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Author: Huaian Diao
Publisher: Springer Nature
ISBN: 3031346157
Category : Mathematics
Languages : en
Pages : 388
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Book Description
Inverse scattering problems are a vital subject for both theoretical and experimental studies and remain an active field of research in applied mathematics. This book provides a detailed presentation of typical setup of inverse scattering problems for time-harmonic acoustic, electromagnetic and elastic waves. Moreover, it provides systematical and in-depth discussion on an important class of geometrical inverse scattering problems, where the inverse problem aims at recovering the shape and location of a scatterer independent of its medium properties. Readers of this book will be exposed to a unified framework for analyzing a variety of geometrical inverse scattering problems from a spectral geometric perspective. This book contains both overviews of classical results and update-to-date information on latest developments from both a practical and theoretical point of view. It can be used as an advanced graduate textbook in universities or as a reference source for researchers in acquiring the state-of-the-art results in inverse scattering theory and their potential applications.
Author: Huaian Diao
Publisher: Springer Nature
ISBN: 3031346157
Category : Mathematics
Languages : en
Pages : 388
Get Book Here
Book Description
Inverse scattering problems are a vital subject for both theoretical and experimental studies and remain an active field of research in applied mathematics. This book provides a detailed presentation of typical setup of inverse scattering problems for time-harmonic acoustic, electromagnetic and elastic waves. Moreover, it provides systematical and in-depth discussion on an important class of geometrical inverse scattering problems, where the inverse problem aims at recovering the shape and location of a scatterer independent of its medium properties. Readers of this book will be exposed to a unified framework for analyzing a variety of geometrical inverse scattering problems from a spectral geometric perspective. This book contains both overviews of classical results and update-to-date information on latest developments from both a practical and theoretical point of view. It can be used as an advanced graduate textbook in universities or as a reference source for researchers in acquiring the state-of-the-art results in inverse scattering theory and their potential applications.
Author: Khosrow Chadan
Publisher: SIAM
ISBN: 0898713870
Category : Mathematics
Languages : en
Pages : 206
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Book Description
Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems.
Author: David Borthwick
Publisher: Birkhäuser
ISBN: 3319338773
Category : Mathematics
Languages : en
Pages : 471
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Book Description
This text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. For the second edition the context has been extended to general surfaces with hyperbolic ends, which provides a natural setting for development of the spectral theory while still keeping technical difficulties to a minimum. All of the material from the first edition is included and updated, and new sections have been added. Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function. The new sections cover the latest developments in the field, including the spectral gap, resonance asymptotics near the critical line, and sharp geometric constants for resonance bounds. A new chapter introduces recently developed techniques for resonance calculation that illuminate the existing results and conjectures on resonance distribution. The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory. This book will serve as a valuable resource for graduate students and researchers from these and other related fields. Review of the first edition: "The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed...The book gathers together some material which is not always easily available in the literature...To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader...would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)
Author: Fioralba Cakoni
Publisher: SIAM
ISBN: 1611974461
Category : Mathematics
Languages : en
Pages : 200
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Book Description
Inverse scattering theory is a major theme of applied mathematics, and it has applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting particular problems in the development of efficient inversion algorithms. Although linearized models continue to play an important role in many applications, an increased need to focus on problems in which multiple scattering effects cannot be ignored has led to a central role for nonlinearity, and the possibility of collecting large amounts of data over limited regions of space means that the ill-posed nature of the inverse scattering problem has become a problem of central importance.? Initial efforts to address the nonlinear and the ill-posed nature of the inverse scattering problem focused on nonlinear optimization methods. While efficient in many situations, strong a priori information is necessary for their implementation. This problem led to a qualitative approach to inverse scattering theory in which the amount of a priori information is drastically reduced, although at the expense of only obtaining limited information about the values of the constitutive parameters. This qualitative approach (the linear sampling method, the factorization method, the theory of transmission eigenvalues, etc.) is the theme of Inverse Scattering Theory and Transmission Eigenvalues.? The authors begin with a basic introduction to the theory, then proceed to more recent developments, including a detailed discussion of the transmission eigenvalue problem; present the new generalized linear sampling method in addition to the well-known linear sampling and factorization methods; and in order to achieve clarification of presentation, focus on the inverse scattering problem for scalar homogeneous media.?
