Comlab

I will address the usage of the elliptic reconstruction technique (ERT) in a posteriori error analysis for fully discrete schemes for parabolic partial differential equations. A posteriori error estimates are effective tools in error control and adaptivity and a mathematical rigorous derivation justifies and improves their use in practical implementations.

The flexibility of the ERT allows a virtually indiscriminate use of various parabolic PDE techniques such as energy methods, duality methods and heat-kernel estimates, as opposed to direct approaches which leave less maneuver room. Thanks to ERT parabolic stability techniques can be combined with different elliptic a posteriori error analysis techniques, such as residual or recovery estimators, to derive a posteriori error bounds. The method has the merit of unifying previously known approaches, as well as providing new ones and providing us with novel error bounds (e.g., pointwise norm error bounds for the heat equation). [These results are based on joint work with Ch. Makridakis and A. Demlow.]

Another feature, which I would like to highlight, of the ERT is its simplifying power. It allows us to derive estimates where the analysis would be very complicated otherwise. As an example, I will illustrate its use in the context of non-conforming methods, with a special eye on discontinuous Galerkin methods. [These are recent results obtained jointly with E. Georgoulis.]

We consider the computational complexity of optimizing various classes

of continuous functions over a simplex, hypercube or sphere. These

relatively simple optimization problems arise naturally from diverse

applications. We review known approximation results as well as negative

(inapproximability) results from the recent literature.

Whether the 3D incompressible Euler or Navier-Stokes equations

can develop a finite time singularity from smooth initial data has been

an outstanding open problem. Here we review some existing computational

and theoretical work on possible finite blow-up of the 3D Euler equations.

We show that the geometric regularity of vortex filaments, even in an

extremely localized region, can lead to dynamic depletion of vortex

stretching, thus avoid finite time blowup of the 3D Euler equations.

Further, we perform large scale computations of the 3D Euler equations

to re-examine the two slightly perturbed anti-parallel vortex tubes which

is considered as one of the most attractive candidates for a finite time

blowup of the 3D Euler equations. We found that there is tremendous dynamic

depletion of vortex stretching and the maximum vorticity does not grow faster

than double exponential in time. Finally, we present a new class of solutions

for the 3D Euler and Navier-Stokes equations, which exhibit very interesting

dynamic growth property. By exploiting the special nonlinear structure of the

equations, we prove nonlinear stability and the global regularity of this class of solutions.

Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be ''close enough'' to the dynamics of the continuous system (which is typically easier to analyze) provided that small -- hence many -- time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought.

In this talk we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency.

Spectral methods are a class of methods for solving PDEs numerically.

If the solution is analytic, it is known that these methods converge

exponentially quickly as a function of the number of terms used.

The basic spectral method only works in regular geometry (rectangles/disks).

A huge amount of effort has gone into extending it to

domains with a complicated geometry. Domain decomposition/spectral

element methods partition the domain into subdomains on which the PDE

can be solved (after transforming each subdomain into a

regular one). We take the dual approach - embedding the domain into

a larger regular domain - known as the fictitious domain method or

domain embedding. This method is extremely simple to implement and

the time complexity is almost the same as that for solving the PDE

on the larger regular domain. We demonstrate exponential convergence

for Dirichlet, Neumann and nonlinear problems. Time permitting, we

shall discuss extension of this technique to PDEs with discontinuous

coefficients.

In this talk we will mainly analyze the vibrations of a simplified 1-d model for a multi-body structure consisting of a finite number of flexible strings distributed along a planar graph. In particular we shall analyze how solutions propagate along the graph as time evolves. The problem of the observation of waves is a natural framework to analyze this issue. Roughly, the question can be formulated as follows: Can we obtain complete information on the vibrations by making measurements in one single extreme of the network? This formulation is relevant both in the context of control and inverse problems.

Using the Fourier development of solutions and techniques of Nonharmonic Fourier Analysis, we give spectral conditions that guarantee the observability property to hold in any time larger than twice the total lengths of the network in a suitable Hilbert that can be characterized in terms of Fourier series by means of properly chosen weights. When the network graph is a tree these weights can be identified.

Once this is done these results can be transferred to other models as the Schroedinger, heat or beam-type equations.

This lecture is based on results obtained in collaboration with Rene Dager.

The "secular equation" is a special way of expressing eigenvalue

problems in a variety of applications. We describe the secular

equation for several problems, viz eigenvector problems with a linear

constraint on the eigenvector and the solution of eigenvalue problems

where the given matrix has been modified by a rank one matrix. Next we

show how the secular equation can be approximated by use of the

Lanczos algorithm. Finally, we discuss numerical methods for solving

the approximate secular equation.