models. These models have never been rigorously tested against experimental observation. We isolate the two most significant components of the mathematical formulation of each model, the “pairing-interaction” component which determines the order in which pairs interact, and the “dominance component” which determines the winner in each interaction. The predictions of these formulations are rigorously evaluated against data collected during the observation of linear hierarchy formation in 14 groups of Leghorn hens. These comparisons indicate that the mathematical formulations of the three models either do not fit the experimental data (at best only capturing a general trend) or that they are only partially accurate but fail to capture some significant aspect of the data. Motivated by the lack of fit between the mathematical formulations and the experimental data, part two examines the basic assumptions on which the dominance components of the models are based, namely: the assumptions of the presence of winner and loser effects in groups; the use of the difference in dominance “score” values to determine winning probability; the lack of individual identification; and the absence of memory of previous encounters. This examination shows that while winner and loser effects have been widely documented in isolated pairs , only one study has looked at these effects in pairs within group s . This study indicated that although cichlid fish evidenced a loser effect in isolated pairs, they did not do so at a rate above chance in socially embedded pairs. While further research is called for, there is at present no evidence that winner and loser effects occur in pairs within groups forming hierarchies. The latter three assumptions (dominance score differences determine winning and losing probabilities, no individual identification, and no memory of past encounters) are intrinsically linked. If animals do identify one another as individuals and if their memories of past encounters do influence subsequent ones, the assumption that contest outcomes are only influenced by differences in dominance scores, cannot be supported. Research across many taxa and social contexts indicates that animals can indeed recognize one another as individuals. In particular, experiments on a broad range of species including mammals, birds, crustaceans, fish, and even certain insects (wasps, ants, and fruit flies) that form dominance relationships demonstrate that individuals can recognize others in subsequent contests, even after only brief prior exposure, and that their memories of one another can last for considerable periods of time. A strict interpretation of some of these experiments might argue that in some species individuals can only identify each other as either “ familiar” or “unfamiliar”, and not as specific individuals. However, this interpretation is enough to invalidate these winner-loser model assumptions since the experiments show that familiar individuals meeting again resume their original relationships while unfamiliar individuals establish new relationships not influenced by their (individual) previous dominance encounters. Given the lack of empirical support for either the mathematical formulations of the models or the assumptions on which they are based, part three of the paper uses new techniques applied to the hen data to uncover several dynamics of hierarchy formation not previously described. These features suggest a very different basis for hierarchy formation than the models, which should be explored in future experimental and theoretical work. These features include bursting (repeated attacks by an animal upon those it already dominates), infrequency of counter-attacks in initial encounters, sequential rank differentiation, the rarity of intransitive dominance relationships, the rapid conversion (i.e. instability) to transitive relationships of those intransitive relationships that do occur, and the stability of transitive relationships. (Linear hierarchies are composed solely of transitive dominance relationships; the larger the number of intransitive relationships among individuals, the further from linearity.) These features suggest that animals forming hierarchies are intensely aware of their own interactions as well as those occurring among other members of their groups. Considerable recent evidence on social cognition in animals supports these suggestions. Conclusions/Significance Our investigations suggest that winner-loser models cannot accurately account for linear hierarchy formation in real animals. The assessments of the core formulations and assumptions common to these models find not simply a lack of empirical support, but that the available evidence argues that the mathematical formulations are flawed and the assumptions are false. Our discovery of a number of new features of hierarchy formation in the hens suggests that they are intensely aware of the interactions among their fellows and of their placement within the group. Beyond animal dominance hierarchies, we believe our results have more general implications for efforts to understand other kinds of social organization in humans and animals. Researchers in both the social sciences and animal behavior have proposed that the structure of different types of social organization are explained by differences in characteristics of individuals. These explanations would seem particularly promising to account for the structure of dominance hierarchies. However, as noted in the Introduction, earlier efforts to explain the form of linear hierarchies as reflections of linear rankings on attributes that animals have prior to joining groups have proven untenable. In this paper, we show that models based upon continual updates to dominance ability through feedback from wins and loses during hierarchy formation also are inadequate. Thus, this research indicates that models based either upon prior or dynamically updated attributes appear to be inadequate to realistically account for linear dominance hierarchies in animals. If explanations based upon differences among individual do not work for dominance hierarchies in animals, it seems likely that they may not work for other kinds of social organization. As we have suggested for hierarchies, accounts based upon social cognition and dynamics of interaction are likely to be more realistic.
talk101007: Free-Boundary Problems in Finance and Singularity-Separating Method You-lan Zhu Dept. of Mathematics and Statistics University of North Carolina at Charlotte There are two types of derivative securities in finance. One is Euporean derivative securities and the other is American derivative securities. A Euporean derivative security can be exercised only at maturity and an American derivative security can be exercised at any time before maturity. Because of this fact, in order to evaluate a Euporean derivative security, a partial differential equation (PDE) problem needs to be solved, but in order to evaluate an American derivative security, a linear complementarity (LC) problem needs to be solved. A LC problem usually involves free boundaries. On one side of a free boundary the solution satisfies a PDE, on the other side the solution is known. Therefore if the locations of free boundaries are not tracked, it is not easy to get high accuracy if a coarse mesh is used. In the financial problem, in the final condition, the first derivative of the function is often discontinuous, a coarse mesh usually does not give a good result near that point. In order to get a good result on a coarse mesh, some treatments are provided and good results can be gotten even on a coarse mesh. On the free boundary the second derivative of the solution is discontinuous and at the point mentioned the first derivative is discontinuous. Therefore the solution we want to compute has some singularities. After the treatment the solution computed is smoother, we thus call our method the singurarity-separating method.
