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Statistical and Computational Inverse Problems de Jari Kaipio

Statistical and Computational Inverse Problems: Applied Mathematical Sciences

Autor Jari Kaipio, E. Somersalo
en Limba Engleză Hardback – dec 2004

În volumul Statistical and Computational Inverse Problems, autorii Jari Kaipio și E. Somersalo propun o deschidere interdisciplinară necesară, situată la intersecția dintre matematica aplicată, inginerie și fizică. Această lucrare, publicată în prestigioasa serie Applied Mathematical Sciences a editurii Springer, reușește să pună în dialog rigoarea statistică și calculul computațional pentru a rezolva problemele de inversie care apar în contextul datelor experimentale incomplete sau zgomotoase.

Putem afirma că elementul distinctiv al acestui curs este trecerea de la viziunea deterministă clasică — unde erorile sunt tratate ca fiind neglijabile sau exact cunoscute — către un cadru în care eroarea este recunoscută ca o variabilă aleatorie. Găsim în această carte o progresie logică: primele patru capitole fundamentează teoria inversiei cu accent pe aspectele computaționale, în timp ce capitolele 3 și 4 introduc metodele nonstaționare, adresându-se celor care stăpânesc deja bazele clasice. Statistical and Computational Inverse Problems este comparabil cu Inverse Problems de Kazufumi Ito în rigurozitate, dar actualizat pentru tratarea incertitudinii prin prisma inferenței bayesiene și a metodelor statistice moderne.

Structura textului, care include de la regularizarea Tikhonov până la studii de caz detaliate, facilitează o experiență de învățare aplicată. Credem că absența citărilor din interiorul textului principal, acestea fiind grupate în secțiuni dedicate de note la finalul capitolelor, oferă o cursivitate rară pentru un tratat de cercetare, permițând cititorului să se concentreze pe fluxul logic al argumentației matematice.

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Specificații

ISBN-13: 9780387220734
ISBN-10: 0387220739
Pagini: 360
Ilustrații: XVI, 340 p.
Dimensiuni: 160 x 241 x 25 mm
Greutate: 0.65 kg
Ediția:2005
Editura: Springer
Seria Applied Mathematical Sciences

Locul publicării:New York, NY, United States

Public țintă

Research

De ce să citești această carte

Această carte este esențială pentru cercetătorii și studenții la master sau doctorat care doresc să treacă de la teoria matematică abstractă la aplicații practice în inginerie. Cititorul câștigă o înțelegere profundă a modului în care incertitudinea și erorile de măsurare pot fi modelate statistic. Este recomandată celor care au nevoie de un fundament solid în metode bayesiene aplicate problemelor de inversie, oferind un echilibru între teorie și studii de caz.

Descriere scurtă

This book is aimed at postgraduate students in applied mathematics as well as at engineering and physics students with a ?rm background in mathem- ics. The ?rst four chapters can be used as the material for a ?rst course on inverse problems with a focus on computational and statistical aspects. On the other hand, Chapters 3 and 4, which discuss statistical and nonstati- ary inversion methods, can be used by students already having knowldege of classical inversion methods. There is rich literature, including numerous textbooks, on the classical aspects of inverse problems. From the numerical point of view, these books concentrate on problems in which the measurement errors are either very small or in which the error properties are known exactly. In real-world pr- lems, however, the errors are seldom very small and their properties in the deterministic sensearenot wellknown.For example,inclassicalliteraturethe errornorm is usuallyassumed to be a known realnumber. In reality,the error norm is a random variable whose mean might be known.

Cuprins

Inverse Problems and Interpretation of Measurements.- Classical Regularization Methods.- Statistical Inversion Theory.- Nonstationary Inverse Problems.- Classical Methods Revisited.- Model Problems.- Case Studies.

Recenzii

From the reviews:
"The book is devoted to the development of the statistical approach to inverse problems … . The content is written clearly and without citations in the main text. Every chapter has a section called ‘Notes and comments’ where the citations and further reading, as well as brief comments on more advanced topics, are provided. The book is aimed at postgraduate students … . The book also will be of interest for many researchers and scientists working in the area of image processing." (Tzvetan Semerdjiev, Zentralblatt MATH, Vol. 1068, 2005)
"Inverse problems are usually ill-posed in the sense that a solution need not exist, need not be unique, and depends in a discontinuous way on the data … . there have been two quite separate communities dealing with such problems, one basing their methods mainly on functional analysis, the other one on statistics. … several attempts have been made to bridge the gap between these two groups. The book under review … is a further, quite successful attempt in this direction." (Heinz W. Engel, SIAM Review, Vol. 48 (1), 2006)

Textul de pe ultima copertă

The book develops the statistical approach to inverse problems with an emphasis on modeling and computations. The framework is the Bayesian paradigm, where all variables are modeled as random variables, the randomness reflecting the degree of belief of their values, and the solution of the inverse problem is expressed in terms of probability densities. The book discusses in detail the construction of prior models, the measurement noise modeling and Bayesian estimation. Markov Chain Monte Carlo-methods as well as optimization methods are employed to explore the probability distributions. The results and techniques are clarified with classroom examples that are often non-trivial but easy to follow. Besides the simple examples, the book contains previously unpublished research material, where the statistical approach is developed further to treat such problems as discretization errors, and statistical model reduction. Furthermore, the techniques are then applied to a number of real world applications such as limited angle tomography, image deblurring, electrical impedance tomography and biomagnetic inverse problems. The book is intended to researchers and advanced students in applied mathematics, computational physics and engineering. The first part of the book can be used as a text book on advanced inverse problems courses.
The authors Jari Kaipio and Erkki Somersalo are Professors in the Applied Physics Department of the University of Kuopio, Finland and the Mathematics Department at the Helsinki University of Technology, Finland, respectively.