scholarly journals Golub–Kahan vs. Monte Carlo: a comparison of bidiagonlization and a randomized SVD method for the solution of linear discrete ill-posed problems

2021 ◽  
Author(s):  
Xianglan Bai ◽  
Alessandro Buccini ◽  
Lothar Reichel
Keyword(s):  
Large Scale ◽  
Krylov Subspace ◽  
Computing Time ◽  
Low Rank ◽  
Large Matrix ◽  
Ill Posed ◽  
Randomized Methods ◽  
Value Decomposition ◽  
Svd Method ◽  
Randomized Method

AbstractRandomized methods can be competitive for the solution of problems with a large matrix of low rank. They also have been applied successfully to the solution of large-scale linear discrete ill-posed problems by Tikhonov regularization (Xiang and Zou in Inverse Probl 29:085008, 2013). This entails the computation of an approximation of a partial singular value decomposition of a large matrix A that is of numerical low rank. The present paper compares a randomized method to a Krylov subspace method based on Golub–Kahan bidiagonalization with respect to accuracy and computing time and discusses characteristics of linear discrete ill-posed problems that make them well suited for solution by a randomized method.

scholarly journals A Flexible Coding Scheme Based on Block Krylov Subspace Approximation for Light Field Displays with Stacked Multiplicative Layers

Sensors ◽  
2021 ◽  
Vol 21 (13) ◽  
pp. 4574
Author(s):  
Joshitha Ravishankar ◽  
Mansi Sharma ◽  
Pradeep Gopalakrishnan
Keyword(s):  
Light Field ◽  
Krylov Subspace ◽  
Motion Parallax ◽  
Low Rank ◽  
Coding Scheme ◽  
Krylov Subspace Approximation ◽  
New Type ◽  
Value Decomposition ◽  
Subspace Approximation ◽  
Inter Frame

To create a realistic 3D perception on glasses-free displays, it is critical to support continuous motion parallax, greater depths of field, and wider fields of view. A new type of Layered or Tensor light field 3D display has attracted greater attention these days. Using only a few light-attenuating pixelized layers (e.g., LCD panels), it supports many views from different viewing directions that can be displayed simultaneously with a high resolution. This paper presents a novel flexible scheme for efficient layer-based representation and lossy compression of light fields on layered displays. The proposed scheme learns stacked multiplicative layers optimized using a convolutional neural network (CNN). The intrinsic redundancy in light field data is efficiently removed by analyzing the hidden low-rank structure of multiplicative layers on a Krylov subspace. Factorization derived from Block Krylov singular value decomposition (BK-SVD) exploits the spatial correlation in layer patterns for multiplicative layers with varying low ranks. Further, encoding with HEVC eliminates inter-frame and intra-frame redundancies in the low-rank approximated representation of layers and improves the compression efficiency. The scheme is flexible to realize multiple bitrates at the decoder by adjusting the ranks of BK-SVD representation and HEVC quantization. Thus, it would complement the generality and flexibility of a data-driven CNN-based method for coding with multiple bitrates within a single training framework for practical display applications. Extensive experiments demonstrate that the proposed coding scheme achieves substantial bitrate savings compared with pseudo-sequence-based light field compression approaches and state-of-the-art JPEG and HEVC coders.

The extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations

2021 ◽  
Vol 8 (3) ◽  
pp. 526-536
Author(s):  
L. Sadek ◽  
◽  
H. Talibi Alaoui ◽  
Keyword(s):  
Matrix Equation ◽  
Large Scale ◽  
Krylov Subspace ◽  
Low Rank ◽  
Lyapunov Equations ◽  
Lanczos Algorithm ◽  
Lyapunov Matrix ◽  
Block Lanczos ◽  
Low Dimensional ◽  
Differential Lyapunov Equations

In this paper, we present a new approach for solving large-scale differential Lyapunov equations. The proposed approach is based on projection of the initial problem onto an extended block Krylov subspace by using extended nonsymmetric block Lanczos algorithm then, we get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix equation is solved by the Backward Differentiation Formula method (BDF) or Rosenbrock method (ROS), the obtained solution allows to build a low-rank approximate solution of the original problem. Moreover, we also give some theoretical results. The numerical results demonstrate the performance of our approach.

A computationally efficient tool for assessing the depth resolution in large-scale potential-field inversion

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. A33-A38 ◽  
Author(s):  
Valeria Paoletti ◽  
Per Christian Hansen ◽  
Mads Friis Hansen ◽  
Maurizio Fedi
Keyword(s):  
Potential Field ◽  
Large Scale ◽  
Computing Time ◽  
Depth Resolution ◽  
Source Distribution ◽  
Volcanic Area ◽  
Computationally Efficient ◽  
Large Scale Problems ◽  
Value Decomposition ◽  
Field Inversion

In potential-field inversion, careful management of singular value decomposition components is crucial for obtaining information about the source distribution with respect to depth. In principle, the depth-resolution plot provides a convenient visual tool for this analysis, but its computational complexity has hitherto prevented application to large-scale problems. To analyze depth resolution in such problems, we developed a variant ApproxDRP, which is based on an iterative algorithm and therefore suited for large-scale problems because we avoid matrix factorizations and the associated demands on memory and computing time. We used the ApproxDRP to study retrievable depth resolution in inversion of the gravity field of the Neapolitan Volcanic Area. Our main contribution is the combined use of the Lanczos bidiagonalization algorithm, established in the scientific computing community, and the depth-resolution plot defined in the geoscience community.

scholarly journals Turbulent transport coefficients in galactic dynamo simulations using singular value decomposition

