Unitary Integrators and Applications to Continuous Orthonormalization Techniques

This is the published version, also available here: http://dx.doi.org/10.1137/0731014.In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coefficient matrices in the linear case and a related structure in the nonlinear case. These skew systems arise in a number of applications, and interest originates from application to continuous orthogonal decoupling techniques. In this case, the matrix system has a cubic nonlinearity. Numerical integration schemes that compute a unitary approximate solution for all stepsizes are studied. These schemes can be characterized as being of two classes: automatic and projected unitary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution; the only ones are in fact the Gauss–Legendre point Runge–Kutta (Gauss RK) schemes. The second class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analysis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and implementation issues are considered, and the methods are tested on a number of examples

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

Error propagation in numerical approximations near relative equilibria

AbstractWe study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of Hamiltonian differential equations with symmetries. First it is shown that taking an initial perturbation of a relative equilibrium, the corresponding solution is related, in a first approximation, to another relative equilibrium, with the parameters perturbed from the original. Then, this is used to prove that, for stable relative equilibria, error growth with respect to the perturbed solution is in general quadratic, but only linear for schemes that preserve the invariant quantities of the problem. In this sense, the conclusion is similar to the one obtained when integrating unperturbed relative equilibria. Numerical experiments illustrate the results

Implicit Runge-Kutta formulae for the numerical integration of ODEs

Imperial Users onl

A polymorphic reconfigurable emulator for parallel simulation

Microprocessor and arithmetic support chip technology was applied to the design of a reconfigurable emulator for real time flight simulation. The system developed consists of master control system to perform all man machine interactions and to configure the hardware to emulate a given aircraft, and numerous slave compute modules (SCM) which comprise the parallel computational units. It is shown that all parts of the state equations can be worked on simultaneously but that the algebraic equations cannot (unless they are slowly varying). Attempts to obtain algorithms that will allow parellel updates are reported. The word length and step size to be used in the SCM's is determined and the architecture of the hardware and software is described

エネルギー関数を持つ発展方程式に対する幾何学的数値計算法

学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 松尾 宇泰, 東京大学教授 中島 研吾, 東京大学准教授 鈴木 秀幸, 東京大学准教授 長尾 大道, 東京大学准教授 齋藤 宣一University of Tokyo(東京大学

Efficient solution of parabolic equations by Krylov approximation methods

Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms

Modellierung, Simulation und Optimierung integrierter Schaltkreise

[no abstract available

Adjoint-based algorithms and numerical methods for sensitivity generation and optimization of large scale dynamic systems

This thesis presents advances in numerical methods for the solution of optimal control problems. In particular, the new ideas and methods presented in this thesis contribute to the research fields of structure-exploiting Newton-type methods for large scale nonlinear programming and sensitivity generation for IVPs for ordinary differential equations and differential algebraic equations. Based on these contributions, a new lifted adjoint-based partially reduced exact-Hessian SQP (L-PRSQP) method for nonlinear multistage constrained optimization problems with large scale differential algebraic process models is proposed. It is particularly well suited for optimization problems which involve many state variables in the dynamic process but only few degrees of freedom, i.e., controls, parameter or free initial values. This L-PRSQP method can be understood as an extension of the work of Schäfer to the case of exact-Hessian SQP methods, making use of directional forward/adjoint sensitivities of second order. It stands hence in the tradition of the direct multiple shooting approaches for differential algebraic equations of index 1 of Bock and co-workers. To the novelties that are presented in this thesis further belong - the generalization of the direct multiple shooting idea to structure-exploiting algorithms for NLPs with an internal chain structure of the problem functions, - an algorithmic trick that allows these so-called lifted methods to compute the condensed subproblems directly based on minor modifications to the user given problem functions and without further knowledge on the internal structure of the problem, - a lifted adjoint-based exact-Hessian SQP method that is shown to be equivalent to a full-space approach, but only has the complexity of an unlifted/single shooting approach per iteration, - new adjoint schemes for sensitivity generation based on Internal Numerical Differentiation (IND) for implicit LMMs using the example of Backward Differentiation Formulas (BDF), - the combination of univariate Taylor coefficient (TC) propagation and IND, resulting in IND-TC schemes which allow for the first time the efficient computation of directional forward and forward/adjoint sensitivities of arbitrary order, - a strategy to propagate directional sensitivities of arbitrary order across switching events in the integration, - a local error control strategy for sensitivities and a heuristic global error estimation strategy for IVP solutions in connection with IND schemes, - the software packages DAESOL-II and SolvIND, implementing the ideas related to IVP solution and sensitivity generation, as well as the software packages LiftOpt and DynamicLiftOpt that implement the lifted Newton-type methods for general NLP problems and the L-PRSQP method in the optimal control context, respectively. The performance of the presented approaches is demonstrated by the practical application of our codes to a series of numerical test problems and by comparison to the performance of alternative state-of-the-art approaches, if applicable. In particular, the new lifted adjoint-based partially reduced exact-Hessian SQP method allows the efficient and successful solution of a practical optimal control problem for a binary distillation column, for which the solution using a direct multiple shooting SQP method with an exact-Hessian would have been prohibitively expensive until now