Author: Jingzhi Li
Publisher: Springer Nature
ISBN: 9819937728
Category : Science
Languages : en
Pages : 373
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Book Description
This book highlights the latest developments on the numerical methods for inverse scattering problems associated with acoustic, electromagnetic, and elastic waves. Inverse scattering problems are concerned with identifying unknown or inaccessible objects by wave probing data, which makes possible many industrial and engineering applications including radar and sonar, medical imaging, nondestructive testing, remote sensing, and geophysical exploration. The mathematical study of inverse scattering problems is an active field of research. This book presents a comprehensive and unified mathematical treatment of various inverse scattering problems mainly from a numerical reconstruction perspective. It highlights the collaborative research outputs by the two groups of the authors yet surveys and reviews many existing results by global researchers in the literature. The book consists of three parts respectively corresponding to the studies on acoustic, electromagnetic, and elastic scattering problems. In each part, the authors start with in-depth theoretical and computational treatments of the forward scattering problems and then discuss various numerical reconstruction schemes for the associated inverse scattering problems in different scenarios of practical interest. In addition, the authors provide an overview of the existing results in the literature by other researchers. This book can serve as a handy reference for researchers or practitioners who are working on or implementing inverse scattering methods. It can also serve as a graduate textbook for research students who are interested in working on numerical algorithms for inverse scattering problems.
Author: Vesselin M. Petkov
Publisher: John Wiley & Sons
ISBN: 1119107660
Category : Mathematics
Languages : en
Pages : 428
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Book Description
This book is a new edition of a title originally published in1992. No other book has been published that treats inverse spectral and inverse scattering results by using the so called Poisson summation formula and the related study of singularities. This book presents these in a closed and comprehensive form, and the exposition is based on a combination of different tools and results from dynamical systems, microlocal analysis, spectral and scattering theory. The content of the first edition is still relevant, however the new edition will include several new results established after 1992; new text will comprise about a third of the content of the new edition. The main chapters in the first edition in combination with the new chapters will provide a better and more comprehensive presentation of importance for the applications inverse results. These results are obtained by modern mathematical techniques which will be presented together in order to give the readers the opportunity to completely understand them. Moreover, some basic generic properties established by the authors after the publication of the first edition establishing the wide range of applicability of the Poison relation will be presented for first time in the new edition of the book.
Author: Pierre H. Berard
Publisher: Springer
ISBN: 3540409580
Category : Mathematics
Languages : en
Pages : 284
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Book Description
Author: Hiroshi Isozaki
Publisher: American Mathematical Soc.
ISBN: 0821834215
Category : Mathematics
Languages : en
Pages : 258
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Book Description
This volume grew out of a workshop on spectral theory of differential operators and inverse problems held at the Research Institute for Mathematical Sciences (Kyoto University). The gathering of nearly 100 participants at the conference suggests the increasing interest in this field of research. The focus of the book is on spectral theory for differential operators and related inverse problems. It includes selected topics from the following areas: electromagnetism, elasticity, the Schrodinger equation, differential geometry, and numerical analysis. The material is suitable for graduate students and researchers interested in inverse problems and their applications.
Author: Stig I. Andersson
Publisher: Birkhäuser
ISBN: 3034889380
Category : Mathematics
Languages : en
Pages : 202
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Book Description
Most polynomial growth on every half-space Re (z) ::::: c. Moreover, Op(t) depends holomorphically on t for Re t> O. General references for much of the material on the derivation of spectral functions, asymptotic expansions and analytic properties of spectral functions are [A-P-S] and [Sh], especially Chapter 2. To study the spectral functions and their relation to the geometry and topology of X, one could, for example, take the natural associated parabolic problem as a starting point. That is, consider the 'heat equation': (%t + p) u(x, t) = 0 { u(x, O) = Uo(x), tP which is solved by means of the (heat) semi group V(t) = e- ; namely, u(·, t) = V(t)uoU· Assuming that V(t) is of trace class (which is guaranteed, for instance, if P has a positive principal symbol), it has a Schwartz kernel K E COO(X x X x Rt, E* ®E), locally given by 00 K(x, y; t) = L>-IAk(~k ® 'Pk)(X, y), k=O for a complete set of orthonormal eigensections 'Pk E COO(E). Taking the trace, we then obtain: 00 tA Op(t) = trace(V(t)) = 2::>- k. k=O Now, using, e. g., the Dunford calculus formula (where C is a suitable curve around a(P)) as a starting point and the standard for malism of pseudodifferential operators, one easily derives asymptotic expansions for the spectral functions, in this case for Op.
Author: Pavel Kurasov
Publisher: Springer Nature
ISBN: 3662678721
Category : Science
Languages : en
Pages : 644
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Book Description
This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph. The book has two central themes: the trace formula and inverse problems. The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book. To solve the inverse problem for Schrödinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions. The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.
