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**Inverse heat conduction. Ill-posed problems.**
*(English)*
Zbl 0633.73120

A Wiley-Interscience Publication. New York etc.: John Wiley & Sons, Inc. XVII, 308 p. (TUB/Stat: B 86233) (1985).

This book presents a study of the Inverse Heat Conduction Problem (IHCP), which is the estimation of the surface heat flux history of a heat conducting body. Transient temperature measurements inside the body are utilized in the calculational procedure. The presence of errors in the measurements as well as the ill-posed nature of the problem lead to “estimates” rather than the “true” surface heat flux and/or temperature.

The specific problem treated is only one of many ill-posed problems but the techniques discussed herein can be applied to many others. The basic objective is to estimate a function given measurements that are “remote” in some sense. Other applications include remote sensing, oil exploration, nondestructive evaluation of materials, and determination of the Earth’s interior structure.

This book is written as a textbook in engineering with numerical examples and exercises for students. These examples will be useful to practicing engineers who use the book to become acquainted with the problem and methods of solution. A companion book, Parameter estimation in engineering and science, by the first author and K. J. Arnold (1977; Zbl 0363.62020), discusses estimation of certain constants or parameters rather than functions as in the IHCP. Though many of the ideas relating to least squares and sensitivity coefficients are present in both books, the present book does not require a mastery of parameter estimation.

Our philosophy in writing this book was to emphasize general techniques rather than specialized procedures unique to the IHCP. For example, basic techniques developed in Chapter 4 can be applied either to integral equation representations of the heat diffusion phenomena or to finite difference (or element) approximations of the heat conduction equation. The basic procedures in Chapter 4 can treat nonlinear cases, multiple sensors, nonhomogeneous media, multidimensional bodies, and many equations, in addition to the transient heat conduction equation.

The two general procedures that are used are called (a) function specification and (b) regularization. A method of combining these (the trial function method) is also suggested. One of the important contributions of this book is the demonstration that all of these methods can be implemented in a sequential manner. The sequential method in some case gives nearly the same result as whole domain estimation and yet is much more computationally efficient. One of our goals was to provide the reader with an insight into the basic procedures that provide analytical tools to compare various procedures. We do this by using the concepts of sensitivity coefficients, basic test cases, and the mean squared error. The reader is also shown that optimal estimation involves the compromise between minimum sensitivity to random measurement errors and the minimum bias.

The specific problem treated is only one of many ill-posed problems but the techniques discussed herein can be applied to many others. The basic objective is to estimate a function given measurements that are “remote” in some sense. Other applications include remote sensing, oil exploration, nondestructive evaluation of materials, and determination of the Earth’s interior structure.

This book is written as a textbook in engineering with numerical examples and exercises for students. These examples will be useful to practicing engineers who use the book to become acquainted with the problem and methods of solution. A companion book, Parameter estimation in engineering and science, by the first author and K. J. Arnold (1977; Zbl 0363.62020), discusses estimation of certain constants or parameters rather than functions as in the IHCP. Though many of the ideas relating to least squares and sensitivity coefficients are present in both books, the present book does not require a mastery of parameter estimation.

Our philosophy in writing this book was to emphasize general techniques rather than specialized procedures unique to the IHCP. For example, basic techniques developed in Chapter 4 can be applied either to integral equation representations of the heat diffusion phenomena or to finite difference (or element) approximations of the heat conduction equation. The basic procedures in Chapter 4 can treat nonlinear cases, multiple sensors, nonhomogeneous media, multidimensional bodies, and many equations, in addition to the transient heat conduction equation.

The two general procedures that are used are called (a) function specification and (b) regularization. A method of combining these (the trial function method) is also suggested. One of the important contributions of this book is the demonstration that all of these methods can be implemented in a sequential manner. The sequential method in some case gives nearly the same result as whole domain estimation and yet is much more computationally efficient. One of our goals was to provide the reader with an insight into the basic procedures that provide analytical tools to compare various procedures. We do this by using the concepts of sensitivity coefficients, basic test cases, and the mean squared error. The reader is also shown that optimal estimation involves the compromise between minimum sensitivity to random measurement errors and the minimum bias.

### MSC:

74A15 | Thermodynamics in solid mechanics |

35R25 | Ill-posed problems for PDEs |

35R30 | Inverse problems for PDEs |

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |

65Z05 | Applications to the sciences |

62N99 | Survival analysis and censored data |

80A23 | Inverse problems in thermodynamics and heat transfer |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

80-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to classical thermodynamics |