Luus Rein

Iterative Dynamic Programming

Description

Dynamic programming is a powerful method for solving optimization problems, but has a number of drawbacks that limit its use to solving problems of very low dimension. To overcome these limitations, author Rein Luus suggested using it in an iterative fashion. Although this method required vast computer resources, modifications to his original scheme have made the computational procedure feasible.
With iteration, dynamic programming becomes an effective optimization procedure for very high-dimensional optimal control problems and has demonstrated applicability to singular control problems. Recently, iterative dynamic programming (IDP) has been refined to handle inequality state constraints and noncontinuous functions.
Iterative Dynamic Programming offers the first comprehensive presentation of this powerful tool. It brings together the results of work carried out by the author and others - previously available only in scattered journal articles - along with the insight that led to its development. The author provides the necessary background, examines the effects of the parameters involved, and clearly illustrates IDP's advantages.

Audience
Professionals in engineering design, operations research, optimization and optimal control, applied mathematics, and related fields

Contents
INTRODUCTION
Fundamental Definitions and Notation
Steady-State System Model
Continuous-Time System Model
Discrete-Time System Model
The Performance Index
Interpretation of Results
Examples of Systems for Optimal Control
Solving Algebraic Equations
Solving Ordinary Differential Equations
STEADY-STATE OPTIMIZATION
Linear Programming
LJ Optimization Procedure
References
DYNAMIC PROGRAMMING
Introduction
Examples
Limitations of Dynamic Programming
ITERATIVE DYNAMIC PROGRAMMING
Construction of Time Stages
Construction of Grid for x
Allowable Values for Control
First Iteration
Iterations with Systematic Reduction in Region Size
Example
Use of Accessibs as Grid Points
Algorithm for IDP
Early Applications of IDP
ALLOWABLE VALUES FOR CONTROL
Introduction
Comparison of Uniform Distribution to Random Choice
EVALUATION OF PARAMETERS IN IDP
Number of Grid Points
Multi-Pass Approach
Further Example
PIECEWISE LINEAR CONTINUOUS CONTROL
Problem Formulation
Algorithm for IDP for Piecewise Linear Control
Numerical Examples
TIME-DELAY SYSTEMS
Problem Formulation
Examples
VARIABLE STAGE LENGTHS
Variable Stage-Lengths when Final Time is Free
Problems where Final Time f is not Specified
Systems with Specified Final Time
SINGULAR CONTROL PROBLEMS
Four Simple-Looking Examples
Yeo's Singular Control Problem
Nonlinear Two-Stage CSTR Problem
STATE CONSTRAINTS
Introduction
Final State Constraints
State Inequality Constraints
TIME OPTIMAL CONTROL
Introduction
Time Optimal Control Problem
Direct Approach to Time Optimal Control
Examples
High Dimensional Systems
NONSEPARABLE PROBLEMS
Problem Formulation
Examples
References
SENSITIVITY CONSIDERATIONS
Introduction
Example: Lee-Ramirez Bioreactor
TOWARD PRACTICAL OPTIMAL CONTROL
Optimal Control of Oil Shale Pyrolysis
Future Directions
APPENDICES: Nonlinear Algebraic Equation Solver. Listing of Linear Programming Program. LJ Optimization Programs. Iterative Dynamic Programming Programs. Listing of
DVERK.
INDEX
Each chapter also contains an introduction and a References section.

Features
A careful exposition of IDP methods - provides a solid working knowledge with solutions for a wide range of problems
A variety of examples from a range of disciplines - demonstrates IDP's applicability
Accessibility - an introduction and chapters on steady-state optimization and dynamic programming lay the necessary foundation
Ready-to-run FORTRAN programs - offers direct experience with computations

Iterative Dynamic Programming
ISBN: 1584881488, No of pages: 344
Publication Date: 01/27/00

Nakamura Gen / Saitoh Saburou
Seo Jin Kean / Yamamoto Masahiro

Inverse Problems and Related Topics

Description

Inverse problems arise in many disciplines and hold great importance to practical applications. However, sound new methods are needed to solve these problems. Over the past few years, Japanese and Korean mathematicians have obtained a number of very interesting and unique results in inverse problems.
Inverse Problems and Related Topics compiles papers authored by some of the top researchers in Korea and Japan. It presents a number of original and useful results and offers a unique opportunity to explore the current trends of research in inverse problems in these countries. Highlighting the existence and active work of several Japanese and Korean groups, it also serves as a guide to those seeking future scientific exchange with researchers in these countries.

