(September 3) Jean-Rodolphe Roche:
Numerical Methods in Shape Optimization and Applications -
Numerical methods of Newton-like type for shape optimization and related
problems are presented. We consider applications where the cost function is
a total energy and the state problem is a partial differential equation. We
formulate a precise description of the first and second order shape
derivatives and we give the technics to compute them. The numerical examples
concerns the electromagnetic shaping in 2-d, 3-d and shape identification. (September 17) Valeria Simoncini:
The Nullspace-Free Eigenvalue problem and
the Inexact Shift-and-Invert Lanczos Method -
Given the large generalized symmetric eigenvalue problem A x =M x,
with A semidefinite and M definite, we are interested in the
approximation of the smallest nonzero eigenpairs, assuming that a sparse
basis for the null space of A is available.
A relevant feature is that the dimension of the
null space is large, possibly up to one third of the problem dimension.
We analyze several algebraic formulations explored in recent years.
We consider the inexact version of the Shift-and-invert Lanczos method,
and
we show that apparently different formulations
provide the same approximation iterates, under some standard hypotheses. (September 24) Yannis G. Kevrekidis:
Equation-free multi scale computation: Enabling Microscopic Simulators to Perform System Level
Tasks -
I will present and discuss a framework for computer-aided multiscale analysis, which enables models at a
"fine" (microscopic/stochastic) level of description to perform modeling tasks at a "coarse" (macroscopic, systems)
level.
These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished
through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains:
"patches" in macroscopic space-time.
Traditional modeling approaches first involve the derivation of macroscopic evolution equations (balances closed
through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such
evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem.
Our equation-free (EF) approach, introduced in PNAS (2000) when successful, can bypass the derivation of the
macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We
discuss how the mathematics-assisted development of a computational superstructure may enable alternative
descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics
(MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks
(integration over relatively large time and space scales, "coarse" bifurcation analysis, but also optimization and control
tasks) directly.
In effect, the procedure constitutes a systems identification based, "closure on demand" computational toolkit,
bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis.
We illustrate these "numerical enabling technology" ideas through examples from chemical kinetics (LB, KMC),
rheology (Brownian Dynamics), homogenization and the computation of "coarsely self-similar" solutions, and discuss
various features, limitations and potential extensions of the approach.
Overview article (October 8) Monique Dauge:
Approximation of Maxwell singularities by nodal finite elements -
Near reentrant corners of a perfectly conducting boundary, electromagnetic
fields have strong singularities that are not in H1. The standard
regularized variational formulation of the time-harmonic Maxwell equations,
when discretized using nodal (C0) finite elements, leads to
non-convergent Galerkin methods. The weighted regularization method [1,2] is a
simple modification of the variational formulation that leads to convergent
nodal finite element methods.
In its hp version, the WRM is particularly efficient. For 2D problems,
exponential convergence can be shown. The method works well for 3D problems,
too.
In the talk, some points from the proof of exponential convergence in 2D
will be presented. The convergence behavior will be illustrated by the the
results of computations in 2D and in 3D. The results in 3D indicate that
good approximation requires an extremely strong geometric mesh refinement.
References:
[1] M. Costabel, M. Dauge:
Weighted regularization of Maxwell equations in polyhedral domains.
Numer. Math Online publication DOI 10.1007/s002110100388 (2002)
[2] M. Costabel, M. Dauge, D. Martin, G. Vial:
Weighted regularization of Maxwell equations - computations in
curvilinear polygons.
Proceedings of ENUMATH01, Springer 2002. (October 15) Fabio Nobile:
Some issues in the mathematical modelling and numerical simulation of
the cardiovascular system -
The simulation of the human cardiovascular system presents many
challenging aspects in both modeling and set up of numerical tools. In
this talk we will address two of them. The first one concerns the
simulation of blood flow in a large artery when the deformation of the
vessel wall is taken into account. We will present recent results on
the so-called "Arbitrary Lagrangian Eulerian" (ALE) formulation,
suitable to simulate fluid flow problems in domains with moving
boundaries, and we will discuss the stability of some coupled
fluid/structure algorithms.
The second issue we will deal with concerns the possibility to achieve a
global description of the circulatory system by combining models of
different complexity and space dimension: what we called the "multiscale
approach". This idea is motivated by the fact that local phenomena, as
the presence of an atherosclerotic plaque or an implanted prosthesis,
may have effects on the whole circulation, which should be predicted by
the numerical simulation. We will give an overview on the different
models available in the literature to describe blood circulation and we
will present strategies to couple them in order to obtain a global model
that accounts at the same time for local and systemic features.
(October 17) Gene H. Golub:
Solution of Non-Symmetric, Real Positive Linear Systems -
The methods we discuss use a Hermitian/skew-Hermitian splitting (HSS)
iteration and its inexact variant, the inexact
Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs
inner iteration processes at each step of the outer HSS
iteration. Theoretical analyses show that the HSS method converges
unconditionally to the unique solution of the system of linear
equations. Moreover, we derive an upper bound of the contraction
factor of the HSS iteration which is dependent solely on the spectrum
of the Hermitian part. Numerical examples are presented to illustrate
the effectiveness of both HSS and IHSS iterations. In addition,
several important generalizations are presented.
