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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems (Hardcover, 1st ed. 2017)
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Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems (Hardcover, 1st ed. 2017)
Series: Springer Monographs in Mathematics
Expected to ship within 12 - 19 working days
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This book is a self-contained account of the method based on
Carleman estimates for inverse problems of determining spatially
varying functions of differential equations of the hyperbolic type
by non-overdetermining data of solutions. The formulation is
different from that of Dirichlet-to-Neumann maps and can often
prove the global uniqueness and Lipschitz stability even with a
single measurement. These types of inverse problems include
coefficient inverse problems of determining physical parameters in
inhomogeneous media that appear in many applications related to
electromagnetism, elasticity, and related phenomena. Although the
methodology was created in 1981 by Bukhgeim and Klibanov, its
comprehensive development has been accomplished only recently. In
spite of the wide applicability of the method, there are few
monographs focusing on combined accounts of Carleman estimates and
applications to inverse problems. The aim in this book is to fill
that gap. The basic tool is Carleman estimates, the theory of which
has been established within a very general framework, so that the
method using Carleman estimates for inverse problems is
misunderstood as being very difficult. The main purpose of the book
is to provide an accessible approach to the methodology. To
accomplish that goal, the authors include a direct derivation of
Carleman estimates, the derivation being based essentially on
elementary calculus working flexibly for various equations. Because
the inverse problem depends heavily on respective equations, too
general and abstract an approach may not be balanced. Thus a direct
and concrete means was chosen not only because it is friendly to
readers but also is much more relevant. By practical necessity,
there is surely a wide range of inverse problems and the method
delineated here can solve them. The intention is for readers to
learn that method and then apply it to solving new inverse
problems.
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