Contents
A Finite Difference Model for Calderón's Boundary Inverse Problem
Inverse Problems for Equations with Memory
Parameter Estimation of Elastic Media
The Probe Method and its Applications
Recent Progress in the Inverse Conductivity Problem with Single Measurement
A Moment Method on Inverse Problems for the Heat Equation
Some Remarks on Free Boundaries of Recirculation Euler Flows with Constant Vorticity
Algorithms for the Identification of Spatially Varying/Invariant Stiffness and Dampings in Flexible Beams
Numerical Solutions of the Cauchy Problem in Potential and Elastostatics
Inverse Source Problems in the Helmholtz Equations
A Numerical Method for a Magnetostatic Inverse Problem using the Edge Element. Exact Controllability Method and Multidimensional Linear Inverse Problems
Impedance Computed Tomo-Electrocardiography
An Inverse Problem for Free Channel Scattering
Surface Impedance Tensor and Boundary Value Problem
Aysmptotics for the Spectral and Weyl Functions of the Operator-Value Sturm-Liouville Problem
Exact Controllability Method and Multidimensional Linear Inverse Problems

Features
Comprises a collection of papers on inverse problems authored by top Japanese and Korean researchers
Offers 10 theoretical works and six on numerical simulation
Presents unique works including memory reconstruction problems, the probe method, Calderon's boundary inverse problem, a moment method for the heat equation, and
inverse source problems in the Helmholtz equations
Provides highly original work, review papers, and perspectives for future research


Inverse Problems and Related Topics
ISBN: 1584881917, No of pages: 248
Publication Date: 02/28/00

Farrell Paul / Hegarty Alan / Miller John J H
O'Riordan Eugene / Shishkin Grigorii I

Robust Computational Techniques for Boundary Layers

Description

Current standard numerical methods are of little use in solving mathematical problems involving boundary layers. In Robust Computational Techniques for Boundary Layers, the authors construct numerical methods for solving problems involving differential equations that have non-smooth solutions with singularities related to boundary layers. They present a new numerical technique that provides precise results in the boundary layer regions for the problems discussed in the book. They show that this technique can be adapted in a natural way to a real flow problem, and that it can be used to construct benchmark solutions for comparison with solutions found using other numerical techniques.
Focusing on robustness, simplicity, and wide applicability rather than on optimality, Robust Computational Techniques for Boundary Layers provides readers with an understanding of the underlying principles and the essential components needed for the construction of numerical methods for boundary layer problems. It explains the fundamental ideas through physical insight, model problems, and computational experiments and gives details of the linear solvers used in the computations so that readers can implement the methods and reproduce the numerical results.

Applied mathematicians and numerical analysts, engineers, physicists

Contents
Introduction to Numerical Methods for Problems with Boundary Layers
Numerical Methods on Uniform Meshes
Layer Resolving Methods for Convection-Diffusion Problems in One Dimension
The Limitations of Non-Monotone Numerical Methods
Convection-Diffusion Problems in a Moving Medium
Convection-Diffusion Problems with Frictionless Walls
Convection-Diffusion Problems with No Slip Boundary Conditions
Experimental Estimation of Errors
Non-Monotone Methods in Two Dimensions
Linear and Nonlinear Reaction-Diffusion Problems
Prandtl Flow past a Flat Plate-Blasius' Method
Prandtl Flow past a Flat Plate-Direct Method
References.

Features

Develops a new numerical technique - the Robust Layer-Resolving Method
Applies the techniques developed to a classical problem in fluid dynamics - incompressible laminar flow over a flat plate
Includes numerous tables and graphics of actual computations
Provides details on the linear solvers used in the computations, allowing readers to implement the methods and reproduce the numerical results

Robust Computational Techniques for Boundary Layers
ISBN: 1584881925, No of pages: 256
Publication Date: 03/30/00

Hummel Kenneth E

Introductory Concepts for Abstract Mathematics

Description

A gap has long existed between basic calculus studies and those on abstract algebra and real analysis. The focus of calculus instruction has become more and more computational, leaving students ill prepared for more advanced, abstract work that requires the ability to understand and construct proofs.
Introduction to Abstract Mathematics helps students grow from calculus to more abstract courses. It teaches them to deal effectively with abstract ideas, to comprehend the logical structure of proofs, and to write mathematics using conventional terminology in an effective, logical, and understandable way.

