Functional a posteriori error estimates for time-periodic parabolic optimal control problems

This paper is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Parallel solution methods and preconditioners for evolution equations

The recent development of the high performance computer platforms shows a clear trend towards heterogeneity and hierarchy. In order to utilize the computational power, particular attention must be paid to finding new algorithms or adjust existing ones so that they better match the HPC computer architecture. In this work we consider an alternative to classical time-stepping methods based on use of time-harmonic properties and discuss solution approaches that allow efficient utilization of modern HPC resources. The method in focus is based on a truncated Fourier expansion of the solution of an evolutionary problem. The analysis is done for linear equations and it is remarked on the possibility to use two- or multilevel mesh methods for nonlinear problems, which can enable further, even higher degree of parallelization. The arising block matrix system to be solved admits a two-by-two block form with square blocks, for which a very efficient preconditioner exists. It leads to tight eigenvalue bounds for the preconditioned matrix and, hence, to a very fast convergence of a preconditioned Krylov subspace or iterative refinement method. The analytical background is shown as well as some illustrating numerical examples

Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks

Matrices or operators in two-by-two block form with square blocks arise in numerous important applications, such as in optimal control problems for PDEs. The problems are normally of very large scale so iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is of crucial importance. Since some time a very efficient preconditioner, the preconditioned square block, PRESB method has been used by the authors and coauthors in various applications, in particular for optimal control problems for PDEs. It has been shown to have excellent properties, such as a very fast and robust rate of convergence that outperforms other methods. In this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs. Under certain conditions, the condition number of the preconditioned matrix is bounded by 2 or even smaller. Furthermore, under certain assumptions the rate of convergence is superlinear

Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods

Computationally Efficient Steady--State Simulation Algorithms for Finite-Element Models of Electric Machines.

The finite element method is a powerful tool for analyzing the magnetic characteristics of electric machines, taking account of both complex geometry and nonlinear material properties. When efficiency is the main quantity of interest, loss calculations can be affected significantly due to the development of eddy currents as a result of Faraday’s law. These effects are captured by the periodic steady-state solution of the magnetic diffusion equation. A typical strategy for calculating this solution is to analyze an initial value problem over a time window of sufficient length so that the transient part of the solution becomes negligible. Unfortunately, because the time constants of electric machines are much smaller than their excitation period at peak power, the transient analysis strategy requires simulating the device over many periods to obtain an accurate steady-state solution. Two other categories of algorithms exist for directly calculating the steady-state solution of the magnetic diffusion equation; shooting methods and the harmonic balance method. Shooting methods search for the steady-state solution by solving a periodic boundary value problem. These methods have only been investigated using first order numerical integration techniques. The harmonic balance method is a Fourier spectral method applied in the time dimension. The standard iterative procedures used for the harmonic balance method do not work well for electric machine simulations due to the rotational motion of the rotor. This dissertation proposes several modifications of these steady-state algorithms which improve their overall performance. First, we demonstrate how shooting methods may be implemented efficiently using Runge-Kutta numerical integration methods with mild coefficient restrictions. Second, we develop a preconditioning strategy for the harmonic balance equations which is robust against large time constants, strong nonlinearities, and rotational motion. Third, we present an adaptive framework for refining the solutions based on a local error criterion which further reduces simulation time. Finally, we compare the performance of the algorithms on a practical model problem. This comparison demonstrates the superiority of the improved steady-state analysis methods, and the harmonic balance method in particular, over transient analysis.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113322/1/pries_1.pd