On Approximation of Inverse Problems for Abstract Hyperbolic
        Equations

Orlovsky, Dmitry; Piskarev, Sergey; Spigler, Renato

doi:10.11650/twjm/1500405911

Cited by 5 publications

(4 citation statements)

References 13 publications

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“…An exception is (Fernandez-Berdaguer et al, 1996), in which (results which imply) a special case of Theorem 2 is established. A large number of articles have appeared on abstract first-or second-order hyperbolic equations and related inverse problems, for example (Lavrentiev et al, 1986;Choulli, 1991;Lorenzi and Ramm, 2001;Ramm and Koshkin, 2001;Awawdeh, 2010;Orlovsky et al, 2010), mostly using the theory of strongly continuous semigroups (or cosine operators, in the second order case) to obtain a hold on well-posedness. Of these, the closest in spirit to the present work is that of Orlovsky et al (2010), which treats second order systems via the cosine operator approach but observes that smoothness in time of data implies that weak solutions are strong solutions, analogous to a crucial intermediate result in our work.…”

Section: Denote By σmentioning

confidence: 99%

“…A large number of articles have appeared on abstract first-or second-order hyperbolic equations and related inverse problems, for example Lavrentiev et al (1986), Choulli (1991), Lorenzi and Ramm (2001), Ramm and Koshkin (2001), Awawdeh (2010), Orlovsky et al (2010), mostly using the theory of strongly continuous semigroups (or cosine operators, in the second order case) to obtain a hold on well posedness. Of these, the closest in spirit to this work is that of Orlovsky et al (2010), which treats second order systems via the cosine operator approach but observes that smoothness in time of data implies that weak solutions are strong solutions, analogous to a crucial intermediate result in our work. Choulli (1991) established that a first-order abstract integro-differential initial value problem, resembling those of our abstract framework to be detailed in the next section, is well posed, and also proved uniqueness for the solution of a related inverse problem.…”

Section: Prior Artmentioning

confidence: 99%

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A mathematical framework for inverse wave problems in heterogeneous media

Blazek¹,

Stolk²,

Symes³

2013

Inverse Problems

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This paper provides a theoretical foundation for some common formulations of inverse problems in wave propagation, based on hyperbolic systems of linear integro-differential equations with bounded and measurable coefficients. The coefficients of these time-dependent partial differential equations respresent parametrically the spatially varying mechanical properties of materials. Rocks, manufactured materials, and other wave propagation environments often exhibit spatial heterogeneity in mechanical properties at a wide variety of scales, and coefficient functions representing these properties must mimic this heterogeneity. We show how to choose domains (classes of nonsmooth coefficient functions) and data definitions (traces of weak solutions) so that optimization formulations of inverse wave problems satisfy some of the prerequisites for application of Newton's method and its relatives. These results follow from the properties of a class of abstract first-order evolution systems, of which various physical wave systems appear as concrete instances. Finite speed of propagation for linear waves with bounded, measurable mechanical parameter fields is one of the by-products of this theory. acoustics or elastodynamics, this property implies that appropriate traces of solutions are welldefined, provided that the right-hand-side or source term is minimally smooth in time (only!).Existence, uniqueness, and regularity of the solution as function of problem data -both right-hand side and (operator) coefficients -also follows from this fact.The abstract theory accomplishes even more than that, as it applies to a much wider range of dynamics than those commonly occurring in continuum mechanics, accommodating first-order systems with operator coefficients. This added generality creates no additional difficulty for the basic theory. It is in fact very useful: as will be explained in the Discussion section, it justifies certain infeasible-model methods for inverse problems in wave propagation.Another useful by-product of the theory is the finite speed of propagation property for hyperbolic systems with bounded, measurable coefficients, a result which so far as we can tell is new. This follows from the similar property for systems with smooth coefficients via the continuous dependence of solutions on the coefficients in the sense of convergence in measure.Precise statements of the main results for symmetric hyperbolic systems (1) appear in the next section, followed by a brief discussion of the related literature. The third section contains the definition of the class of abstract systems studied in this paper, and statements of the main results to be established concerning it. Proofs of these results follow in the next three sections.The seventh section shows how symmetric hyperbolic systems fit into the abstract framework, and contains proofs of the main theorems stated in the second section. The eighth section treats the hyperbolic case of linear viscoelasticity as an instance of the theory developed earlier. We end with a disc...

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Section: Denote By σmentioning

confidence: 99%

Section: Prior Artmentioning

confidence: 99%

A mathematical framework for inverse wave problems in heterogeneous media

Blazek¹,

Stolk²,

Symes³

2013

Inverse Problems

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“…Therefore, certain additional conditions that allows us to reduce problem (10) to a first order system without use of the space 1 are of interest [24].…”

Section: Definition 4 Function (⋅)mentioning

confidence: 99%

Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

Ashyralyev

Pastor

Piskarev

et al. 2015

Abstract and Applied Analysis

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“…The role played by coercive inequalities in the study of local boundary value problems for elliptic and parabolic differential equations is well-known [2][3][4]. Theory, applications and methods of solutions of BitsadzeSamarskii type nonlocal boundary value problems for elliptic differential equations have been studied extensively by many researchers [5][6][7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of a third order of accuracy difference scheme for Bitsadze-Samarskii type multi-point NBVPs

Ashyralyev¹,

Tetikoğlu²

2014

AIP Conference Proceedings

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A multi-point nonlocal boundary value problem for a multidimensional elliptic equation with Bitsadze-Samarskii condition is considered. A third order of accuracy stable difference scheme for the approximate solution of this problem is presented. The stability, the almost coercive stability and the coercive stability estimates for the solution of this difference scheme are obtained. A numerical analysis is given to justify the theoretical part.

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On Approximation of Inverse Problems for Abstract Hyperbolic Equations

Cited by 5 publications

References 13 publications

A mathematical framework for inverse wave problems in heterogeneous media

A mathematical framework for inverse wave problems in heterogeneous media

Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

Well-posedness of a third order of accuracy difference scheme for Bitsadze-Samarskii type multi-point NBVPs

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