**The Carus Mathematical Monographs**

Volume: 32;
2016;
320 pp;
Hardcover

Print ISBN: 978-0-88385-141-8

Product Code: CAR/32

List Price: $63.00

AMS Member Price: $47.25

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**Electronic ISBN: 978-1-61444-029-1
Product Code: CAR/32.E**

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# Linear Inverse Problems and Tikhonov Regularization

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*Mark S. Gockenbach*

MAA Press: An Imprint of the American Mathematical Society

Inverse problems occur frequently in science
and technology, whenever we need to infer causes from effects that we
can measure. Mathematically, they are difficult problems because they
are unstable: small bits of noise in the measurement can completely
throw off the solution. Nevertheless, there are methods for finding
good approximate solutions.

Linear Inverse Problems and Tikhonov
Regularization examines one such method: Tikhonov regularization for
linear inverse problems defined on Hilbert spaces. This is a clear
example of the power of applying deep mathematical theory to solve
practical problems. Beginning with a basic analysis of Tikhonov
regularization, this book introduces the singular value expansion for
compact operators, and uses it to explain why and how the method
works. Tikhonov regularization with seminorms is also analyzed, which
requires introducing densely defined unbounded operators and their
basic properties. Some of the relevant background is included in
appendices, making the book accessible to a wide range of
readers.

# Table of Contents

## Linear Inverse Problems and Tikhonov Regularization

- Cover cov11
- Half title i3
- Copyright ii4
- Title iii5
- Series iv6
- Contents vii9
- Preface xi13
- 1 Introduction to inverse problems 117
- 2 Well-posed, ill-posed, and inverse problems 1531
- 3 Tikhonov regularization 4965
- 3.1 Existence of the Tikhonov solution 5268
- 3.2 Convergence of Tikhonov regularization 6076
- 3.3 Convergence for noisy data 7086
- 3.4 Rates of convergence 7389
- 3.5 Parameter choice rules 84100
- 3.6 Converse results 98114
- 3.6.1 Preliminary results 100116
- 3.6.2 The convergence of Tikhonov regularization can be arbitrarily slow 102118
- 3.6.3 Tikhonov regularization can fail to converge if the regularization parameter is not chosen properly 103119
- 3.6.4 Converse results about the rate of convergence 114130
- 3.6.5 The rate of convergence in the discrepancy principle 118134
- 3.6.6 Summary 121137

- 4 Compact operators and the singular value expansion 123139
- 4.1 Finite-dimensional problems and the singular value decomposition 123139
- 4.2 Compact operators 137153
- 4.3 The spectral theorem for a compact self-adjoint operator 143159
- 4.4 The singular value expansion of a compact operator 163179
- 4.5 The generalized inverse in terms of the SVE 182198
- 4.6 Tikhonov regularization in terms of the SVE 197213
- 4.7 Convergence of Tikhonov regularization via the SVE 202218
- 4.8 General regularization methods 210226

- 5 Tikhonov regularization with seminorms 225241
- Epilogue 271287
- A Basic Hilbert space theory 279295
- B Sobolev spaces 307323
- Bibliography 317333
- Index 319335
- About the Author 321337