Invited Speakers
The current list of invited speakers is:
Abstracts
|
Louisiana State University |
In this talk we will discuss several new
nonconforming finite element methods for electromagnetic
problems formulated in subspaces of the intersection of
H(curl) and H(div), where the problems become elliptic.
Both theoretical and numerical results will be presented.
Free-surface flows in fluid mechanics are among the most difficult problems to solve mathematically, and they have engaged researchers for at least the last 150 years. Their difficulty comes from the fact that they are inherently non-linear, since the shape of the region occupied by the fluid is itself part of the problem to be solved.
In past two decades, some steady-state (time independent) free-surface problems in fluid mechanics have been able to be solved numerically. Typically, the fluid is assumed to be ideal (not viscous), so that Laplace's equation for a velocity potential holds throughout the fluid. The problem is nevertheless still highly non-linear, since the free-surface location remains unknown initially. In recent years, however, it has become apparent that steady-state solutions leave many questions unanswered. In particular, it is sometimes not clear if those solutions would be stable to small disturbances, or if they could even be reached at all in a time-dependent problem that starts from sensible initial conditions.
In this talk, we will present a new spectral method for solving some free-surface flows in ideal fluids. The method makes use of a basic Fourier-series type representation of the solution with time-dependent coefficients. It is then possible to derive elegant identities involving the Fourier coefficients along the free surface, and these lead to a highly efficient set of ordinary differential equations for the coefficients, that can be solved to high accuracy. We will illustrate the method by reference to some practical problems in the draining of a tank of water, and resonant sloshing in a tank forced horizontally. We will also consider the famous Rayleigh-Taylor instability, in which a heavy fluid lies above a light fluid. Spectral methods have been developed for this problem in both the ideal and the viscous fluid cases, and reveal some interesting behaviour as the instability develops and grows.
Sustained floating-point rates on real applications, as tracked by the Gordon Bell Prize, have increased by over five orders of magnitude from 1988, when 1 Gigaflop/s was reported on a structural simulation, to the 200 Teraflop/s recently reported on a molecular dynamics simulation. Various versions of Moore's Law over the same interval provide only two to three orders of magnitude of improvement for an individual processor; the remaining factor comes from concurrency, which is of order 100,000 for the BlueGene/L computer, the platform of choice for the majority of recent Bell Prize finalists. As the semiconductor industry begins to slip relative to its own roadmap for silicon-based logic and memory, concurrency has played an increasing role in attaining the next order of magnitude, the long-awaited milepost of 1 Petaflop/s (one million Gigaflop/s), which occurred in June 2008 for the ScaLAPACK benchmark on Roadrunner at Los Alamos National Laboratory, and should be demonstrated on a practical application during 2009.
Simulations based partial differential equations (PDEs), such as fluid dynamics or electromagnetics, can be among the first applications to take advantage of petascale capabilities, but not the way most are presently being pursued. Only "weak scaling" can get around the fundamental limitation expressed in Amdahl's Law (1967) and only optimal implicit formulations can get around another limitation on scaling that is an immediate consequence of Courant-Friedrichs-Lewy stability theory (1928) under weak scaling of a PDE. Many PDE-based applications and other lattice-based applications with petascale roadmaps,such as quantum chromodynamics, will likely be forced to adopt optimal implicit solvers. However, even this narrow path to petascale simulation is made treacherous by the imperative of dynamic adaptivity, which drives us to consider algorithms and queueing policies that are less synchronous than those in common use today. Drawing on the SCaLeS report (2003-04), the latest ITRS roadmap, some back-of-the-envelope estimates, and numerical experiences with PDE-based codes on recently available platforms, we will attempt to project the pathway to Petaflop/s for representative applications.
Circadian clocks govern daily behaviors of organisms in all kingdoms of life. In mammals, the master clock resides in the suprachiasmatic nucleus (SCN) of the hypothalamus. It is composed of thousands of neurons, each of which contains a sloppy oscillator - a molecular clock governed by a transcriptional feedback network. Via intercellular signaling, the cell population synchronizes spontaneously, forming a coherent oscillation. This multi-oscillator is then entrained to its environment by the daily light/dark cycle.