talk101707: Simulation of Turbulent Flows With Strong Shocks Bruce Dryxell Los Alamos National Laboratory Computation of turbulent flows with strong shocks is a very challenging problem, since the requirements for a method to produce accurate results for turbulence are orthogonal to those needed to treat shocks properly. In order to prevent an unphysical rate of decay of the turbulent structures, it is necessary to use a method with very low numerical dissipation. Because of this, central difference schemes are widely used. However, computing strong shocks with a central difference scheme can produce unphysical post-shock oscillations that corrupt the entire flow unless additional dissipation is added. This dissipation can be difficult to localize to the area near the shock and can lead to inaccurate treatment of the turbulence. Modern high-resolution shock-capturing methods usually use upwind algorithms to provide the dissipation necessary to stabilize shocks. However, this upwind dissipation can also lead to an unphyical rate of decay of the turbulence. This talk will discuss a hybrid method for simulating turbulent flows with strong shocks that couples a high-order central difference scheme with a high-resolution shock-capturing method. The shock-capturing method is used only in the vicinity of discontinuities in the flow, while the central difference scheme is used in the remainder of the computational domain. Results of this new method will be shown for a variety of test problems, including Richtmyer-Meshkov instabilities and the interaction of a shock with a turbulent flow field.
talk103107: Coupled atomistic-continuum methods for fluid Weiqin Ren Courant Institute of Mathematical Sciences New York University Abstract This talk consists of two parts. In the first part, I present a multiscale method for the study of fluid systems with unknown constitutive relations and/or boundary conditions. The multiscale method captures the macroscale behavior of the fluid system using molecular dynamics. In the multiscale method, the contiuuum and atomistic models are coupled in a seamless may that does not require going back and forth between the macro and micro states of the system. I will discuss the details of the coupling scheme, its application to complex fluids, and also the major difficulties in implementation. In the second part of the talk, I will discuss the moving contact line problem. The difficulty in this problem comes from the fact that the hydrodynamics with the no-slip boundary condition predicts an non-integrable viscous stress at the moving contact line. I will present a detailed study of the physical processes and various forces in the contact line region by molecular dynamics (MD). A continuum model for the boundary condition is formulated based on the results of MD.
talk110707: Thomas L. Jackson Rocket Center
talk112807: Hamiltonian Systems and Liouville Equations with Discontinous Hamiltonians: Computation of High Frequency Waves in Heterogeneous Media Shi Jin, Professor Department of Mathematics University of Wisconsin Madison, WI 53706, USA We introduce Eulerian methods that are efficient in computing high frequency waves through heterogeneous media. The method is based on the classical Liouville equation in phase space, with discontinous Hamiltonians (or singular coefficients) due to the barriers or material interfaces. We provide physically relavant interface conditions consistent with the correct transmissions and reflections, and then build the interface conditions into the numerical fluxes. This method allows the resolution of high frequency waves without numerically resolving the small wave lengths, and capture the correct transmissions and reflections at the interface. Moreover, we extend the method to include diffraction, and quantum barriers. Applications to semiclassical limit of linear Schrodinger equation, geometrical optics, elastic waves, and semiconductor device modeling, will be discussed.
talk022008: The Risk of Bankruptcy in Long-term Investment Qiang Zhang Department of mathematics, and Department of Economics and Finance City University of Hong Kong In recent years, various continuous-time strategies in portfolio management have been developed with different objectives. However the risks associated with these strategies are not well understood. We focus on one particular measure of risk in this talk, namely the probability of bankruptcy occurring while applying these strategies. We demonstrate that if the target return rate is set above certain critical value, then the probability of being in bankruptcy will be one hundred percent for a long term investor. This is a joint work with Minjie Yu and Dennis Yang.
