A class of Lagrange-Newton-SQP methods is investigated for optimal control problems governed by semilinear parabolic initial- boundary value problems. Distributed and boundary controls are given, restricted by pointwise upper and lower bounds. The convergence of the method is discussed in appropriate Banach spaces. Based on a weak second order sufficient optimality condition for the reference solution, local quadratic convergence is proved. The proof is based on the theory of Newton methods for generalized equations in Banach spaces.

We study two interacting particles in a random potential chain by means of the transfer matrix method. The dependence of the two-particle localization length lampta_2 on disorder and interaction strength is investigated. Our results demonstrate that the recently proposed enhancement of lampta_2 as compared to the results for single particles is entirely due to the finite size of the systems considered. This is shown for a Hubbard-like onsite interaction and also a long-range interaction.

We use the density matrix renormalization group to study the quantum transitions that occur in the half-filled one-dimensional fermionic Hubbard model with onsite potential disorder. We find a transition from the gapped Mott phase with algebraic spin correlations to a gapless spin-disordered phase beyond a critical strength of the disorder 1 c ss U= 2. Both the transitions in the charge and spin sectors are shown to be coincident. We also establish the finite-size corrections to the charge gap and the spin-spin correlation length in the presence of disorder and using a finite-size-scaling analysis we obtain the zero temperature phase diagram of the various quantum phase transitions that occur in the disorder-interaction plane.

An error in the Gaussian random variate generator by Kinderman and Ramage is described that results in the generation of random variates with an incorrect distribution. An additional statement that corrects the original algorithm is given. / Series: Preprint Series / Department of Applied Statistics and Data Processing

In this paper we compare the performance, scalability, and robustness of different parallel algorithms for the numerical solution of nonlinear boundary value problems arising in the magnetic field computation and in solid mechanics. These problems are discretized by using the finite element method with triangular meshes and piecewise linear functions. The nonlinearity is handled by a nested Newton solver, and the linear systems of algebraic equations within each Newton step are solved by means of various iterative solvers, namely multigrid methods and conjugate gradient methods with preconditioners based on domain decomposition, multigrid, or BPX techniques, respectively. The basis of the implementation of all solvers is a non-overlapping domain decomposition data structure such that they are well-suited for parallel machines with MIMD architecture.

In this paper we present two hierarchically preconditioned methods for the fast solution of mesh equations that approximate 2D-elliptic boundary value problems on unstructured quasi uniform triangulations. Based on the fictitious space approach the original problem can be embedded into an auxiliary one, where both the hierarchical grid information and the preconditioner by decomposing functions on it are well defined. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as the BPX-preconditioned cg-iteration having optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods which can be of importance in the industrial engineering, where often only the nodal coordinates and the element connectivity of the underlying (fine) discretization are available.

In this paper introduce new approach for unified theory for continuous and discrete time (optimal) control problems based on the generalized Cayley transformation. We also relate the associated discrete and continuous generalized algebraic Riccati equations. We demonstrate the potential of this new approach proving new result for discrete algebraic Riccati equations. But we also discuss where this new approach as well as all other approaches still is non-satisfactory. We explain a discrepancy observed between the discrete and continuous cse and show that this discrepancy is partly due to the consideration of the wrong analogues. We also present an idea for a metatheorem that relates general theorems for discrete and continuous control problems.

A collection of benchmark examples is presented for the numerical solution of continuous-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.

This is the second part of a collection of benchmark examples for the numerical solution of algebraic Riccati equations. After presenting examples for the continuous-time case in Part I, our concern in this paper is discrete-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.

The experimental program ¨SPC-PM Po 3D¨ is part of the ongoing research of the Chemnitz research group Scientific Parallel Computing (SPC) into finite element methods for problems over three dimensional domains. The package in its version 2.0 is documented in two manuals. The User's Manual provides an overview over the program, its capabilities, its installation, and handling. Moreover, test examples are explained. The aim of the Programmer's Manual is to provide a description of the algorithms and their realization. It is written for those who are interested in a deeper insight into the code, for example for improving and extending. In Version 2.0 the program can solve the Poisson equation and the Lam\'e system of linear elasticity with in general mixed boundary conditions of Dirichlet and Neumann type. The domain $\Omega\subset\R^3$ can be an arbitrarily bounded polyhedron. The input is a coarse mesh, a description of the data and some control parameters. The program distributes the elements of the coarse mesh to the processors, refines the elements, generates the system of equations using linear or quadratic shape functions, solves this system and offers graphical tools to display the solution. Further, the behavior of the algorithms can be monitored: arithmetic and communication time is measured, the discretization error is measured, different preconditioners can be compared. We plan to extend the program in the next future by including a multigrid solver, an error estimator and adaptive mesh refinement, as well as the treatment of coupled thermo-elastic problems. The program has been developed for MIMD computers; it has been tested on Parsytec machines (GCPowerPlus-128 with Motorola Power PC601 processors and GCel-192 on transputer basis) and on workstation clusters using PVM. The special case of only one processor is included, that means the package can be compiled for single processor machines without any change in the source files.