Exponential Data Fitting and its Applications


by

Victor Pereyra, Godela Scherer

DOI: 10.2174/97816080504821100101
eISBN: 978-1-60805-048-2, 2010
ISBN: 978-1-60805-345-2



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Indexed in: Scopus

Real and complex exponential data fitting is an important activity in many different areas of science and engineering, ranging from Nu...[view complete introduction]

Table of Contents

Foreword

- Pp. i

Michael Saunders

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Preface

- Pp. ii-iii (2)

Victor Pereyra, Mountain View and Godela Scherer

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Contributors

- Pp. iv-vi (3)

V. Pereyra and G. Scherer

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Exponential data fitting

- Pp. 1-26 (26)

Victor Pereyra and Godela Scherer

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Computational aspects of exponential data fitting in Magnetic Resonance Spectroscopy

- Pp. 27-51 (25)

Diana M. Sima, Jean-Baptiste Poullet and Sabine Van Huffel

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Recovery of relaxation rates in MRI T2-weighted brain images via exponential fitting

- Pp. 52-70 (19)

Marco Paluszny, Marianela Lentini, Miguel Martin-Landrove, Wuilian Torres and Rafael Martin

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Exponential time series in lattice quantum field theory

- Pp. 71-93 (23)

Saul D. Cohen, George T. Fleming and Huey-Wen Lin

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Solving separable nonlinear least squares problems with multiple datasets

- Pp. 94-109 (16)

Linda Kaufman

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Sum-of-exponentials models for time-resolved spectroscopy data

- Pp. 110-127 (18)

Katharine M. Mullen and Ivo H. M. van Stokkum

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Two exponential models for optically stimulated luminescence

- Pp. 128-144 (17)

Per Christian Hansen, Hans Bruun Nielsen, Christina Ankjærgaard and Mayank Jain

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Modelling type Ia supernova light curves

- Pp. 145-164 (20)

Bert W. Rust, Dianne P. O’Leary and Katharine M. Mullen

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Accurate calculations of the high-frequency impedance matrix for VLSI interconnects and inductors above a multi-layer substrate:

- Pp. 165-192 (28)

Navin Srivastava, Roberto Suaya and Victor Pereyra

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Index

- Pp. 193-195 (3)

V. Pereyra and G. Scherer

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Foreword

Exponential data fitting has a long history theoretically, algorithmically, and in its applications. Despite its widespread use and success there has been no unified presentation of the subject available. This book attempts to fill the gap. The problem and some of the most successful and proven algorithms for its solution are presented in detail. Then a number of experts in different fields describe their applications and specific models. This combination will alert many other researchers to what is out there and how their problems relate to various applications across disciplines. The editors have done a commendable job in assembling this material. They are experts in the field. We thank them for sharing their knowledge.

Michael Saunders
Stanford University


Preface

Exponential approximations are ubiquitous in scientific computing. There are many reasons for their success in modeling various phenomena. Processes that are modeled by linear differential equations with constant coefficients and Fourier expansions are just two such reasons. In particular, wave propagation problems in their many expressions are heavily represented because wave fronts in the far field are plane, and plane waves are described by complex exponentials.

In this volume we consider the least squares approximation of data by linear combinations of real or complex exponentials. In the first chapter, Pereyra and Scherer discuss the problem from a mathematical and numerical point of view, giving an historical perspective and detailed description of some of the most successful and frequently used methods. The Variable Projection (VP) method, introduced in 1973 by Golub and Pereyra, extending the initial contribution of Gutman, Pereyra and Scolnik, is carefully discussed as the nonlinear least squares method of choice. Prony’s method and its more numerically stable variants are also discussed, as they are also frequently used in many applications. Some comparisons and discussion of advantages and shortcomings are included.

Various applications are then considered by experts in their respective fields. A common thread in these chapters is a fairly comprehensive presentation of the physical problem, then a discussion of the methods used by that particular community. Among these methods we find prominently the ones described in Chapter 1 and some of their variations.

In Chapter 2, Sima, Poullet and Van Huffel consider the problem of fitting Nuclear Magnetic Resonance (NMR) spectroscopic data from in vivo and in vitro biological specimens. They include a particularly detailed description of the NMR phenomenon and give important information on data preprocessing that experience through the years has indicated is necessary for handling the various sources of noise and complications inherent in this type of data.

The multi-disciplinary team of Paluzny, Lentini, M. Martin, R. Martin and Torres in Chapter 3 go into a different application of NMR, namely Imaging, as it pertains to brain tumor segmentation and identification.

Leaving Biology and Medicine we jump to High Energy Physics in Chapter 4, where Cohen, Fleming and Lin consider an application in Lattice Quantum Field Theory. Historically it is interesting to find that Fleming discovered some of the tools used in this approach from studying the NMR literature.

We then switch to a more generic algorithmic contribution in Chapter 5 by Kaufman, which extends the Variable Projection techniques to problems with multiple data sets, where a common set of nonlinear parameters is desired and where careful implementation can produce large savings in storage and computing time.

In Chapter 6, Mullen and van Stokkum consider in detail the problem of Time- Resolved Spectroscopy data. For multi-spectral data, tensor approximations are a natural tool and we find the type of problems considered by Kaufman in the previous chapter.

Hansen, Nielsen, Ankj{\ae}rgaard, & Jain in Chapter 7 attack the modeling of Optically Stimulated Luminescense by linear combinations of exponentials, sharing their experience and comparing two different models and two different techniques: VP and a Fredholm integral equation of the first kind approach.

