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(January 27) Prof. Ricardo Duran:
Numerical approximation of parabolic problems with blowing up solutions -
For many differential equations involving non linear reaction terms
the solutions can become unbounded in finite time. This phenomenon
is known as blow up and has been extensively studied from a theoretical
point of view. Our objective is to analyze numerical aproximations of these problems.
First we consider standard semi-discretizations in space and analyze
whether the system of ordinary differential equations obtained
reproduces the behavior of the original problem. We show that this is
not always true: for example, there are cases in which the solutions
of the original problem blow up while the corresponding semi-discrete
approximations do not. For some model problems we show that, under appropriate conditions,
the system of ode's blows up at some time which approximates the
blow up time of the original problem when the mesh size goes to zero. We also analyze the blow up rate of the semi-discretization and compare
it with that of the original equation for different model problems. Finally we consider the discretization in time. We introduce an adaptive
time step procedure and prove that it allows us to approximate
the blow up time. We also analyze the localization of blow up for
the discrete problem and its relation with that of the continuous one. (February 3) Prof. Peter Wolfe:
Stability of stationary and translating cables -
We study the stability of equilibrium configurations of cables which are
stationary as well as translating (as in a ski-lift). Much of the theoretical
basis for the work is due to S. Antman but there is also a large body of work on
this in the engineering literature. One of the leaders in this field is C.D.
Mote, who has, among other things done some experimental work in the field. Our
work is not yet complete but we intend to compare our numerical results with the
results of President Mote's experiments. (February 10) Prof. Dianne O'Leary:
Computing and Displaying Confidence Intervals for Images -
We consider the problem of deblurring an image in the case when a
small part of the image is of special interest. For example, the
subregion may correspond to a particular star cluster in an astronomical
image, or a potential tumor in a diagnostic image. Our contribution is two
fold. First, we show how sophisticated but expensive techniques involving
linear algebra and optimization can be applied to subimages, even when
the cost is too great to be used on the complete image. Second, we
demonstrate the usefulness of confidence intervals for the pixel values.
We show how to compute such intervals and how to display them in a novel
way that gives the scientist valuable information. (February 17) Prof. Christoph Schwab:
hp-Discontinuous Galerkin time stepping for parabolic problems -
The Discontinuous Galerkin Finite Element Method (DGFEM) for the
time stepping in parabolic problems is analyzed in the context of
the hp-version of the Galerkin method. Error bounds which are
explicit in the time step as well as in the approximation order are
derived and it is shown that the hp-DGFEM gives spectral convergence
in problems with smooth time dependence. In conjunction with geometric time partitions it is proved that the
hp-DGFEM results in exponential rates of convergence (in terms of
the number of spatial problems to be solved) even in the presence
of analytic, but possibly incompatible initial data or piecewise
analytic forcing terms. The analysis also shows that for fixed order
r>= 0, algebraically graded time partitions are determined that
give the optimal algebraic convergence rates, again for incompatible
initial data. A fully discrete scheme is discussed. Here, the use of
certain mesh-design principles for the spatial discretizations
yields exponential rates of convergence in time and space. Numerical
examples confirm the theoretical results. (February 24) Dr. Hans Johnston:
A Novel Finite Difference Method for the Incompressible
Navier-Stokes Equations in the Velocity-Pressure Formulation -
We present a second order numerical method for the
time dependent incompressible Navier-Stokes in the
velocity-pressure formulation implemented on non-staggered grids.
The key to the method lies in a consistent and accurate approximation
of the Neumann boundary condition for the pressure Poisson equation.
With this numerical boundary condition we are able to treat the
pressure as a dynamic variable in the time evolution of the flow,
thus recovering the true physical pressure at each time step. When
coupled with a high order explicit time stepping scheme, the overall
method is simple to implement and highly efficient. Numerical
examples will be presented for both constant and variable density
flows to demonstate the scheme's clean accuracy and convergence
properties. Extension of the overall scheme to 3D flows will
also be discussed.
(March 2) Prof. Irad Yavneh:
Multigrid Methods for Incompressible Flows -
Multigrid computational methods were introduced in the 1970's as
efficient iterative solvers for elliptic boundary-value problems.
Since then, the range of problems treated successfully by multigrid
(more generally, multi-level) techniques has been expanded appreciably.
In particular, these methods have been applied to discretized equations
of incompressible flow by many researchers and practitioners since
the late 1970's. While excellent performance was observed for
flows at low Reynolds numbers, convergence rates were found to
deteriorate severely for high Reynolds numbers (as well as other
related singular-perturbation problems.)
The main difficulties associated with multigrid solution
of incompressible flows at high Reynolds numbers were recognized
nearly two decades ago. Methods for dealing with these problems
were proposed by several researchers, mainly in recent years.
