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Adjoint Equations and Perturbation Algorithms in Nonlinear Problems





ISBN 9780367448585
Published June 30, 2020 by CRC Press
288 Pages

 
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Book 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

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    Author(s)

    Biography

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