In Chapter 8, Rust, O’Leary and Mullen study the modeling of Supernova Light curves using exponentials.

Finally, in Chapter 9, Srivastava, Suaya, Pereyra and Banerjee consider the application of exponential approximations to problems of electromagnetic wave propagation in modern chip design technology.

It is remarkable how in this apparently disparate ensemble of applications, rather few numerical algorithms have proven to be the tools of choice. Thus we hope that other practitioners will benefit from the insights given in this book and find that the same tools are useful in their own fields of application.

Taking advantage of the electronic nature of this book, some of the authors have added unconventional material that can be downloaded. Hyperlinks and brief descriptions are given in appendices to the chapters. This material includes computer programs, scripts, and data that will allow reproduction of some of the results. In general, colored boxes around some material indicate active links to Chapters, Sections, figures, cross-references, citations, external URL’s or to the above mentioned additional materials.

We would like to thank all who have made it possible to complete this collaborative effort, and most specially Per Christian Hansen for his help in editing several of the chapters.

Victor Pereyra, Mountain View, CA
Godela Scherer, Reading, GB
September 2009

List of Contributors

Editor(s):
Victor Pereyra
San Diego State University
USA


Godela Scherer
University of Reading
UK




Contributor(s):
Christina Ankjærgaard
Radiation Research Division, Riso National Laboratory for Sustainable Energy, Technical University of Denmark
Roskilde, DK-4000
Denmark


Kaustav Banerjee
Department of Electrical and Computer Engineering
University of California
Santa Barbara
CA, 93106
USA


Saul D. Cohen
Thomas Jefferson National Accelerator Facility Newport News
VA, 23606
USA


George T. Fleming
Department of Physics
Yale University
New Haven
CT, 06520
USA


Per Christian Hansen
Department of Informatics and Mathematical Modelling
Technical University of Denmark Kgs
Lyngby, DK-2800
Denmark


Mayank Jain
Radiation Research Division, Riso National Laboratory for Sustainable Energy, Technical University of Denmark
Roskilde, DK-4000
Denmark


Linda Kaufman
Computer Science Dept
William Patterson University
Coach House, Room 213
Wayne
NJ
USA


Marianela Lentini
Escuela de Matematicas, Universidad Nacional de Colombia, Sede Medellin
Colombia


Huey-Wen Lin
Department of Physics
University of Washington
Seattle
WA, 98195
USA


Rafael Martín
Centro de Fisica Molecular y Medica Escuela de Fisica, Facultad de Ciencias , Universidad Central de Venezuela
Caracas
Venezuela


Miguel Martín-Landrove
Centro de Fisica Molecular y Medica, Escuela de Fisica Facultad de Ciencias Universidad Central de Venezuela y Centro de Diagnóstico Docente Las Mercedes
Caracas
Venezuela


Katharine M. Mullen
Ceramics Division National Institute of Standards and Technology (NIST)
100 Bureau Drive, M/S 8520
Gaithersburg
MD, 20899
USA


Dianne P. O'Leary
Computer Science Department
Institute for Advanced Computer Studies University of Maryland, College Park, MD 20742; and National Institute of Standards and Technology
Gaithersburg
MD, 20899-8910



Hans Bruun Nielsen
Department of Informatics and Mathematical Modelling
Technical University of Denmark
DK-2800 Kgs
Lyngby
Denmark


Marco Paluszny
Escuela de Matematicas Universidad Nacional de Colombia
Medellín
Colombia


Victor Pereyra
Weidlinger Associates Inc. (retired)
399 W. El Camino Real, #200 Mountain View, CA , USA
CA, 94040
USA
/
Computational Sciences Research Institute,San Diego State University
San Diego
CA
USA


Jean-Baptiste Poullet
Formerly with Department of Electrical Engineering ESAT-SCD
Katholieke Universiteit Leuven
Kasteelpark Arenberg 10, 3001 Leuven-Heverlee
Belgium


Bert W. Rust
Mathematical and Computational Sciences Division National Institute of Standards and Technology (NIST)
100 Bureau Drive, MS 8910
Gaithersburg
MD, 20899-8910
USA


Godela Scherer
Mathematics Department
University of Reading
UK


Diana M. Sima
Department of Electrical Engineering ESAT-SCD
Katholieke Universiteit Leuven Kasteelpark Arenberg 10
Leuven-Heverlee, 3001
Belgium


Navin Srivastava
Mentor Graphics Corporation
8005 SW Boeckman Rd
Wilsonville
Oregon, 97070
USA


Roberto Suaya
Mentor Graphics Corporation
110 rue Blaise Pascal
St Ismier Cedex, 38334
France


Wuilian Torres
Centro de Procesamiento Digital de Imagenes, Instituto de Ingenieria y, Laboratorio de Computacion Grafica y Geometria Aplicada, Escuela de Matematica, Facultad de Ciencias,Universidad Central de Venezuela
Caracas
Venezuela


Sabine Van Huffel
Department of Electrical Engineering ESAT-SCD
Katholieke Universiteit Leuven
Kasteelpark Arenberg 10, 3001 Leuven-Heverlee
Sabine
Belgium


Ivo H. M. van Stokkum
Department of Physics and Astronomy
Faculty of Sciences, Vrije Universiteit Amsterdam, de Boelelaan 1081
1081 HV Amsterdam
The Netherlands




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