Some of these approaches will be described, along with an assessment
of their efficiency and robustness, both theoretical and practical. (March 9) Prof. Z.C. Shi:
Cascadic Multigrid Methods -
Recently a new type of multigrid method, the so-called Cascadic
Multigrid has been developed. It performs only iterations on each grid
level without any coarse grid correction that may be viewed as a
"one-way" multigrid. In this talk we will discuss the convergence
property and the computational complexity of the method for elliptic
and parabolic equations using different conforming and nonconforming
finite approximations. The effect of smoothers is also mentioned. (March 16) Prof. Zdenek Strakos:
Convergence of GMRES -
Consider a linear algebraic system
Ax = b where A is a given n by n nonsingular matrix and b
an n-dimensional vector.
Iterative methods aim to contruct
approximate solutions x0, x1,..., xk converging
to the solution x; convergence may not be
monotonic. Many modern methods (such as those based on Krylov subspaces)
guarrantee (in exact arithmetic) xk = x for some k <= n.
This finite termination property has, however, a little practical effect.
In real world applications n may be of order of millions,
but we need to achieve convergence to an acceptable accuracy in
much fewer steps. Therefore it is important to analyze convergence
of iterative methods in dependence on some particular characteristics
of the problem. We also need to understand effects of finite precision
arithmetic (rounding errors). We will focus on convergence of the generalized minimal residual
method (GMRES). We will briefly recall and discuss some published work
and present several recent results.
(March 16) Prof. Vidar Thomée:
A Parallel Method for Time-Discretization of Parabolic Problems Based on Laplace Transformation and Quadrature -
The solution of an initial boundary value
problem for a parabolic equation
ut + Au = f with
a suitable forcing term f is represented as an integral
in the complex left halfplane with an integrand containing
the resolvent of the elliptic operator A.
The integral is approximated by a quadrautre rule which
reduces the calculation to the solution of a number of
independent elliptic problems which may be solved
in parallel. Using the finite element method for the
approximate solution of the elliptic problems yields a fully
discrete method for the original problem.
(March 30) Prof. Oliver Ernst:
Acceleration Strategies for Restarted Minimal Residual Methods -
Restarted minimum residual (MR) methods are a popular solution approach for
solving non-Hermitian linear systems, but one that is hampered by the
fundamental difficulty that convergence of the restarted method may be
considerably slower than compared with the full (unrestarted) method.
We provide an overview of existing strategies which compensate for this
deterioration in convergence due to restarting for the class of MR Krylov
subspace methods. The key theoretical device for comparing different
strategies is their abstract formulation as repeated orthogonal projections
with respect to general correction spaces. We further evaluate the popular
practice of using nearly invariant subspaces to either augment Krylov
subspaces or to construct preconditioners which invert on these subspaces.
In the case where these spaces are exactly invariant, the augmentation
approach is shown to be superior. Moreover, we show how a strategy recently
introduced by de Sturler for truncating the approximation space of an MR
method can be interpreted as a controlled loosening of the condition for
global MR approximation based on the canonical angles between subspaces.
This is joint work with Michael Eiermann and Olaf Schneider (TU Freiberg) (April 6) Dr. Alfred S. Carasso:
Direct Blind Deconvolution and Levy Probability Densities -
Blind deconvolution seeks to deblur an image without knowing the
cause of the blur. Iterative methods are commonly applied to that
problem, but the iterative process is slow, uncertain, and often
ill-behaved. This talk considers a significant but limited class
of blurs that can be expressed as convolutions of 2-D symmetric
Levy `stable' probability density functions. This class includes
and generalizes Gaussian and Lorentzian distributions. For such
blurs, methods are developed that can detect the point spread
function from 1-D Fourier analysis of the blurred image. A separate
image deblurring technique uses this detected point spread function
to deblur the image. Each of these two steps uses direct non-iterative
methods, and requires interactive adjustment of parameters. As a
result, blind deblurring of 512X512 images can be accomplished in
minutes of CPU time on current desktop workstations. Numerous blind
experiments on synthetic data show that for a given blurred image,
several distinct point spread functions may be detected that lead to
useful, yet visually distinct reconstructions. Application to real
blurred images will also be demonstrated. (April 10) Prof. Monique Dauge:
A new method for using nodal finite elements in Maxwell computations for domains
with non-convex corners -
It is now well known that using nodal finite elements in a conformal variational formulation involving the
regularized bilinear form rot-rot + div-div with essential boundary conditions leads to definitely wrong
results in the presence of non-convex corners. We present a new way of regularizing the rot-rot operator
by a temperate divergence term, with the help of a weight. We will provide theoretical arguments and
numerical results, based on a joint work with Martin Costabel and on a code developped by Daniel Martin. (April 13) Dr. Bruce Hendrickson:
Devising Effective Parallel Algorithms -
The field of parallel computing seems to be in a continual state
of flux with rapid evolution in architectures, programming models
and languages. The one constant is the overriding importance of
good algorithms. Unfortunately, devising effective parallel
algorithms for scientific applications remains more of an art
than a science. In this talk I will describe new parallel
algorithms for two important classes of scientific computations:
simulating the motions of molecules, and modeling car crashes.
