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August 5, 2011
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We prove logarithmic convergence rate of the Levenberg–Marquardt method in a Hilbert space if a logarithmic source condition is satisfied. This method is applied to an inverse potential problem. Numerical implementations demonstrate the convergence rate.
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August 5, 2011
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In this paper, we consider an inverse time problem for a spherically symmetric heat equation. The problem is ill-posed. A spectral method is applied to formulate a regularized solution which is stably convergent to the exact ones. A quite sharp error estimate for the regularized solution is obtained with suitable choice of regularization parameter.
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August 5, 2011
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In this paper, we present some abstract results giving a general connection between null-controllability and several inverse problems for a class of parabolic equations. We obtain some conditional stability estimates for the inverse problems consisting of determining the initial condition and the source term, from interior or boundary measurements. We apply this framework for Stokes system with interior and boundary observations, for a coupling of two Stokes system and a linear fluid-structure system.
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We introduce new classes of sets extending the class of convex bodies. We show strong inclusions between these classes of bodies. In the case of bodies in Euclidean spaces, we obtain a new characterization of sets with positive reach, prove the Helly type theorem for them, and find applications to geometric tomography. We investigate the problem of determination of sets with positive reach by their projection-type images, and generalize corresponding stability theorems by H. Groemer.
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We report on solving the inverse problem of finding the kernel of an asymptotic singular integral operator under which monomial signals that are compactly supported over the closed unit interval [0, 1 – ] are asymptotic generalized fixed points over the semi-closed unit interval (0, 1]. This operator defines a certain Even-Hilbert Riemann–Lebesgue transformation with a kernel that is double parameterized over a certain momentum Hilbert space. The inversion of this singular transformation is proved to be in the form of an associated Odd-Hilbert Riemann–Lebesgue transformation. The paper contains also proofs for a number of operational properties of this transform, with an identified area for potential applicability in solving certain functional initial-value problems.
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August 5, 2011
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We consider the Gel'fand inverse problem and continue studies of Mandache (Inverse Problems 17: 1435–1444, 2001). We show that the Mandache-type instability remains valid even in the case of Dirichlet-to-Neumann map given on the energy intervals. These instability results show, in particular, that the logarithmic stability estimates of Alessandrini (Appl. Anal. 27: 153–172, 1988), Novikov and Santacesaria (J. Inverse Ill-Posed Probl., 2010) and especially of Novikov (2010) are optimal (up to the value of the exponent).
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A generalization of a simplified form of the continuous regularized Gauss–Newton method has been considered for obtaining stable approximate solutions for ill-posed operator equations of the form F ( x ) = y , where F is a nonlinear operator defined on a subset of a Hilbert space ℋ 1 with values in another Hilbert space ℋ 2 . Convergence of the method for exact data is proved without assuming any specific source condition on the unknown solution. For the case of noisy data, order optimal error estimates based on an a posteriori as well as an a priori stopping rule are derived under a general source condition which includes the classical source conditions such as the Hölder-type and logarithmic type, and certain nonlinearity assumptions on the operator F .
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August 5, 2011
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Theorems of the existence and uniqueness for one class of direct problems are presented. Moreover, an algorithm for the reconstruction of the acoustic impedance is given when the form of the “initial” condition for the direct problem is known.
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This paper is devoted to a class of inverse coefficient problems for parabolic differential equations. These problems play a very important role in many branches of engineering and sciences. Based on a variational iteration technique, we show the rapid convergence of the approximated solution sequence constructed to the exact solution. To show the efficiency of the present method, one example is displayed.
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