Inverse Problems for Partial Differential Equations
Springer Science & Business Media, 29 Haz 2013 - 286 sayfa
This book describes the contemporary state of the theory and some numerical aspects of inverse problems in partial differential equations. The topic is of sub stantial and growing interest for many scientists and engineers, and accordingly to graduate students in these areas. Mathematically, these problems are relatively new and quite challenging due to the lack of conventional stability and to nonlinearity and nonconvexity. Applications include recovery of inclusions from anomalies of their gravitational fields; reconstruction of the interior of the human body from exterior electrical, ultrasonic, and magnetic measurements, recovery of interior structural parameters of detail of machines and of the underground from similar data (non-destructive evaluation); and locating flying or navigated objects from their acoustic or electromagnetic fields. Currently, there are hundreds of publica tions containing new and interesting results. A purpose of the book is to collect and present many of them in a readable and informative form. Rigorous proofs are presented whenever they are relatively short and can be demonstrated by quite general mathematical techniques. Also, we prefer to present results that from our point of view contain fresh and promising ideas. In some cases there is no com plete mathematical theory, so we give only available results. We do not assume that a reader possesses an enormous mathematical technique. In fact, a moderate knowledge of partial differential equations, of the Fourier transform, and of basic functional analysis will suffice.
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IllPosed Problems and Regularization
Uniqueness and Stability in the Cauchy Problem
Single Boundary Measurements
Many Boundary Measurements
Integral Geometry and Tomography
Diğer baskılar - Tümünü görüntüle
additional algorithm analytic applied assume assumptions boundary measurements boundary value problem Cauchy data Cauchy problem coefficients compact conclude conductivity equation conformal mapping consider constant continuous convergence convex Corollary data g defined depend derivatives differential equation Dirichlet data Dirichlet problem Dirichlet-to-Neumann map domain Q eigenvalues elliptic equation elliptic operator equation 2.0 Exercise formula given gravimetry heat equation Helmholtz equation Hölder hyperbolic equation hyperbolic problem ill-posed problems implies inequality initial boundary value initial data integral equation integral geometry inverse conductivity problem inverse problem Isakov lateral boundary data Lemma linear Lipschitz maximum principle method Neumann data nonlinear norm numerical observe obtain operator paper parabolic equations parabolic problem plane potential proof is complete proof of Theorem Prove uniqueness Radon transform regularization respect right side Section smooth solves space stability estimate subtracting Theorem Theorem 4.1 Uhlmann uniquely determined uniqueness and stability uniqueness results