2019 ◽  
Vol 491 (3) ◽  
pp. 3870-3883 ◽  
Author(s):  
Abhijit B Bendre ◽  
Kandaswamy Subramanian ◽  
Detlef Elstner ◽  
Oliver Gressel
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Turbulent Transport ◽  
Transport Coefficients ◽  
Mean Field ◽  
Dynamo Model ◽  
The Mean ◽  
Value Decomposition ◽  
Svd Method ◽  
Mean Field Dynamo

ABSTRACT Coherent magnetic fields in disc galaxies are thought to be generated by a large-scale (or mean-field) dynamo operating in their interstellar medium. A key driver of mean magnetic field growth is the turbulent electromotive force (EMF), which represents the influence of correlated small-scale (or fluctuating) velocity and magnetic fields on the mean field. The EMF is usually expressed as a linear expansion in the mean magnetic field and its derivatives, with the dynamo tensors as expansion coefficients. Here, we adopt the singular value decomposition (SVD) method to directly measure these turbulent transport coefficients in a simulation of the turbulent interstellar medium that realizes a large-scale dynamo. Specifically, the SVD is used to least-square fit the time series data of the EMF with that of the mean field and its derivatives, to determine these coefficients. We demonstrate that the spatial profiles of the EMF reconstructed from the SVD coefficients match well with that taken directly from the simulation. Also, as a direct test, we use the coefficients to simulate a 1D mean-field dynamo model and find an overall similarity in the evolution of the mean magnetic field between the dynamo model and the direct simulation. We also compare the results with those which arise using simple regression and the ones obtained previously using the test-field method, to find reasonable qualitative agreement. Overall, the SVD method provides an effective post-processing tool to determine turbulent transport coefficients from simulations.

scholarly journals DATA ANALYSIS OF GRAVITATIONAL WAVES USING A NETWORK OF DETECTORS

2008 ◽  
Vol 17 (07) ◽  
pp. 1095-1104 ◽  
Author(s):  
LINQING WEN
Keyword(s):  
Source Localization ◽  
Large Scale ◽  
Design Sensitivity ◽  
Singular Value ◽  
Consistency Check ◽  
Term Operation ◽  
Stable Solutions ◽  
Value Decomposition ◽  
Svd Method

Several large-scale gravitational wave (GW) interferometers have achieved long term operation at design sensitivity. Questions arise on how to best combine all available data from detectors of different sensitivities for detection, consistency check or veto, localization and waveform extraction. We show that these problems can be formulated using the singular value decomposition (SVD)1 method. We present techniques based on the SVD method for (1) detection statistic, (2) stable solutions to waveforms, (3) null-stream construction for an arbitrary number of detectors, and (4) source localization for GWs of unknown waveforms.

scholarly journals Computationally Efficient Optimal Control for Unstable Power System Models

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Mahtab Uddin ◽  
M. Monir Uddin ◽  
Md. Abdul Hakim Khan
Keyword(s):  
Optimal Control ◽  
Power System ◽  
Large Scale ◽  
Krylov Subspace ◽  
Feedback Stabilization ◽  
Descriptor Systems ◽  
Low Rank ◽  
Computationally Efficient ◽  
System Models ◽  
Alternating Direction

In this article, the focus is mainly on gaining the optimal control for the unstable power system models and stabilizing them through the Riccati-based feedback stabilization process with sparsity-preserving techniques. We are to find the solution of the Continuous-time Algebraic Riccati Equations (CAREs) governed from the unstable power system models derived from the Brazilian Inter-Connected Power System (BIPS) models, which are large-scale sparse index-1 descriptor systems. We propose the projection-based Rational Krylov Subspace Method (RKSM) for the iterative computation of the solution of the CAREs. The novelties of RKSM are sparsity-preserving computations and the implementation of time-convenient adaptive shift parameters. We modify the Low-Rank Cholesky-Factor integrated Alternating Direction Implicit (LRCF-ADI) technique-based nested iterative Kleinman–Newton (KN) method to a sparse form and adjust this to solve the desired CAREs. We compare the results achieved by the Kleinman–Newton method with that of using the RKSM. The applicability and adaptability of the proposed techniques are justified numerically with MATLAB simulations. Transient behaviors of the target models are investigated for comparative analysis through the tabular and graphical approaches.

scholarly journals Augmented Arnoldi-Tikhonov Regularization Methods for Solving Large-Scale Linear Ill-Posed Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yiqin Lin ◽  
Liang Bao ◽  
Yanhua Cao
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Large Scale ◽  
Krylov Subspace ◽  
Dimensional Subspace ◽  
Tikhonov Regularization Method ◽  
Rough Approximation ◽  
Ill Posed ◽  
Low Dimensional

We propose an augmented Arnoldi-Tikhonov regularization method for the solution of large-scale linear ill-posed systems. This method augments the Krylov subspace by a user-supplied low-dimensional subspace, which contains a rough approximation of the desired solution. The augmentation is implemented by a modified Arnoldi process. Some useful results are also presented. Numerical experiments illustrate that the augmented method outperforms the corresponding method without augmentation on some real-world examples.

scholarly journals Accelerated Randomized Methods for Receiver Design in Extra-Large Scale MIMO Arrays

2021 ◽  
pp. 1-1
Author(s):  
Victor Croisfelt Rodrigues ◽  
Abolfazl Amiri ◽  
Taufik Abrao ◽  
Elisabeth De Carvalho ◽  
Petar Popovski
Keyword(s):  
Large Scale ◽  
Receiver Design ◽  
Randomized Methods ◽  
Mimo Arrays
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