Contents
Logical and Propositional Calculus
Sets and Set Operations
Cartesian Products, Relations, and Functions
Algebraic and Order Properties
Transfinite Cardinal Numbers
The Axiom of Choice and Ordinal Numbers
Reading List
Solutions to Selected Problems
Index

Features

Provides the foundation required for advanced and abstract studies, including a thorough development of sets, relations, and functions
Imparts other important concepts, such as the development of real numbers, least upper bounds, and concepts related to transfinite cardinal arithmetic
Teaches the writing of mathematics and the construction of proofs

Introductory Concepts for Abstract Mathematics
ISBN: 1584881348, No of pages: 344
Publication Date: 03/23/00

Liebeck M W
Imperial College, UK

A Concise Introduction to Pure Mathematics

Description

Written in a relaxed, readable style, A Concise Introduction to Pure Mathematics provides a robust bridge to university mathematics. In nineteen brief chapters, it covers the range of topics needed to build a firm foundation for the study of the higher mathematics. These include sets and proofs, real numbers and decimals, rational numbers, an introduction to analysis, complex numbers, polynomial equations, induction, Euler's formula and Platonic solids, integers and prime numbers, counting methods, functions, infinite sets, and countability.

Contents
Preface
Sets and Proofs
Number Systems
Decimals
Inequalities
nth Roots and Rational Powers
Complex Numbers
Polynomial Equations
Induction
Euler's Formula and Platonic Solids
Introduction to Analysis
The Integers
Prime Factorization
More on Prime Numbers
Congruence of Integers
Cpounting and Choosing
More on Sets
Equivalence Relations
Functions
Infinity
Further Reading
Index

Publication Pricing

A Concise Introduction to Pure Mathematics
ISBN: 1584881933, No of pages: 176
Publication Date: 03/24/00

Griffiths D F / University of Dundee
Watson G A / University of Dundee

Numerical Analysis 1999

Description

Of considerable importance to numerical analysts, this text contains the proceedings of the 18th Dundee Biennial Conference on Numerical Analysis, featuring eminent analysts and current topics. The papers cover everything from partial differential equations to linear algebra and approximation theory and contain contributions from the leading experts in the field. The applications range from image processing and molecular dynamics to superconductivity. If you rely on numerical methods, Numerical Analysis 1999 will serve as an essential guide to the direction of current research.

Audience

Postgraduate students and researchers in numerical analysis. Engineers and Scientists who use numerical methods

Contents

Mixed hp-Finite Element Methods for Incompressible Flow -M. Ainsworth, P. Coggins, and B. Senior
Adaptive Finite Element Methods for Optimization Problems -R. Becker, H. Kapp and R. Rannacher
Variational PDE Models and Methods for Image Processing -P. Blomgren, T. F. Chan, P. Mulet, L. Vese, and W.L. Wan
Interacting with the Subgrid World -F. Brezzi
The Computing Power of Geometry -F. Chaitin-Chatelin
Numerical Approximation of Vortex Density Evolution in a Superconductor -C.M.Elliott and V.M.Styles
Krylov Subspace Methods for Radial Basis Function Interpolation -A.C. Faul and M.J.D. Powell
A Homotopy Method for Mixed Complementarity Problems Based on the PATH Solver
-M. Ferris, T. S. Munson, and D. Ralph
Energy Conservation by Störmer-Type Numerical Integrators -E. Hairer and Ch. Lubich
New Results on FETI Methods for Elliptic Problems with Discontinuous Coefficients
-A. Klawonn and O. B. Widlund
The Search for Good Polynomial Interpolation Points on the Sphere -I. H. Sloan and R. S. Womersley
Gramian Based Model Reduction of Large-Scale Dynamical Systems -P. Van Dooren
Solving Data Fitting Problems in lp Norms with Bocertainties in the Data -G. A. Watson

Features

Contains invited papers from leading international experts in the field
Provides a valuable guide to the direction of current research in many areas of numerical analysis
Covers areas from ordinary and partial differential equations to linear algebra and approximation theory
Includes applications to optimization, fluid dynamics, and acoustics

Numerical Analysis 1999
ISBN: 1584880201, No of pages: 288
Publication Date: 03/27/00