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Computational engineering/science uses a blend of applications, mathematical models and computations. Mathematical models require accurate approximations of their parameters, which are often viewed as solutions to inverse problems. Thus, the study of inverse problems is an integral part of computational engineering/science. This book presents several aspects of inverse problems along with needed prerequisite topics in numerical analysis and matrix algebra. If the reader has previously studied these prerequisites, then one can rapidly move to the inverse problems in chapters 4-8 on image restoration, thermal radiation, thermal characterization and heat transfer.

“This text does provide a comprehensive introduction to inverse problems and fills a void in the literature”.

Robert E White, Professor of Mathematics, North Carolina State University



Road Map

We shall begin by presenting some areas of application of inverse problems and the content of book’s chapters.
Francisco Duarte Moura Neto, Antônio José da Silva Neto

Mathematical Modeling

In this chapter we present some of the aspects of the interface between mathematics and its applications, the so-called mathematical modeling. This is neither mathematics nor applied mathematics and is usually performed by scientists and engineers. We do this in the context of linear algebra, which allows easy comparison between direct and inverse problems.We also show some of the modeling stages and encourage the use of a least-squares method. An application to the study of flow in a porous medium is presented. Some key issues pertaining to the use of models in practical situations are discussed.
Francisco Duarte Moura Neto, Antônio José da Silva Neto

Fundamental Concepts in Inverse Problems

The final answer to several problems can be reduced to evaluating a function–the solution function or the solution operator–and in the case of inverse problems it is not different. This is the point of view of a mathematician – insisting in the use of the notion of function. Not that one can always come about with the solution operator explicitly, but we can think abstractly on it and deduce its properties. This justifies the treatment that we present in this chapter of some of the aspects and complications that arise in the evaluation of functions. Next, we discuss some general aspects of mathematical models and inverse problems. A few classification schemes of inverse problems, illuminating different aspects, are presented. These classifications are used in subsequent chapters.
Francisco Duarte Moura Neto, Antônio José da Silva Neto

Spectral Analysis of an Inverse Problem

In this chapter we treat the solution of systems of linear equations in finite dimension spaces. This can be seen as examples of inverse reconstruction problems of Type I.We present a mathematical analysis of these linear inverse problems of finite dimension based on the spectral theorem.We study several aspects and behaviour of well-established methods for solving inverse problems. In particular, the concept of regularization, a very important notion in the area of inverse problems, will be dealt with. The analysis presented here is elementary – it depends on notions of linear algebra and convergence of numerical series. A similar study of regularization for problems in infinite dimensional spaces depends on functional analysis and is beyond the scope of this book. The interested reader is encouraged to consult [44, 29].
Francisco Duarte Moura Neto, Antônio José da Silva Neto

Image Restoration

Assume we have access to a certain image obtained by means of a device that, in the process of acquisition, causes a degradation, such as blurring. The objective of this chapter is to show how to restore the image from the degraded image, considering specific examples.
Francisco Duarte Moura Neto, Antônio José da Silva Neto

Radiative Transfer and Heat Conduction

We show in this chapter how to estimate the value of several material properties, such as single scattering albedo and thermal conductivity, present in the heating process of a material with thermal radiation.
Francisco Duarte Moura Neto, Antônio José da Silva Neto

Thermal Characterization

The development of new materials is a research intensive area, which is fueled by ever-increasing technological demands. With relevant applications, both in engineering and medicine, there is an obvious need for using adequate techniques for the characterization of new materials to verify if they meet the physical and chemical properties specified in the design phase, such as viscosity, elasticity, density, etc. In this context, the meaning of characterization differs from the meaning we have established in Chapters 1 and 2.
Francisco Duarte Moura Neto, Antônio José da Silva Neto

Heat Conduction

We present in this chapter estimates of the intensity of thermal sources with spatial and temporal dependence in heat transfer by conduction. This is an inverse reconstruction problem, and it is classified as a Type IV inverse problem. In otherwords, we consider a function estimation problem in an infinite dimensional model.
Francisco Duarte Moura Neto, Antônio José da Silva Neto

A General Perspective

This chapter introduces a more general perspective of direct and inverse problems.
At the outset, we must observe that, from a mathematical standpoint, there is no reason to justify the distinction between direct and inverse problems. As pointed out by Keller [43], it is adequate to say that two problems are mutually inverse if the formulation of one involves partly or completely the solution of the other. Our presentation emphasises this.
Francisco Duarte Moura Neto, Antônio José da Silva Neto


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