# Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems

## Description

This monograph is based on the authors' studies carried out to investigate one of the most promising trends in the theory of ill-posed problems: namely, iterative regularization and its application to inverse heat transfer problems. Effective methods for solving inverse problems have allowed researchers to simplify experiments considerably, and to increase the accuracy and confidence of results in experimental data processing. The authors discuss a broad range of problems concerned with both the theory of regularizing gradient algorithms and peculiarities of their application to the most often encountered inverse problems of reconstruction of external heat fluxes and the identification of mathematical models for heat transfer processes.

306 pages,
© 1995

## Table of Contents:

Foreword to the English Language Edition

1 Identification and Inverse Problems in the Studies of Thermophysical Processess

1.1 On Mathematical Modeling of Physical Processes

1.2 Examples of Inverse Problem Statements

1.3 Identification of Mathematical Models for Physical Processes

1.4 On Applications of Inverse Problem Methods

1.5 Correctness of Inverse Problem Statement

1.6 Uniqueness of Inverse Heat Conduction Problem Solutions

2 Iterative Regularization of Ill-Posed Problems

2.1 Structure of Gradient Methods

2.3 Regularizability Conditions for Iterative Methods

2.5 Generalized Residual Criterion

2.6 Regularization of the Stationarization Method

2.7 Account of Apriori Information in Gradient Algorithms

2.8 Modification of Gradient Algorithms for Solving Multiparametric Problems

2.2 Convergence of Gradient Methods with Exact Initial Data

3 Construction of Gradient Algorithms for Solving Inverse Heat Transfer Problems

3.1 Construction of Residual Gradient in Linear Inverse Heat Transfer Problems. Problem Statement

3.2 Some Properties of Boundary-Value Problem Solutions for Second-Order Parabolic Equations

3.3 Adjoint Operator for Linear Boundary Inverse Heat Conduction Problems

3.4 Adjoint Operator for Linear Coefficient Inverse Heat Conduction Problems

3.5 Adjoint Operators for Inverse Heat Conduction Problems of Determining Thermal Resistances

3.6 Adjoint Operators for Retrospective Inverse Heat Conduction Problems

3.7 Adjoint Operator for Inverse Problems with Several Measurements. Nonuniqueness of Adjoint Boundary-Value Problems

3.8 Differentiability of Some Nonlinear Operators

3.9 Residual Gradient in Nonlinear Problems

3.10 Account of Apriori Information in Solving Coefficient Inverse Heat Conduction Problems

3.11 Some Heuristic Techniques for Choosing the Regularization Parameter

3.12 Application of Regularizing Gradient Algorithms to Experimental Data Smoothing and Differentiation

4 Iterative Solution of Boundary Inverse Heat Conduction Problems

4.1 Linear Inverse Problems in the Domain with Fixed Boundaries

4.2 Solution of Linear Inverse Problem

4.3 Algorithm Construction for Solving Boundary Inverse Problems with N Measurements

4.4 Linear Inverse Problem in the Domain with Moving Boundaries

4.5 Solution of Nonlinear Inverse Problems

4.6 Multidimensional Inverse Heat Conduction Problems

5 Algorithms for Solving Coefficient Inverse Problems

5.1 Inverse Problems for the Quasilinear Parabolic Equation

5.2 Reconstruction of Parameters in the Generalized Heat Conduction Equation with Constant Coefficients

5.3 Solution of Inverse Problems for Homogeneous Heat Conduction Equation in the Domain with Moving Boundaries

6 Design of Experiments for Solving Inverse Heat Conduction Problems

6.1 General Problem Statement for the Design of Thermal Experiments

6.2 Some Peculiarities of Thermal Experiments

6.3 Influence of Measurement Scheme on Convergence and Accuracy of Coefficient Inverse Heat Conduction Problem Solutions

6.4 Optimal Design of Measurements

6.5 Algorithms for Numerical Construction of Optimal Measurement Designs for One-Dimensional Inverse Problems

6.6 Optimization of Measurement Schemes in Inverse Heat Conduction Problems

Appendix Some Information from the Theory of Operators in Hilbert Spaces