These two, very different, examples will be used as case studies
to address broader questions concerning tools, techniques and
philosophy underlying the development of effective parallel
algorithms and applications. (April 18) Dr. Francesca Fierro:
An adaptive approach to the minimization of functionals with
regularized total variation -
We address the minimization of a functional F
defined on the space BV(Omega;{-1,1}) of the characteristic
functions of sets with finite perimeter in a given bounded domain
Omega in R2.
The functional includes the total variation \int_\Omega |\nabla \chi|.
Such minimization problems arise in the time discretization
of the mean curvature flow as suggested in [1,3].
In order to solve this minimization problem we follow the approach
proposed in [2]. This approach is based on the minimization in
BV(Omega ; [-1,1]) of convex, regularized approximations
Fepsilon to the original functional F.
We present a posteriori error estimates for regular minimum
points of Fepsilon, which turn out to be an essential
tool for the implementation of an adaptive minimization algorithm.
These estimates are robust in the sense that their constants
do not depend on the regularization parameter epsilon.
Finally, we present some numerical simulations. [1] F.Almgren J.E.Taylor and L.Wang ``Curvature driven flows:
a variational approach"
SIAM J. Control Opt., 31 (1993), pp 387-437 [2] G. Bellettini M.Paolini C.Verdi `` Convex approximation of
functionals with curvature"
Atti Accad. Naz.Lincei Cl.Sci.Fis.Mat. Natur. Rend. Lincei (9)
Mat.Appl., 1 (1991), pp 297-306 [3] S.Luckhaus T. Sturzenheker ``Implicit time discretization for
the mean curvature flow equation''
(1994), Preprint n.334 SFB 256 Bonn
(April 27) Dr. Andreas Veeser:
Efficient and Reliable A Posteriori Error Estimatos
for Elliptic Obstacle Problems -
A posteriori error estimators are derived for linear finite element
approximations to elliptic obstacle problems. These estimators yield
global upper and local lower bounds everywhere for the discretization
error. Here, discretization error means the sum of two contributions:
the distance between continuous and discrete solution in the energy-norm
and some quantity that is related to the distance of continuous and
discrete contact set in some weak sense. Another important property of
these estimators is that, in the interior of the discrete coincidence
set, they reduce to expressions that measure only data resolution. (May 1) Prof. Carsten Carstensen:
Finite Element Methods for Nonconvex Minimisation Problems
and an Application in Material Science -
The rapidly developing field of the mathematical modeling of microstructure has important applications in
material science (advanced materials), micromagnetism, homogenization and optimization. Typically, the
continuous problem (P) lacks classical solutions. There exist minimizing sequences in (P) that have a
weak limit, but non-(quasi-)convexity implies, in general, that the weak limit u is {\em not} a solution of the
problem (P). In experiments, we observe oscillations of strains which form a macroscopic or averaged
quantity u. The efficient numerical simulation on the macroscopic level aims to compute the weak limit u as
a solution of a related {\em Relaxed Problem} (RP) while the microscopic mechanism can be detected by a
direct finite element minimization of (P) or, more sophisticated, by a generalised formulation (GP).
The presentation will focus on adaptive strategies for relaxed problems such as the double-well problem
in one-dimension (Young's example), in higher dimensions, or in linearised elasticity and on related topics
in micromagnetism and homogenization problems. (May 4) Dr. John Mackenzie:
On the Solution of Phase Change Problems using Adaptive Moving Meshes -
Adaptive moving mesh methods are developed for
the numerical solution of an enthalpy formulation of
one and two-dimensional heat conduction problems with a phase change.
The grids are obtained from a global mapping of the physical to the
computational domain which is designed to cluster mesh points
around the interface between the two phases of the material.
The enthalpy equation is discretised using a semi-implicit
Galerkin finite element method using linear basis functions.
The moving finite element method is applied to problems
where the phase front is cusp shaped and where the interface
changes topology.
(May 8) Prof. Walter Gander:
Adaptive Quadrature - Art or Science? -
First, the basic principles of adaptive quadrature are
reviewed. Adaptive quadrature programs being recursive by nature,
the choice of a good termination criterion is given particular
attention. Two Matlab quadrature programs are presented. The first
is an implementation of the well-known adaptive recursive Simpson's
rule; the second is new and is based on a four-point Gauss-Lobatto
formula and two successive Kronrod extensions. Comparative test
results are described and attention is drawn to serious
deficiencies in the adaptive routines quad and quad8
provided by Matlab. (May 18) Prof. Charalampos Makridakis:
A class of finite element methods for hyperbolic conservation laws -
A class of finite element methods for hyperbolic
conservation laws is considered.
Their regularization mechanism on shocks
is a combination of appropriate mesh refinement
and regularization by wave operators.
An adaptive algorithm motivated by a posteriori estimates
on model cases is proposed.
These finite element methods are stable
(compactness in H-1) and high-order accurate
(high-order convergence rates before shocks).
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