An inexact Newton method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is an application of Newton's method to a nonlinear differential operator equation. Since the linear boundary-value problem to be solved at each iteration must be discretized, it is natural to consider quasilinearization in the framework of an inexact Newton method. More importantly, each linear problem is only a local model of the nonlinear problem, and so it is inefficient to try to solve the linear problems to full accuracy. Conditions on size of the relative residual of the linear differential equation can then be specified to guarantee rapid local convergence to the solution of the nonlinear continuous problem. If initial-value techniques are used to solve the linear boundary-value problems, then an integration step selection scheme is proposed so that the residual criteria are satisfied by the approximate solutions. Numerical results are presented that demonstrate substantial computational savings by this type of economizing on the intermediate problems.This work was supported in part by DOE Contract DE-AS05-82-ER13016 and NSF Grant RII-89-17691 and was part of the author's doctoral thesis at Rice University. It is a pleasure to thank the author's thesis advisors, Professor R. A. Tapia and Professor J. E. Dennis, Jr.     

Spline approximations for nonlinear hereditary control systems

A spline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.This work was supported in part by the Air Force Office of Scientific Research under Contract No. AFOSR-76-3092D, in part by the National Science Foundation under Grants Nos. NSF-MCS-79-05774-05 and NSF-MCS-82-00883, and in part by the US Army Research Office under Contract No. ARO-DAAG29-79-C-0161. The results reported here are a portion of the author's doctoral dissertation written under the supervision of Professor H. T. Banks, Brown University. The author is indebted to Professor Banks for his many valuable comments and suggestions during the course of this work.Part of this research was completed while the author was a visitor at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virginia.     

Geometric methods for nonlinear optimal control problems

It is the purpose of this paper to develop and present new approaches to optimal control problems for which the state evolution equation is nonlinear. For bilinear systems in which the evolution equation is right invariant, it is possible to use ideas from differential geometry and Lie theory to obtain explicit closed-form solutions.The author wishes to thank Professor A. Krener for many stimulating discussions and in particular for suggesting Theorem 3.3. Also, special thanks are due to the author's thesis advisor Professor R. W. Brockett under whose direction most of the research was done. Finally, the author thanks two anonymous referees for suggestions which have improved the exposition.     

Use of the method of particular solutions in nonlinear,two-point boundary-value problems

The nonlinear, two-point boundary-value problems associated with two differential systems, one uncontrolled and one controlled, are solved. First, quasilinearization techniques are used to replace each nonlinear system with one that is linear. Then, the method of particular solutions is used to solve the linear problem. The procedure is employed iteratively, and it is shown to converge rapidly to a solution for both the uncontrolled system and the controlled system.This research, supported by the NASA-Manned Spacecraft Center, Grant No. NGR-44-006-089, is a condensed version of the investigations described in Refs. 1–2.The author is indebted to Professor Angelo Miele for suggesting the topic and stimulating discussion.     

Parameter Identification for Nonlinear Abstract Cauchy Problems Using Quasilinearization

A quasilinearization approach to parameter identification in nonlinear abstract Cauchy problems in which the parameter appears in the nonlinear term, is presented. This approach has two main advantages over the classical one: it is much more intuitive and the derivation of the algorithm is done without need of the sensitivity equations on which classical quasilinearization is based. Sufficient conditions for the convergence of the algorithm are derived in terms of the regularity of the solutions with respect to the parameters. A comparison with the standard approach is presented and an application is included in which the nonphysical parameters in a mathematical model for shape memory alloys are estimated.     

Spline Techniques for Solving Singularly-Perturbed Nonlinear Problems on Nonuniform Grids

A numerical method based on cubic splines with nonuniform grid is given for singularly-perturbed nonlinear two-point boundary-value problems. The original nonlinear equation is linearized using quasilinearization. Difference schemes are derived for the linear case using a variable-mesh cubic spline and are used to solve each linear equation obtained via quasilinearization. Second-order uniform convergence is achieved. Numerical examples are given in support of the theoretical results.     

Extensions in quasilinearization techniques for optimal control

This paper presents several extensions in quasilinearization techniques for optimal control problems. Techniques are developed that facilitate the application of quasilinearization to control problems where bounds on the controls exist. Toward this end, quadratic convergence for bounded continuous control is shown. A method for extending the region over which the method converges is presented, and the theoretical advantage of the extended method is shown. The work of Long is modified to provide more accurate integration while preserving its usefulness in solving problems where the final time is free. A companion paper presents computational results.This research was supported in part by the Air Force Office of Scientific Research, Grant No. AF-AFOSR-699-67.     

Initial-value methods for second-order singularly perturbed boundary-value problems

In this paper, we present a numerical method for solving linear and nonlinear second-order singularly perturbed boundary-value-problems. For linear problems, the method comes from the well-known WKB method. The required approximate solution is obtained by solving the reduced problem and one or two suitable initial-value problems, directly deduced from the given problem. For nonlinear problems, the quasilinearization method is applied. Numerical results are given showing the accuracy and feasibility of the proposed method.This work was supported in part by the Consiglio Nazionale delle Ricerche (Contract No. 86.02108.01 and Progetto Finalizzatto Sistemi Informatia e Calcolo Paralello, Sottoprogetto 1), and in part by the Ministero della Pubblica Istruzione, Rome, Italy.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

Smooth partitions of unity in preduals of WCG spaces

This paper is based on part of the author's Ph.D. thesis written under the supervision of Professor V. Zizler and has been supported in part by a Province of Alberta Graduate Fellowship     

Haar wavelet solutions of nonlinear oscillator equations

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.     