talk021108: Fluid-Structure Interaction in Blood Flow Prof. Suncica Canic Department of Mathematics University of Houston The focus of this talk will be on the analysis and computation of fluid-structure interaction in blood flow. Understanding solutions to moving-boundary problems describing fluid-structure interaction between blood flow and arterial walls is important in understanding the mechanisms leading to various complications in cardiovascular function. Although fascinating progress has been made in some areas of modeling and simulation of the human cardiovascular system many of the basic difficulties remain open and will continue to present major challenges in the years to come. The speaker will give an overview of the main problems and difficulties associated with the study of fluid-structure interaction in blood flow. Recent results in the analysis of solutions to the benchmark problem in blood flow will be presented and recent developments in the numerical algorithm design will be mentioned. Applications involving certain cardiovascular interventions will be shown. Colalborators: Dr. Z. Krajcer and Dr. D. Rosenstrauch (Texas Heart Institute), Dr. C. Hartley (Baylor College of Medicine), Prof. R. Glowinski, Prof. T.W. Pan, Prof. G. Guidoboni (University of Houston), Prof. A. Mikelic (University of Lyon 1, FR), Prof. J. Tambaca (University of Zagreb, CRO)
talk022008: Studying cosmic baryon fluid with cosmological hydrodynamical simulation Li-Zhi Fang University of Arizona The cosmic gravitational field is dominated by dark matter and dark energy, and therefore, the evolution of cosmic baryon fluid is dynamically governed by the underlying dark matter field. In linear regime, the velocity and density fields of baryon matter follow the dark matter point-by-point. However, once the nonlinear and stochastic nature should be considered, the dynamical behavior of the baryon fluid is significant different from those of the dark matter. To understand this evolution, high quality simulation of cosmological hydrodynamics is critical. The WENO/N-body code is found to be effective to reveal important features of the cosmic baryon fluid, including the statistical decoupling of baryon fluid from dark matter field; the intermittency, and the turbulence-like scaling of velocity and density fields, etc. These results have successfully been applied to explain observational data.
talk031908: Hydrodynamics and Radiation Hydrodynamics with Astrophysical Applications Paul Drake University of Michigan, Ann Arbor, MI 48105 We emphasize experiments that are both at the forefront of High-Energy-Density Physics (HEDP) and relevant to issues in astrophysics. Our primary nonlinear hydrodynamics experiments, using the Omega laser, have been exploring blast-wave-driven instabilities of relevance to supernova explosions. In this area we are preparing for experiments having a modal structure based on the spectrum of modes present in 3D calculations of presupernova stellar structure. In experiments preparing for this we observed penetration of some denser material into less dense material far beyond the distances anticipated from simulations, a result we believe to be of profound significance. We are also developing an experiment design for the National Ignition Facility aimed at the behavior of diverging, 3D, multi-interface explosions. Our radiation hydrodynamic experiments are focused on radiative shocks, in the specific case for which radiative losses in the upstream direction lead to large increases in the post-shock density. We have developed a prototype radiative-shock system over several years. One current project is aimed at applying new diagnostic methods, and has worked with UV Thomson scattering, streaked optical pyrometry, and X-ray Thomson scattering. The other project is aimed at understanding the lateral structure we see, through a combination of theory, radiation-hydrodynamic simulations, and experiments using multi-directional radiography. Our goal is to thoroughly understand the behavior of such systems, and to produce data suitable for code benchmarking. We assemble our own targets for all these experiments, at times using components or subassemblies from General Atomics, LLNL, or industrial suppliers. The experiments benefit very strongly from a wide range of collaborations, to be directly cited in the talk. We collaborate with scientists from LLNL, LLE, and NRL on all aspects of our work in various specific contexts, notably including experiment design and target fabrication. We collaborate on theory and simulation with researchers from Florida State U., Chicago, Arizona, Texas, and Stony Brook. This research was sponsored by NNSA Stewardship Sciences Academic Alliances through DOE Research Grant DE-FG52-04NA00064 and by other grants and contracts.
talk040208: Adaptive hierarchical sparse grid collocation methods for the solution of stochastic differential equations Professor Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering Cornell University, Ithaca, NY 14853-3801 http://mpdc.mae.cornell.edu/ Several critical phenomena like contaminant spread, nuclear waste disposal and oil recovery rely on accurate analysis and prediction of multiscale phenomena. Such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the underlying system. In this talk, we will introduce a methodology that accounts for the stochastic and multiscale nature exhibited by such systems. In particular, we will discuss: (1) A data driven strategy to incorporate limited experimental data into the stochastic analysis, (2) Effective computational strategies to solve the resulting stochastic partial differential equations (SPDEs) using adaptive hierarchical multi-linear sparse grid collocation techniques and (3) A stochastic variational multiscale formulation to incorporate uncertain multiscale features. A number of examples will be presented to demonstrate the various techniques discussed. These include problems related to long-term integration and stochastic discontinuity, and flow in random heterogeneous media.
talk051408: Wave-wave interactions of a gasdynamic type Liviu Dinu Institute of Mathematics of the Romanian Academy Some Burnat type "algebraic" genuinely nonlinear approaches [centered on a duality connection between the hodograph character and the physical character] and Martin type "differential" approaches [centered on a Monge-Amp`ere type representation] will be overviewed to begin with. A parallel will be then considered between these two types of approaches [making evidence of some nontrivial contrasts] regarding their contribution to describing some nondegenerate gasdynamic regular interaction solutions. The two mentioned constructions show some distinct, complementary valences. The genuinely nonlinear Burnat type approaches appear to be essential for some isentropic multidimensional extensions (simple waves solutions, regular interactions of simple waves solutions) with a classifying potential. The Martin type approaches appear, in their turn, to be essential for an anisentropic extension in two independent variables [unsteady one-dimensional, steady two-dimensional]. The two types of interactions constructed parallel, from a local and regular prospect, some details [interactions of simple waves solutions] of the Zhang and Zheng global and irregular construction. This is a joint work with Marina Ileana Dinu.