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31
A WeylTitchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control
, 2007
"... The paper defines the WeylTitchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator col ..."
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Cited by 9 (1 self)
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The paper defines the WeylTitchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator colligations, and the linear systems theory and suggest an approach to the study of boundary value problems based on the open systems theory due to M. S. Livˇsic. Examples of boundary value problems for partial differential equations and calculations of their WeylTitchmarsh functions are offered as illustration. In particular, we give an independent derivation of the WeylTitchmarsh function for the three dimensional Schrödinger operator introduced by W. O. Amrein and D. B. Pearson. Relationships to the Schrödinger operator with singular potential supported by the unit sphere are clarified and other possible applications of the developed approach in mathematical physics are noted. Introduction and notation One of the mathematical folklore beliefs is the possibility to interpret the WeylTitchmarsh function known in the theory of SturmLiouville equation [40] as a transfer function of some linear
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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The Calderon problem for twodimensional manifolds by the BCmethod
, 2002
"... This paper is a corrected and extended version of the preprint [3]. Our results are the following: ffl we show a relationship between the Calderon problem and Function Algebras and give a new proof of Theorem 1 exploiting this relationship; ffl a simple formula (see (1.6)) linking the DNmap to the ..."
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Cited by 6 (0 self)
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This paper is a corrected and extended version of the preprint [3]. Our results are the following: ffl we show a relationship between the Calderon problem and Function Algebras and give a new proof of Theorem 1 exploiting this relationship; ffl a simple formula (see (1.6)) linking the DNmap to the Euler characteristic of the manifold is derived
Dynamical inverse problem for the Schrödinger equation (the BCmethod)
, 2002
"... this paper we use important results by R. Triggiani and P.F. Yao [12] on the exact boundary controllability of the Schrodinger system ..."
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Cited by 3 (1 self)
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this paper we use important results by R. Triggiani and P.F. Yao [12] on the exact boundary controllability of the Schrodinger system
Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm, Inverse Problems, accepted for publication
, 2010
"... The validity of a synthesis of a globally convergent numerical method with the adaptive FEM technique for a coefficient inverse problem is verified on time resolved experimental data. Refractive indices, locations and shapes of dielectric abnormalities are accurately imaged. 1 ..."
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Cited by 3 (3 self)
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The validity of a synthesis of a globally convergent numerical method with the adaptive FEM technique for a coefficient inverse problem is verified on time resolved experimental data. Refractive indices, locations and shapes of dielectric abnormalities are accurately imaged. 1
Inverse scattering on matrices with boundary conditions
 J. Phys. A: Math. Gen
"... We describe inverse scattering for the matrix Schrödinger operator with general selfadjoint boundary conditions at the origin using the Marchenko equation. Our approach allows the recovery of the potential as well as the boundary conditions. It is easily specialised to inverse scattering on starsha ..."
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Cited by 3 (0 self)
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We describe inverse scattering for the matrix Schrödinger operator with general selfadjoint boundary conditions at the origin using the Marchenko equation. Our approach allows the recovery of the potential as well as the boundary conditions. It is easily specialised to inverse scattering on starshaped graphs with boundary conditions at the node. 1
Boundary control and inverse problem for the dynamical Maxwell system: the recovering of velocity in regular zone.
, 1998
"... The paper deals with an approach to the Inverse Problems based upon their relations to the Boundary Control Theory (the BCmethod). A possibility to recover a velocity c = (") \Gamma1=2 via response operator of the Maxwell dynamical system is discussed. The main result is that the operator (plus v ..."
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Cited by 2 (2 self)
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The paper deals with an approach to the Inverse Problems based upon their relations to the Boundary Control Theory (the BCmethod). A possibility to recover a velocity c = (") \Gamma1=2 via response operator of the Maxwell dynamical system is discussed. The main result is that the operator (plus values of c; @c @ at a boundary) determines velocity uniquely in the maximal nearboundary layer in which the semigeodesical coordinates are regular. The recovering is timeoptimal: the response operator given for time 2T determines a velocity in nearboundary layer of optical thickness T lying in the regular zone. 0 Introduction Let\Omega ae R 3 be a bounded domain with smooth boundary \Gamma; "; be smooth positive functions (permittivities) in\Omega ; function c := (") \Gamma1=2 is a velocity. Consider the Maxwell system: "e t = rot h; h t = \Gammarot e in\Omega \Theta (0; T ); ej t=0 = 0; hj t=0 = 0 in \Omega\Gamma e ` j \Gamma\Theta[0;T ] = f where (\Delta) ` is a tangential ...
A globally accelerated numerical method for optical tomography with continuous wave source
 J. Inverse IllPosed Probl
"... A new numerical method for an inverse problem for an elliptic equation with unknown potential is proposed. In this problem the point source is running along a straight line and the sourcedependent Dirichlet boundary condition is measured as the data for the inverse problem. A rigorous convergence a ..."
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Cited by 2 (2 self)
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A new numerical method for an inverse problem for an elliptic equation with unknown potential is proposed. In this problem the point source is running along a straight line and the sourcedependent Dirichlet boundary condition is measured as the data for the inverse problem. A rigorous convergence analysis shows that this method converges globally, provided that the socalled tail function is approximated well. This approximation is verified in numerical experiments, so as the global convergence. Applications to medical imaging, imaging of targets on battlefields and to electrical impedance tomography are discussed. 1
FORWARD AND INVERSE SCATTERING ON MANIFOLDS WITH ASYMPTOTICALLY CYLINDRICAL ENDS
, 905
"... Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange perturbation of the metric of the form (dy) 2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codi ..."
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Cited by 2 (0 self)
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Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange perturbation of the metric of the form (dy) 2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to (dy) 2 + h(x, dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energy, we show that these two manifolds are isometric. 1.
A GLOBALLY CONVERGENT NUMERICAL METHOD FOR SOME COEFFICIENT INVERSE PROBLEMS WITH RESULTING SECOND ORDER ELLIPTIC EQUATIONS
"... Abstract. A new globally convergent numerical method is developed for some multidimensional Coefficient Inverse Problems for hyperbolic and parabolic PDEs with applications in acoustics, electromagnetics and optical medical imaging. On each iterative step the Dirichlet boundary value problem for a s ..."
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Cited by 2 (2 self)
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Abstract. A new globally convergent numerical method is developed for some multidimensional Coefficient Inverse Problems for hyperbolic and parabolic PDEs with applications in acoustics, electromagnetics and optical medical imaging. On each iterative step the Dirichlet boundary value problem for a second order elliptic equation is solved. The global convergence is rigorously proven and numerical experiments are presented. 1.