The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

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Inverse problems: A Bayesian perspective

Stochastic spectral methods for efficient Bayesian solution of inverse problems

Chapter 22 – inverse radiative heat transfer

A General Optimization Method using Adjoint Equation for Solving Multidimensional Inverse Heat Conduction

Design sensitivity plays a critical role in inverse and identification studies, as well as numerical optimization, and reliability analysis. Herein, we review the state of design sensitivity analysis as it applies to linear elliptic systems. Both first- and second-order sensitivities are derived as well as first-order sensitivities for symmetric positive definite eigenvalue systems. Although these results are not new, some of the derivations offer a different perspective than those previously presented. This article is meant as a tutorial, and as such, a simple two-degree-of-freedom spring system is employed to exemplify the sensitivity analyses. However, the concepts presented in this trivial example may be readily extended to compute sensitivities for complex systems via numerical techniques such as the finite element, boundary element, and finite difference methods.

Design sensitivity analysis: Overview and review

Description of the Inverse Heat Conduction Problem Exact Solutions of the Inverse Heat Conduction Problem Approximate Methods for Direct Heat Conduction Problems Inverse Heat Conduction Estimation Procedures Inverse Convolution Procedures Difference Methods for Solution of the One Dimensional Inverse Heat Conduction Problem Multiple Heat Flux Estimation Heat Transfer Coefficient Estimation Index

Inverse Heat Conduction: Ill-Posed Problems

This book discusses the engineering problem of determining surface heat flux and temperature history based on interior temperature measurements. It also provides the analytical techniques needed to arrive at otherwise difficult solutions, summarizing the findings of the last ten years. Topics include the steady state solution, Duhammel's Theorem, ill-posed problems, single future time step, and more.

Charles R. St. Clair

Papers

Inverse Heat Conduction: Ill-Posed Problems

Inverse Heat Conduction