SIGEST

Abstract

Previous article Next article Full AccessSIGESThttps://doi.org/10.1137/SIREAD000046000003000475000001BibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Since the "new" SIAM Review began in 1999, inverse problems have appeared in the Survey and Review, Problems and Techniques, and Education sections, but not, until now, in SIGEST. Given the growing importance of and interest in inverse problems, the SIREV editors are pleased to present a SIGEST paper that addresses a basic question in inverse problems. As implied by its name, an inverse problem is closely related to another kind of problem, most often called either the "forward" or "direct" problem. Consider a mathematical model that is intended to represent a real-world system or process. Except in trivial cases, mathematical models contain parameters, each of which characterizes a property, such as conductivity or elasticity, of the system being modeled. The definition of "parameter" in this context is very general, and may include initial conditions, boundary data, or the shape of the boundary. Given a complete specification of the model, including values for all the needed parameters, the forward problem is to compute "the answers," namely, the outputs or responses of the system, often by solving ordinary or partial differential equations. (Ideally, the solution to the forward problem will match measured data from the system.) The inverse problem arises when the parameters of the model are unknown and we wish to deduce or estimate them from a (necessarily limited) set of measurements. Inverse problems typically have the unpleasant mathematical feature of being ill-posed, i.e., solutions need not exist, are not necessarily unique, and are not stable under perturbations of the data. The last property is significant in practice because experimental data are almost certain to contain noise. Inverse problems arise in a multiplicity of scientific, engineering, and medical applications, with new ones constantly arising as mathematical understanding and computational methods advance. "Classical" applications of inverse problems include electromagnetic and acoustic radiation (for example, radar), quantum mechanics, and geophysics. Nondestructive testing and evaluation lead to inverse problems in which measurements taken on the exterior of a body during application of energy (electrical, thermal, or mechanical) are used to deduce properties of the interior, such as the presence and nature of defects (e.g., holes or cracks). In medical imaging, the lively area of elastography---measurement of the elastic properties of tissue---is producing promising strategies for in vivo tumor detection, generalizing centuries-old diagnosis by hand palpation that relies on the difference in stiffness between normal soft tissue and tumors. This issue's SIGEST paper, "Detecting an Inclusion in an Elastic Body by Boundary Measurements," by G. Alessandrini, A. Morassi, and E. Rosset, first appeared in 2002 in the SIAM Journal on Mathematical Analysis (SIMA). Every publication in SIMA must contain innovative analytical techniques and be related to a model of natural phenomena. Precisely in this dual spirit, our SIGEST paper addresses the extremely difficult problem of estimating the size of an unknown inclusion in an elastic isotropic body from boundary measurements. The authors' approach features hard analysis, mainly quantitative estimates related to the size of the zero set of the solution to the relevant Lame system of linearized elasticity. The paper also includes discussion of how the estimates derived could be used as a decision tool in quality control and in designing experiments for concrete applications. Previous article Next article FiguresRelatedReferencesCited byDetails Firing Thoreau: Conscience and At-Will EmploymentSSRN Electronic Journal Cross Ref Volume 46, Issue 3| 2004SIAM Review History Published online:04 August 2006 InformationCopyright © 2004 Society for Industrial and Applied Mathematics Article & Publication DataArticle DOI:10.1137/SIREAD000046000003000475000001Article page range:pp. 475-475ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics