Adjoint Equations and Perturbation Algorithms in Nonlinear Problems: 1st Edition (Hardback) book cover

Adjoint Equations and Perturbation Algorithms in Nonlinear Problems

1st Edition

By Guri I. Marchuk, Valeri I. Agoshkov, Victor P. Shutyaev

CRC Press

288 pages

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Hardback: 9780849328718
pub: 1996-05-07
$145.00
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pub: 2018-04-24
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Description

Sparked by demands inherent to the mathematical study of pollution, intensive industry, global warming, and the biosphere, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems is the first book ever to systematically present the theory of adjoint equations for nonlinear problems, as well as their application to perturbation algorithms. This new approach facilitates analysis of observational data, the application of adjoint equations to retrospective study of processes governed by imitation models, and the study of computer models themselves. Specifically, the book discusses:

  • Principles for constructing adjoint operators in nonlinear problems

  • Properties of adjoint operators and solvability conditions for adjoint equations

  • Perturbation algorithms using the adjoint equations theory for nonlinear problems in transport theory, quasilinear motion, substance transfer, and nonlinear data assimilation

  • Known results on adjoint equations and perturbation algorithms in nonlinear problems

    This groundbreaking text contains some results that have no analogs in the scientific literature, opening unbounded possibilities in construction and application of adjoint equations to nonlinear problems of mathematical physics.

  • Table of Contents

    Principles of Construction of Adjoint Operators in Non-Linear Problems

    Dual Spaces and Adjoint Operators

    Construction of Adjoint Operators Based on Using the Lagrange Identity

    Definition of Adjoint Operators Based on Using Taylor's Formula

    Operators of the Class D and their Adjoint Operators

    Properties of Adjoint Operators Constructed on the Basis of Various Principles

    General Properties of Main and Adjoint Operators Corresponding to Non-Linear Operators

    Properties of Operators of the Class D

    Properties of Adjoint Operators Constructed with the Use of the Taylor Formula

    Solvability of Main and Adjoint Equations in Non-Linear Problems

    Main and Adjoint Equations. Problems

    Solvability of the Equation F(u) = y

    Solvability of the Equation A(u)v = y

    Solvability of the Equation Ã(u)v = y

    Solvability of the Equation A*(u)w = p

    Solvability of the Equation Ã*(u)w = p

    Transformation Groups, Conservation Laws and Construction of the Adjoint Operators in Non-Linear Problems

    Definitions. Non-Linear Equations and Operators. Conservation Laws

    Transformation of Equations

    Adjoint Equations

    Relationship between Different Adjoint Operators

    General Remarks on Constructing the Adjoint Equations with the Use of the Lie Groups and Conservation Laws

    Construction of Adjoint Operators with Prescribed Properties

    The Noether Theorem, Conservation Laws and Adjoint Operators

    On Some Applications of Adjoint Equations

    Perturbation Algorithms in Non-Linear Problems

    Perturbation Algorithms for Original Non-Linear Equations and Equations Involving Adjoint Operators

    Perturbation Algorithms for Non-Linear Functionals Based on Using Main and Adjoint Equations

    Spectral Method in Perturbation Algorithms

    Justification of the N-th Order Perturbation Algorithms

    Convergence Rate Estimates for Perturbation Algorithms. Comparison with the Successive Approximation Method

    Justification of Perturbation Algorithms in Quasi-Linear Elliptic Problems

    Adjoint Equations and the N-th Order Perturbation Algorithms in Non-Linear Problems of Transport Theory

    Some Problems of Transport Theory

    The N-th Order Perturbation Algorithms for an Eigenvalue Problem

    A Problem of Control and its Approximate Solution with the Use of Perturbation Algorithms

    Investigation and Approximate Solution of a Non-Linear Problem for the Transport Equation

    Adjoint Equations and Perturbation Algorithms for a Quasilinear Equation of Motion

    Statement of the Problem. Basic Assumptions. Operator Formulation

    Transformation of the Problem. Properties of the Non-Linear Operator

    Adjoint Equation

    An Algorithm for Computing the Functional

    The Problem on Chemical Exchange Processes

    Adjoint Equations and Perturbation Algorithms for a Non-Linear Mathematical Model of Mass Transfer in Soil

    Mathematical Models of Mass Transfer in Soil

    Formulation of a Non-Linear Mathematical Model

    Transformation of the Problem. Properties of the Non-Linear Operator

    Perturbation Algorithm. Adjoint Equation

    Approximate Solution of the Problem on Finding an Effective Dispersion Coefficient

    An Algorithm for Solving the Problem

    Applications of Adjoint Equations in Science and Technology

    Adjoint Equations in Data Assimilation Problems

    Application of Adjoint Equations for Solving the Problem of Liquid Boundary Conditions in Hydrodynamics

    Shape Optimization Using Adjoint Equation Approaches

    Global Transport of Pollutants

    Problems of Climate Change Sensitivity in Various Regions of the World

    Subject Categories

    BISAC Subject Codes/Headings:
    MAT003000
    MATHEMATICS / Applied
    MAT007000
    MATHEMATICS / Differential Equations