Galerkin's procedure,quasilinearization, and nonlinear boundary-value problems

Among the popular and successful techniques for solving boundary-value problems for nonlinear, ordinary differential equations (ODE) are quasilinearization and the Galerkin procedure. In this note, it is demonstrated that utilizing the Galerkin criterion followed by the Newton-Raphson scheme results in the same iteration process as that obtained by applying quasilinearization to the nonlinear ODE and then the Galerkin criterion to each linear ODE in the resulting sequence. This equivalence holds for only the Galerkin procedure in the broad class of weighted-residual methods.This work was supported in part by the National Science Foundation, Grant No. GJ-1075.     

An iterative method for generalized nonlinear complementarity problems

An iterative method for solving generalized nonlinear complementarity problems (Ref. 1) involving stronglyK-copositive operators is introduced. Conditions are presented which guarantee the convergence of the method; in addition, the sequence of iterates is used to prove the existence of a solution to the problem under conditions not included in the previous study. Separate consideration is given to the generalized linear complementarity problem.This research was partially supported by National Science Foundation, Grant No. GP-16293. This paper constitutes part of the junior author's doctoral thesis written at Rensselaer Polytechnic Institute. Research support was provided by an NDEA Fellowship and an RPI Fellowship.     

Integrated system modeling and optimization via quasilinearization

System modeling and system optimization are two coupled and strongly related concepts in the modern approach to large-scale systems. Yet, they have been treated as two separate problems in the literature. The identification of system parameters, often referred to as system modeling, is essential in order to obtain an optimal control policy. This work considers the two problems jointly and provides a computational methodology in tackling the integrated problem formulation. This is done by viewing one of the objective functions in the bicriterion problem formulation as a constraint. A computational strategy such as quasilinearization is employed for the solution of the integrated problem. An example problem is introduced, and numerical results using an IBM 360/91 digital computer are presented.The authors are very grateful to Professor C. T. Leondes for his invaluable assistance, guidance, and comments. This research was supported in part by the Air Force Office of Scientific Research, Grant No. 699-67, and in part by the National Science Foundation, Grant No. GK-4086.     

Local convergence of the diagonalized method of multipliers

In this study, we consider a modification of the method of multipliers of Hestenes and Powell in which the iteration is diagonalized, that is, only a fixed finite number of iterations of Newton's method are taken in the primal minimization stage. Conditions are obtained for quadratic convergence of the standard method, and it is shown that a diagonalization where two Newton steps are taken preserves the quadratic convergence for all multipler update formulas satisfying these conditions.This work constitutes part of the author's doctoral dissertation in the Department of Mathematical Sciences, Rice University, under the direction of Professor R. A. Tapia and was supported in part by ERDA Contract No. E-(40-1)-5046.The author would like to thank Professor Richard Tapia for his comments, suggestions, and discussions on this material.     

Infinitary Jónsson functions and elementary embeddings

Summary We give an extender characterization of a very strong elementary embedding between transitive models of set theory, whose existence is known as the axiom I2. As an application, we show that the positive solution of a partition problem raised by Magidor would refute it.Mathematics subject classifications (1991): 03E55, 03E05This work is part of the author's thesis written under the direction of Professor K. Eda, to whom he is very grateful. He also wishes to thank the referee and Professor A. Blass for their careful reading and helpful suggestions. This research was partially supported by Grant-in-Aid for Scientific Research (No. 04302009), Ministry of Education, Science and Culture     

The general quadratic optimization problem

A principal pivoting algorithm is given for finding local minimizing points for general quadratic minimization problems. The method is a generalization of algorithms of Dantzig, and Van de Panne and Whinston for convex quadratic minimization problems.This paper is based on part of the author's doctoral dissertation written under Dr. Robert M. Thrall at the University of Michigan. The author was partially supported by funds from contract number DA-ARO-D-31-124-0767 with the U.S. Army Research Office, Durham.     

Quasilinearization Method for Nonlinear Elliptic Boundary-Value Problems

The quasilinearization method is developed for strong solutions of semilinear and nonlinear elliptic boundary-value problems. We obtain two monotone, L^p-convergent sequences of approximate solutions. The order of convergence is two. The tools are some results on the abstract quasilinearization method and from weakly–near operators theory.     

Quasilinearization and the methods of finite difference and initial values

Methods for the solution of nonlinear boundary-value problems for ordinary differential equations are discussed and classified as either finite-difference methods or initial-value methods. Within this framework, two algorithms, which are generated using the quasilinearization method, are presented and shown to be representative of these two methods. Consequently, both of the most widely used techniques for the solution of these problems can be formulated within the framework of the quasilinearization method. The computational properties of these algorithms are also discussed.     

Continuation in quasilinearization

A continuation method is described for extending the applicability of quasilinearization to numerically unstable two-point boundary-value problems. Since quasilinearization is a realization of Newton's method, one might expect difficulties in finding satisfactory initial trialpoints, which actually are functions over the specified interval that satisfy the boundary conditions. A practical technique for generating suitable initial profiles for quasilinearization is described. Numerical experience with these techniques is reported for two numerically unstable problems.