Conjugate Decomposition and Its Applications

Wang, Liping; Yuan, Jinyun

doi:10.1007/s40305-013-0008-9

Cited by 2 publications

(2 citation statements)

References 17 publications

(19 reference statements)

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“…( 18)), as can be seen in Figure 8c. Furthermore, the corresponding posterior covariance can be estimated by a low-rank representation, where the rank is defined by the number of (inner) CG iterations (Wang & Yuan 2013).…”

Section: Efficient Implementation Of the Proposed Vb Schemementioning

confidence: 99%

A reduced-order variational Bayesian approach for efficient subsurface imaging

Urozayev

Ait‐El‐Fquih

Hoteit

et al. 2021

Geophysical Journal International

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Summary This work considers the reconstruction of a subsurface model from seismic observations, which is known to be a high-dimensional and ill-posed inverse problem. Two approaches are combined to tackle this problem: the Discrete Cosine Transform (DCT) approach, used in the forward modeling step, and the Variational Bayesian (VB) approach, used in the inverse reconstruction step. VB can provide not only point estimates but also closed forms of the full posterior probability distributions. To efficiently compute such estimates of the full joint posterior distributions of large-scale seismic inverse problems, we resort to a DCT order-reduction scheme with a VB approximation of the posteriors, avoiding the need for costly Bayesian sampling methods. More specifically, we first reduce the model parameters through truncation of their DCT coefficients. This helps regularizing our seismic inverse problem and alleviates its computational complexity. Then, we apply a VB inference in the reduced-DCT space to estimate the dominant (retained) DCT coefficients together with the variance of the observational noise. We also present an efficient implementation of the derived VB-based algorithm for further cost reduction. The performances of the proposed scheme are evaluated through extensive numerical experiments for both linear and non-linear forward models. In the former, the subsurface reflectivity model was reconstructed at a comparable estimation accuracy as the optimal weighted-least squares solution. In the latter, the main structural features of the squared slowness model were well reconstructed.

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Section: Efficient Implementation Of the Proposed Vb Schemementioning

confidence: 99%

A reduced-order variational Bayesian approach for efficient subsurface imaging

Urozayev

Ait‐El‐Fquih

Hoteit

et al. 2021

Geophysical Journal International

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“…To avoid the possible breakdowns in original LCG , several techniques were proposed, such as regularization, updating, permutation in Yuan et al (2004), augmenting in Dai and Yuan (2004), and block technique in Wang and Dai (2008). Moreover, such conjugacy-like idea has been extended to conjugate decomposition of matrix A and applied to harmonic estimation in power system (Wang and Yuan 2013).…”

Section: Introductionmentioning

confidence: 99%

Hybrid methods based on LCG and GMRES

Wang

Zhu

2015

Comp. Appl. Math.

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Recently, the left conjugate gradient (LCG) method has been developed and applied in many fields, such as computing physics and thermal pollution. As a typical Galerkin method, LCG is easy to get implemented and performs efficiently. LCG iteratively generates approximate solutions with residuals orthogonal to the Krylov subspaces. On the other hand, LCG lacks residual norm-minimization which is different from the famous generalized minimal residual method (GMRES). Although LCG and GMRES are based on different numerical principles, both algorithms exhibit similar behaviors in extensive computational experiments. To probe into the relationship between LCG and GMRES, we first demonstrate the mathematical equivalence of LCG and the full orthogonalization method (FOM). Then, the residual norm connection of LCG and GMRES is established as an easy consequence of relation between FOM and GMRES. Moreover, this paper presents two hybrid algorithms based on the LCG and GMRES. To unify the directness of LCG and optimality of GMRES, LCG residual norm-minimization (LCGR) algorithm is proposed. If orthogonalization is considered in the hybridization, LCG orthogonalization (LCGO) is designed. Numerical experiments show that two hybrid schemes do improve the performance of the standard LCG and GMRES, as well as the comparable methods such as nested GMRES (GMRESR) and generalized conjugate residual orthogonalization (GCRO).

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Conjugate Decomposition and Its Applications

Cited by 2 publications

References 17 publications

A reduced-order variational Bayesian approach for efficient subsurface imaging

A reduced-order variational Bayesian approach for efficient subsurface imaging

Hybrid methods based on LCG and GMRES

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