Computational Mathematics and Applications (past)

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.]

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.

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.

In this talk we shall be looking at recent and forthcoming developments in the widely used LAPACK and ScaLAPACK numerical linear algebra libraries.

Improvements include the following: Faster algorithms, better numerical methods, memory hierarchy optimizations, parallelism, and automatic performance tuning to accommodate new architectures; more accurate algorithms, and the use of extra precision; expanded functionality, including updating and downdating and new eigenproblems; putting more of LAPACK into ScaLAPACK; and improved ease of use with friendlier interfaces in multiple languages. To accomplish these goals we are also relying on better software engineering techniques and contributions from collaborators at many institutions.

After an overview, this talk will highlight new more accurate algorithms; faster algorithms, including those for pivoted Cholesky and updating of factorizations; and hybrid data formats.

This is joint work with Jim Demmel, Jack Dongarra and the LAPACK/ScaLAPACK team.

Several formulations of structural optimization problems based on linear and nonlinear semidefinite programming will be presented. SDP allows us to formulate and solve problems with difficult constraints that could hardly be solved before. We will show that sometimes it is advantageous to prefer a nonlinear formulation to a linear one. All the presented formulations result in large-scale sparse (nonlinear) SDPs. In the second part of the talk we will show how these problems can be solved by our augmented Lagrangian code PENNON. Numerical examples will illustrate the talk.

Joint work with Michael Stingl.