Comlab

Radial basis function (RBF) methods have been successfully used to approximate functions in multidimensional complex domains and are increasingly being used in the numerical solution of partial differential equations. These methods are often called meshfree numerical schemes since, in some cases, they are implemented without an underlying grid or mesh.

The focus of this talk is on the class of RBFs that allow exponential convergence for smooth problems. We will explore the dependence of accuracy and stability on node locations of RBF interpolants. Because Gaussian RBFs with equally spaced centers are related to polynomials through a change of variable, a number of precise conclusions about convergence rates based on the smoothness of the target function will be presented. Collocation methods for PDEs will also be considered.

Adaptive discretizations and iterative multilevel solvers are nowadays well established techniques for the numerical solution of PDEs.

The development of efficient multilevel techniques in the context of PDE-constrained optimization methods is an active research area that offers the potential of reducing the computational costs of the optimization process to an equivalent of only a few PDE solves.

We present a general class of inexact adaptive multilevel SQP-methods for PDE-constrained optimization problems. The algorithm starts with a coarse discretization of the underlying optimization problem and provides

1. implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and

2. implementable accuracy requirements for iterative solvers of the PDE and adjoint PDE on the current grid

such that global convergence to the solution of the infinite-dimensional problem is ensured.

We illustrate how the adaptive refinement strategy of the multilevel SQP-method can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equation. Moreover, we discuss the efficient handling of control constraints and describe how efficent multilevel preconditioners can be constructed for the solution of the arising linear systems.

Numerical results are presented that illustrate the potential of the approach.

This is joint work with Jan Carsten Ziems.

The evaluation of oscillatory integrals is often considered to be a computationally challenging problem. However, in many cases, the exact opposite is true. Oscillatory integrals are cheaper to evaluate than non-oscillatory ones, even more so in higher dimensions. The simplest strategy that illustrates the general approach is to truncate an asymptotic expansion, where available. We show that an optimal strategy leads to the construction of certain unconventional Gaussian quadrature rules, that converge at twice the rate of asymptotic expansions. We examine a range of one-dimensional and higher dimensional, singular and highly oscillatory integrals.

A new primal-dual augmented Lagrangian merit function is proposed that may be minimized with respect to both the primal and dual variables. A benefit of this approach is that each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of classical primal methods are given: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual l1 linearly constrained Lagrangian (pdl1-LCL) method.

Model reduction (also called system reduction, order reduction) is an ubiquitous tool in the analysis and simulation of dynamical systems, control design, circuit simulation, structural dynamics, CFD, etc. In the past decades many approaches have been developed for reducing the complexity of a given model. In this introductory talk, we will survey some of the most prominent methods used for linear systems, compare their properties and highlight similarities. In particular, we will emphasize the role of recent developments in numerical linear algebra in the different approaches. Efficiently using these techniques, the range of applicability of some of the methods has considerably widened.

The performance of several approaches will be demonstrated using real-world examples from a variety of engineering disciplines.

The modelling of the elastoplastic behaviour of single

crystals with infinite latent hardening leads to a nonconvex energy

density, whose minimization produces fine structures. The computation

of the quasiconvex envelope of the energy density involves the solution

of a global nonconvex optimization problem. Previous work based on a

brute-force global optimization algorithm faced huge numerical

difficulties due to the presence of clusters of local minima around the

global one. We present a different approach which exploits the structure

of the problem both to achieve a fundamental understanding on the

optimal microstructure and, in parallel, to design an efficient

numerical relaxation scheme.

This work has been carried out jointly with Carsten Carstensen

(Humboldt-Universitaet zu Berlin) and Sergio Conti (Universitaet

Duisburg-Essen)

First, I'll give a very brief update on our nonlinear Krylov accelerator for the usual Modified Newton's method. This simple accelerator, which I devised and Neil Carlson implemented as a simple two page Fortran add-on to our implicit stiff ODEs solver, has been robust, simple, cheap, and automatic on all our moving node computations since 1990. I publicize further experience with it here, by us and by others in diverse fields, because it is proving to be of great general usefulness, especially for solving nonlinear evolutionary PDEs or a smooth succession of steady states.

Second, I'll report on some recent work in computerized tomography from X-rays. With colored computer graphics I'll explain how the standard "filtered backprojection" method works for the classical 2D parallel beam problem. Then with that backprojection kernel function H(t) we'll use an integral "change of variables" approach for the 2D fan-beam geometry. Finally, we turn to the tomographic reconstruction of a 3D object f(x,y,z) from a wrapped around cylindical 2D array of detectors opposite a 2D array of sources, such as occurs in PET (positron-emission tomography) or in very-wide-cone-beam tomography with a finely spaced source spiral.