(October 29) C. D. Levermore:
The convergence of numerical transfer schemes in diffusive regimes -
In regimes with small mean-free-paths the transfer equation with
anisotropic boundary conditions has an asymptotic behavior that is
governed by a diffusion equation and boundary conditions obtained
through a matched asymptotic boundary layer analysis. A numerical
scheme for solving this problem has a contribution to the truncation
error that generally gives rise to a nonuniform consistency with the
transfer equation as the mean-free-path vanishes, thus degrading its
performance in diffusive regimes. We show that whenever certain
numerical schemes have the correct diffusion limit, both in the
interior and at the boundaries, their solutions converge to the
solution of the transport equation uniformly in mean-free-path.
The proof of this convergence is based on an asymptotic diffusion
expansion and requires error estimates on a matched boundary layer
approximation to the solution of the descrete equation. (joint work with Shi Jin and Francois Golse) (November 5) F. Seydou:
An Inverse Problem for Anisotropic Objects -
We discuss the problem of finding an
anisotropic profile for the scattering of
electromagnetic waves with fixed frequencies. This type
of problem is one of the most important and interesting
problems arising in mathematical physics and has
applications in geophysics, technology, medicine and
nondestructive testing. The difficulty in solving such
problems is due to two unpleasant facts: They are
nonlinear and, more seriously, they are ill-posed.
In the previous works most algorithms were for TM and
TE wave illuminations whereas much less was reported
for the more complicated anisotropic case. We shall
consider a simpler case of anisotropic medium. In
Particular we shall assume the contrast to be orthotropic, but
the problem is still much more complicated compared to
the scalar case. We consider an optimization method to
deal with the numerical approximation and discuss an
iterative (the simplified Newton) method to try and solve
the problem.
(November 12) Guowei Wei:
A local spectral method for scientific computing -
There has been an ongoing interest in computational
methodology by numerous researchers in every field of
science and engineering. Most effort has been centered in
developing either global methods or local methods for
solving a variety of problems. Global methods, such as
Fourier or Chebyshev spectral methods, are usually highly
accurate. But local methods, such as finite element and
finite difference methods, are much more flexible for
handling complex boundaries and geometries. We introduce
a wavelet-like approach for achieving global methods'
accuracy and local methods' flexibility in solving problems
with complex boundaries and geometries. The mathematical
foundation of the proposed method is the theory of
distributions. Example applications are discussed to fluid
dynamics, electromagnetics, solid mechanics and nonlinear
waves.
(November 19) Zhiming Chen:
A Convergent Adaptive Finite Element Algorithm with Reliable And
Efficient Error Control for Linear Parabolic Problems -
An efficient and reliable a posteriori error estimator is derived for linear
parabolic equations which does not depend on any regularity assumption on the
underlying elliptic operator. A convergent adaptive algorithm with variable
time-step sizes and space meshes is proposed and studied which, at each time step,,
delays the mesh coarsening until the final iteration of the adaptive procedure,
allowing only mesh and time-step size refinements before. The key ingredient in
the convergence analysis is a new coarsening strategy. Numerical results are
presented. This is a joint work with Feng Jia.
(November 26) Ming-Jun Lai:
Multivariate Spline Method for Numerical Solution of PDE -
I will explain how to use multivariate splines, piecewise polynomials
of arbitrary degrees and smoothness over triangulation to numerically
solve partial differential equations. The method does not require
finite elements or locally supported basis functions and is very easy
to implement. I will show how to use the stream function formulation
for 2D Navier-Stokes equations and how to use the divergence free
spline spaces for 3D Navier-Stokes equations. Some numerical
experiments on standard cavity driven flows, backward facing step
flows, flows around obstacles will be demonstrated.
(December 5) John Ball:
Varying volume fractions -
In elasticity models of materials which can undergo phase transformations
involving a change of shape, different phases or variants of the material
are associated with energy wells comprising corresponding sets of
deformation gradients. Zero-energy microstructures can then be identified
with gradient Young measures supported on these sets and the corresponding
phase fractions are the masses of these measures restricted to each well. The talk will describe a general result for gradient Young measures that
allows one to vary the volume fractions of such microstructures, together
with applications, open questions and related results.
(December 10) Susan Minkoff:
Coupled Flow and Mechanics for Time-Lapse Seismic Modeling -
Accurate prediction of oil and gas production in weak-formation
reservoirs requires coupled flow and mechanical deformation modeling.
We describe a loosely-coupled, staggered-in-time simulator which
combines multiphase flow and advanced mechanics to model changes
in reservoir properties (permeability and porosity) in compactible
reservoirs during production. Numerical simulations indicate that these
changes in flow parameters have a visible impact on time-dependent seismic
models of the field despite large differences in scale between flow parameters
and typical seismic wavelengths. (Joint work with Mike Stone, Sandia National Labs,
Steve Bryant, Malgo Peszynska, Mary Wheeler, University of Texas at Austin)
(January 21) Ian Sloan:
Good approximations on the sphere, with applications to geodesy and
the scattering of sound -
The theme of this talk is that polynomial approximations on the sphere are important for
applications, but that successful applications of high-degree polynomials need a good understanding of
underlying approximation properties. We illustrate with two case studies. First, for applications in geodesy, there is good reason to use cubature rules that have a high degree of
polynomial accuracy. The stability, and even the computability, of such rules depends critically on the
properties of the underlying polynomial interpolants. Second, a recent spectral approach to the scattering of
sound by three dimensional objects needs for its analysis good approximation properties of the
hyperinterpolation polynomial approximation scheme. In the course of this talk the existing state of knowledge
for both interpolation and hyperinterpolation will be reviewed.
|