Both at the cellular and tissular levels, the most important feature of the clock is its ability not simply to keep time, but to adjust its time, or phase, to signals. We present the parametric impulse phase response curve (pIPRC), an analytical analog to the phase response curve (PRC) used experimentally. We use the pIPRC to understand both the consequences of intercellular signaling and the light entrainment process. Further, we determine which model components determine the phase response behavior of a single oscillator by using a novel model reduction technique. We reduce the number of model components while preserving the pIPRC and then incorporate the resultant model into a couple SCN tissue model. Emergent properties, including the ability of the population to synchronize spontaneously are preserved in the reduction. Finally, we present some mathematical tools for the study of synchronization in a network of coupled, noisy oscillators.
| University of Erlangen-Nuremberg
|
The Lattice Boltzmann Method (LBM) is based on a
discretization of the Boltzmann equation and results in a cellular
automaton that is relatively simple, easy to extend, and well
suited for parallelization.
In the past few years, the LBM method has been established as an
alternative method in computational fluid dynamics. Here, we will
discuss extensions of the LBM to compute flows in complex geometries
such as blood vessels or porous media, fluid structure interaction
problems with moving obstacles, and free surface flows. We will
outline the implementation in the waLBerla software framework and
speedup results for current parallel hardware. This includes
Heterogeneous multicore CPUs, such as the IBM Cell processor,
and the implementation on massively parallel systems with
Thousands of processor cores.
|
University of New South Wales |
Richard Bellmann coined the phrase "the curse of
dimensionality" to describe the extraordinarily rapid increase in
the difficulty of most problems as the number of variables increases. A typical problem is numerical multiple integration. It is clear that the cost of every integration formula of product type
rises exponentially with the number of dimensions. Nevertheless,
problems with hundreds or even thousands of variables do arise, and are now being tackled successfully. In this talk I will touch briefly on recent advances in understanding and constructing high dimensional integration rules, but much of the focus will be on applications, in diverse fields such as mathematical finance, linear models in statistics, and flow through porous media. A general theme is that high-dimensional problems present an enduring challenge for numerical analysis.
We present h-and p-adaptive procedures for the symmetric coupling of finite element and
boundary element methods for linear and nonlinear interface problems [1,2]. (Here, h
denotes the mesh size and p the polynomial degree of the Galerkin solution). Our coupling
approach reduces the interface problems to differential equations in a bounded domain Ω
together with (weekly singular, singular, and hyper singular) boundary integral operators
on the boundary of Ω. We give two types of a posteriori error estimates a) using
hierarchical basis techniques with easily computable local error indicators and b) residual
based estimates. The corresponding error controlled adaptive procedures give good meshes
and polynomial degree distributions for the h-p version of the coupling, allowing
independent refinements of finite elements and boundary elements, leading to efficient
numerical procedures. Furthermore, we propose efficient preconditioners for the minimum
residual method to solve the indefinite, symmetric systems of equations resulting from the
FE/BE coupling yielding (almost) bounded numbers of iterations. Here we use Multigrid, BPX,
and Schwarz methods both for finite and boundary elements [3]. We demonstrate the power and
the applicability of the FE/BE coupling with the 3D eddy current problem in
electromagnetics [4] and the contact problem with friction for two elastic bodies [5,6]. In
all cases numerical results are presented which support the theory.
[1] E. P. Stephan: Coupling of Boundary Element Methods and Finite Element Methods,
Encyclopedia of Computational Mechanics, Edited by Erwin Stein, René de Borst and Thomas
J.R. Hughes. Vol. 1, Chapter 13: Fundamentals. 2004 John Wiley \& Sons.
[2] P. Mund, E.P. Stephan: An adaptive two-level method for the coupling of nonlinear
FEM-BEM equations, SIAM J. Numer. Anal. 36 (1999) 1001-1021
[3] N. Heuer, M. Maischak, E.P. Stephan: Preconditioned Minimum Residual Iteration for the
h-p Version of the Coupled FEM-BEM with Quasi-uniform Meshes, Numer. Linear Algebra. Appl.
6 (1999) 435-456
[4] E.P. Stephan, M. Maischak, F. Leydecker: An hp-adaptive finite element/boundary element
coupling method for electromagnetic problems, Comput. Mech. 39 (2007)673-680
[5] M. Maischak, E. P. Stephan: A FEM-BEM coupling method for a nonlinear transmission
problem modelling Coulomb Friction Contact, Comp. Meth. Appl. Mech. Eng. 194 (2005), no.
2-5, 453-466
