R-pseudoinverses for compact operators in Hilbert spaces: existence and stability

Cheverda, V. A.; Kostin, V. I.

doi:10.1515/jiip.1995.3.2.131

Cited by 28 publications

(18 citation statements)

References 5 publications

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“…If one keeps r singular values the inverse is called r-pseudoinverse and the solution obtained this way is called the r-solution Cheverda & Kostin (1995). Here we will refer to particular results from Cheverda & Kostin (1995) (not only matrices).…”

Section: Appendix C: On Stability Of Inversion By Svd Trancationmentioning

confidence: 99%

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Waveform inversion for orthorhombic anisotropy with P waves: feasibility and resolution

Kazei

Alkhalifah

2018

Geophysical Journal International

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Summary Various parameterizations have been suggested to simplify inversions of first arrivals, or P −waves, in orthorhombic anisotropic media, but the number and type of retrievable parameters have not been decisively determined. We show that only six parameters can be retrieved from the dynamic linearized inversion of P −waves. These parameters are different from the six parameters needed to describe the kinematics of P −waves. Reflection-based radiation patterns from the P − P scattered waves are remapped into the spectral domain to allow for our resolution analysis based on the effective angle of illumination concept. Singular value decomposition of the spectral sensitivities from various azimuths, offset coverage scenarios, and data bandwidths allows us to quantify the resolution of different parameterizations, taking into account the signal-to-noise ratio in a given experiment. According to our singular value analysis, when the primary goal of inversion is determining the velocity of the P −waves, gradually adding anisotropy of lower orders (isotropic, vertically transversally isotropic, orthorhombic) in hierarchical parameterization is the best choice. Hierarchical parametrization reduces the tradeoff between the parameters and makes gradual introduction of lower anisotropy orders straightforward. When all the anisotropic parameters affecting P −wave propagation need to be retrieved simultaneously, the classic parameterization of orthorhombic medium with elastic stiffness matrix coefficients and density is a better choice for inversion. We provide estimates of the number and set of parameters that can be retrieved from surface seismic data in different acquisition scenarios. To set up an inversion process, the singular values determine the number of parameters that can be inverted and the resolution matrices from the parameterizations can be used to ascertain the set of parameters that can be resolved.

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Section: Appendix C: On Stability Of Inversion By Svd Trancationmentioning

confidence: 99%

“…Here we will refer to particular results from Cheverda & Kostin (1995) (not only matrices). Moreover Cheverda & Kostin (1995) show that the relative perturbation δm est of the r-solution is bounded in the following way:…”

Section: Appendix C: On Stability Of Inversion By Svd Trancationmentioning

confidence: 99%

Waveform inversion for orthorhombic anisotropy with P waves: feasibility and resolution

Kazei

Alkhalifah

2018

Geophysical Journal International

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“…Proximity of r-solutions of the initial equation and the perturbed equation may be estimated by means of the following theorem (see, [9]). …”

Section: N Dzmentioning

confidence: 99%

“…It consists in truncating of singular value decomposition (SVD) of a compact operator (for details see [9]). We believe that its main advantage is a possibility to investigate the stability, information contents and resolving ability of the Born inversion for an arbitrary vertically inhomogeneous containing medium.…”

Section: Introductionmentioning

confidence: 99%

Linearized inversion of data of multi-offset data for vertically inhomogeneous background

Cheverda¹,

Clément²,

Khaidukov³

et al. 1998

Jiip

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One of the most widespread problems in seismology is the necessity to adjust some a priori given medium structure. As a rule, such a priori information is vertically inhomogeneous component of wave propagation velocity. Such theoretical aspects of the problem as its uniqueness estimates of conditional stability are studied rather well. There axe also a variety of algorithms for its numerical solution. Therefore, in this paper the main attention is paid to numerical analysis of resolving ability and information contents of linearized inversion of multi-offset data.*Domaine de Voluceau, The work was supported by RFBR grants No. 96-01-01515 and 97-05-65280; and by the France-Russian Center of Appl. Math, and Inf. grant "Reliable numerical methods of solutions of nonlinear continually correct problems in application to inverse problems of wave propagation theory". Brought to you by | University of Arizona Authenticated Download Date | 5/31/15 6:15 AM * As we shall see from the presentation below, the case of half-space z > 0 does not have serious changes except the calculations which are more tedious. Brought to you by | University of Arizona Authenticated Download Date | 5/31/15 6:15 AM

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“…В первом разделе содержится математическая постановка обратной задачи, проводится построение оператора чувствительности и алгоритма ньютоновского типа на основе r-псевдообратных операторов [8,9] увеличивающейся размерности для построенных операторов чувствительности. Во втором разделе представлены результаты численных экспериментов.…”

Section: Introductionunclassified

Numerical Algorithms for Diffusion Coefficient Identification in Problems of Tissue Engineering

Penenko¹

2016

Math.Biol.Bioinf.

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ВВЕДЕНИЕПрактически все разрабатываемые технологии тканевой инженерии включают этап приготовления заготовки импланта определенной формы из подходящего материала. Существенной характеристикой такого материала является его пористость -наличие развитой системы каналов с диаметрами от долей до сотен микрон. По каналам, заполненным питательным раствором, могут мигрировать клетки и распространяться химические вещества. Заготовка заселяется клетками и полученный протоимплант инкубируется в специально подобранных условиях с целью получения тканеинженерного импланта [1,2]. Естественно ожидать, что транспортные процессы диффузионного типа играют решающую роль в тканевой инженерии. От них зависит скорость заселения заготовок имплантов клетками за счет спонтанной подвижности 1 a.penenko@yandex.ru 2 nikolaev@bionet.nsc.ru 3 prorector1@nsu.ru

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R-pseudoinverses for compact operators in Hilbert spaces: existence and stability

Cited by 28 publications

References 5 publications

Waveform inversion for orthorhombic anisotropy with P waves: feasibility and resolution

Waveform inversion for orthorhombic anisotropy with P waves: feasibility and resolution

Linearized inversion of data of multi-offset data for vertically inhomogeneous background

Numerical Algorithms for Diffusion Coefficient Identification in Problems of Tissue Engineering

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