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Cholesky factorization

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Journal Article
Journal: Geophysics
Published: 16 November 2015
Geophysics (2016) 81 (1): V17–V30.
... a matrix-matrix multiplication and matrix inversion in every iteration step. The inversion of the symmetrical positive-definite matrix ( ( W ( m − 1 ) H ) T ( W ( m − 1 ) H ) + α I ) can be computed effectively by Cholesky factorization or the conjugate gradient (CG...
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Journal Article
Journal: Geophysics
Published: 22 June 2018
Geophysics (2018) 83 (4): R345–R359.
... from the fact that the matrix C i , j ( m ) differs from source to source and frequency to frequency. First, to apply the Cholesky method, we need to calculate the Cholesky factorization for each i and j , i.e., for each source and frequency. As a result...
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Journal Article
Journal: Geophysics
Published: 15 June 2010
Geophysics (2010) 75 (3): C25–C35.
... the feasibility, potential, and limitation of the blocky inversion method. The laterally correlated blocky inversion method requires a large, sparse matrix to be inverted. For our examples, we used a Cholesky factorization procedure, described in Appendix A . The mathematical properties...
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Journal Article
Journal: Geophysics
Published: 26 May 2015
Geophysics (2015) 80 (4): E213–E224.
.... The computational cost of each time-stepping process can be further reduced by reusing a previous complete/incomplete Cholesky (iChol) factorization when the current time-step size is the same as the previous one. However, major drawbacks of the FETD method are the fact that (1) the factorization requires a large...
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Journal Article
Journal: Geophysics
Published: 20 December 2012
Geophysics (2013) 78 (1): E47–E57.
... and a Cholesky decomposition can be performed with the work distributed over an array of processors. The forward modeling is quickly carried out using the factored operator. Time savings are considerable and they make 3D inversion of large ground or airborne data sets feasible. This is illustrated by using...
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Journal Article
Journal: Geophysics
Published: 03 July 2008
Geophysics (2008) 73 (4): R59–R69.
... be obtained by solving the linear system (A-3) Q μ = λ . We solve equation A-3 using a Cholesky factorization of the sparse Q matrix. This entails finding a lower triangular matrix L so that Q = LL ′ , and μ is obtained by solving (A-4) Lw = λ , L ′ μ = w...
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Journal Article
Journal: Geophysics
Published: 01 January 2003
Geophysics (2003) 68 (3): 922–930.
..., solutions for F and f are obtained by standard CG algorithms with an incomplete Cholesky factorization (ICF) used as preconditioner. Numerical instabilities in the factorization of P for some problems motivate the use of a “shifted” Cholesky decomposition ( Manteuffel, 1980 ) whereby the ICF of P...
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Journal Article
Journal: Geophysics
Published: 23 October 2018
Geophysics (2018) 83 (6): R629–R647.
... the Cholesky factorization of the Hessian H post — i.e., H post = R post ⊤ R post — as follows ( Rue, 2001 ): m s = m * + R post − 1 r , (20) where the matrix R post is an upper triangular matrix and the vector r is a random vector...
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Journal Article
Published: 26 September 2017
Bulletin of the Seismological Society of America (2017) 107 (6): 2792–2808.
... the spatial correlation among grid points. As shown by Bensi et al. (2011b) , representing the dependency among grid points is facilitated by a Cholesky factorization of the correlation matrix. Let us assume a grid of n points, in which the variability of the intraevent term is represented...
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Journal Article
Journal: Geophysics
Published: 11 December 2012
Geophysics (2013) 78 (1): G1–G13.
... formulated with Cholesky’s decomposition, we can verify that the computation time required for building the linear system and for solving the linear inverse problem can be reduced by as many as three and four orders of magnitude, respectively. Applications to synthetic and real data show that our method...
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Journal Article
Journal: Geophysics
Published: 13 November 2007
Geophysics (2008) 73 (1): F1–F8.
... as a single preconditioning step for BiCGStab2 to ensure the invariance of the preconditioner. For the results in that paper, we used a generic subroutine for solving complex-valued band matrices. Here, we replaced this routine by a nonstandard Cholesky decomposition. The standard decomposition factors...
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Journal Article
Journal: Geophysics
Published: 20 July 2022
Geophysics (2022) E291–E306.
... from the observed total-field anomaly data d o is solving the least-squares normal equation: A T A p = A T d o . (9) Equation  9 is usually solved by first computing the Cholesky factor G of the positive-definite matrix A T A and then using...
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Journal Article
Journal: Geophysics
Published: 08 June 2022
Geophysics (2022) 87 (4): E253–E265.
... successive overrelaxation (SSOR) approximation for the stiffness matrix, which is obtained through numerical discretization of Maxwell’s equations based on the coupled potentials. We use an incomplete Cholesky decomposition ( Saad, 2003 ) of the stiffness matrix and construct an improved preconditioner...
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Journal Article
Journal: Geophysics
Published: 03 June 2011
Geophysics (2011) 76 (4): F227–F238.
... sparse Cholesky factorization package ( Chen et al., 2009 ). As a result, we obtain the potential distribution u h for each node within the mesh as well as the sought potential values U ℓ for each of the electrodes. The latter can be read directly and requires neither interpolation nor...
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Journal Article
Journal: Geophysics
Published: 29 September 2021
Geophysics (2021) 86 (6): E383–E389.
... for the factorization of the system matrix, they are at the same level for the proposed method and the BE method. Specifically, the proposed method uses 17.1 s and 6562 MB, whereas FETD-BE uses 15.3 s and 6677 MB using the Cholesky factorization in MATLAB. The second example is a land CSEM model, which has...
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Journal Article
Journal: Geophysics
Published: 13 December 2018
Geophysics (2019) 84 (1): E23–E35.
... inversion algorithm uses a limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm ( Nocedal and Wright, 2006 ). Inside the L-BFGS, a Cholesky factor for equation  4 is reused to compute a search direction vector. Accordingly, one inversion iteration requires only one new factorization...
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Series: Society of Exploration Geophysicists
Published: 01 January 2017
EISBN: 9781560803423
... equations should be defined with the SPD property. Such a system is given by Cholesky factorization of M = EE T , in which E is a lower triangular matrix chosen so that the condition number of ( EE T ) −1 H is small. Therefore, multiplying Hx = g by E −1 gives (4.25...
Journal Article
Journal: Geophysics
Published: 01 January 2000
Geophysics (2000) 65 (4): 1154–1161.
... equation ( 8 ), then we update Q , and so on. For well-posed problems, convergence is reached after 10–20 iterations. System ( 7 ), which is the order of the number of data, is symmetric, and complex Cholesky factorization is appropiate for its resolution. In the cases where data are given at the points...
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Journal Article
Published: 01 April 1984
Bulletin of the Seismological Society of America (1984) 74 (2): 559–568.
..., residual weighting, and downweighting S arrivals relative to P arrivals. Distance weighting. Beginning on the first iteration, stations which have (esti- mated) epicentral distances in the range of 1 to 2 focal depths acquire a multipli- cative weighting factor of 0.75; between 2 and 4 focal depths...
Journal Article
Journal: Geophysics
Published: 23 August 2007
Geophysics (2007) 72 (5): SM177–SM184.
... should stress that C is not Hermitian. A Cholesky-type decomposition without pivoting is used to factor this matrix into C = L L T , where L is a lower-triangular matrix with six diagonals, and T means transposition. Note that this approach is different from the standard Cholesky...
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