Regularization with randomized SVD for large-scale discrete inverse problems

Xiang, Hua; Zou, Jun

doi:10.1088/0266-5611/29/8/085008

Cited by 60 publications

(92 citation statements)

References 51 publications

(110 reference statements)

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“…Calculating such a summation is not easy in a quantum computer since σ i is unknown, and each measurement only returns one singular value of A. For most of practical problems, A is of low rank, so we can consider an approximate SVD to calculate the GCV function [26]. That is, we can use a rank-r approximation of A and compute a partial summation r i=1 (σ 2 i + µ 2 ) −1 with the first r largest singular values.…”

Section: Gcv Functionmentioning

confidence: 99%

Quantum regularized least squares solver with parameter estimate

Shao

Xiang

2020

Quantum Inf Process

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In this paper we propose a quantum algorithm to determine the Tikhonov regularization parameter and solve the ill-conditioned linear equations. For regularized least squares problem with a fixed regularization parameter, we use the HHL algorithm and work on an extended matrix with smaller condition number. For the determination of the regularization parameter, we combine the classical Lcurve and GCV function, and design quantum algorithms to compute the norms of regularized solution and the corresponding residual in parallel and locate the best regularization parameter by Grover's search. The quantum algorithm can achieve a quadratic speedup in the number of regularization parameters and an exponential speedup in the dimension of problem size.

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Section: Gcv Functionmentioning

confidence: 99%

Quantum regularized least squares solver with parameter estimate

Shao

Xiang

2020

Quantum Inf Process

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“…If λ is a relatively small value, the inverted impedance strongly oscillates and is accompanied by trails like noodles in the presence of noise, and the spatial continuities of the inverted impedance are not ideal. So far, a variety of methods have been applied to determine λ (e.g., Tikhonov and Glasko, 1965;Hansen and O'Leary, 1993;van Wijk et al, 2002;Wang and Sacchi, 2007;Brezinski et al, 2008;Xiang and Zou, 2013). We decided the optimal λ value for the field data by testing on small-scale 2D seismic data, and it was calibrated with well-log data.…”

Section: Zðtþmentioning

confidence: 99%

Simultaneous multitrace impedance inversion with transform-domain sparsity promotion

et al. 2015

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The impedance inversion technique plays a crucial role in seismic reservoir properties prediction. However, most existing impedance inversion methods often suffer from spatial discontinuities and instability because each vertical profile is processed independently in the inversion. We tested a transform-domain sparsity promotion simultaneous multitrace impedance inversion method to address this issue. The approach was implemented through minimizing a data misfit term and a transform-domain sparsity constraint term that incorporates the (2D or 3D) structural information into the inversion processing. A 2D synthetic data example was applied to mainly explain the roles of the transform-domain sparsity constraint. We determined that the transform-domain sparsity constraint can help in stabilizing the inversion, reducing the influence of high-wavenumber noise on the inverted result, and exploring spatial continuities of structures. Furthermore, a 3D field data example was used to examine the effectiveness of the proposed method for dealing with the real data and to reveal the difference between the results from the 3D simultaneous inversion and the section-by-section inversion. We found that the inverted results roughly matched a lowpass-filtered version of impedance curves derived from well log data. Also, it has been demonstrated that the 3D simultaneous inversion technique provided a better estimation than the section-by-section inversion technique in terms of guaranteeing more spatial continuities of geologic features in the impedance model.

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“…That is, after solving (16) and (19), we have Using (20), we can achieve that  = b − α α p (22) So solving linear equations (3) is equivalent to solving both (16) and (19), whose coefficient matrices are the same. Here we solve (16) and (19) by low-rank approximation.…”

Section: Approximate Linear Solversmentioning

confidence: 99%

“…Here we solve (16) and (19) by low-rank approximation. If we have * ≈ K UΣV , which can be known by randomized SVD, then (16) and (19) can be approximated by…”

Section: Approximate Linear Solversmentioning

confidence: 99%

See 1 more Smart Citation

An approximate linear solver in least square support vector machine using randomized singular value decomposition

Liu

Xiang

2015

Wuhan Univ. J. Nat. Sci.

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In this paper, we investigate the linear solver in least square support vector machine (LSSVM) for large-scale data regression. The traditional methods using the direct solvers are costly. We know that the linear equations should be solved repeatedly for choosing appropriate parameters in LSSVM, so the key for speeding up LSSVM is to improve the method of solving the linear equations. We approximate large-scale kernel matrices and get the approximate solution of linear equations by using randomized singular value decomposition (randomized SVD). Some data sets coming from University of California Irvine machine learning repository are used to perform the experiments. We find LSSVM based on randomized SVD is more accurate and less time-consuming in the case of large number of variables than the method based on Nyström method or Lanczos process.

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Regularization with randomized SVD for large-scale discrete inverse problems

Cited by 60 publications

References 51 publications

Quantum regularized least squares solver with parameter estimate

Quantum regularized least squares solver with parameter estimate

Simultaneous multitrace impedance inversion with transform-domain sparsity promotion

An approximate linear solver in least square support vector machine using randomized singular value decomposition

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