scholarly journals A geometric method for eigenvalue problems with low-rank perturbations

2017 ◽  
Vol 4 (9) ◽  
pp. 170390 ◽  
Author(s):  
Thomas J. Anastasio ◽  
Andrea K. Barreiro ◽  
Jared C. Bronski
Keyword(s):  
Eigenvalue Problems ◽  
Low Rank ◽  
Spectrum Of An Operator ◽  
Link Type ◽  
Classical Differential Geometry ◽  
Family Of Curves ◽  
Rank One ◽  
Non Local ◽  
Integrator Model ◽  
Appl Math

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelope of a family of curves) to completely analyse the spectrum. We use these techniques to analyse three problems of this form: a model of the oculomotor integrator due to Anastasio & Gad (2007 J. Comput. Neurosci. 22 , 239–254. ( doi:10.1007/s10827-006-0010-x )), a continuum integrator model, and a non-local model of phase separation due to Rubinstein & Sternberg (1992 IMA J. Appl. Math. 48 , 249–264. ( doi:10.1093/imamat/48.3.249 )).

scholarly journals Mutual information for low-rank even-order symmetric tensor estimation

2020 ◽  
Author(s):  
Clément Luneau ◽  
Jean Barbier ◽  
Nicolas Macris
Keyword(s):  
Mutual Information ◽  
Finite Rank ◽  
Interpolation Method ◽  
Symmetric Tensor ◽  
Low Rank ◽  
Tensor Factorization ◽  
Adaptive Interpolation ◽  
Odd Order ◽  
Rank One ◽  
Even Order

Abstract We consider a statistical model for finite-rank symmetric tensor factorization and prove a single-letter variational expression for its asymptotic mutual information when the tensor is of even order. The proof applies the adaptive interpolation method originally invented for rank-one factorization. Here we show how to extend the adaptive interpolation to finite-rank and even-order tensors. This requires new non-trivial ideas with respect to the current analysis in the literature. We also underline where the proof falls short when dealing with odd-order tensors.

Non-local Low-rank Point Cloud Denoising for 3D Measurement Surfaces

2022 ◽  
pp. 1-1
Author(s):  
Dingkun Zhu ◽  
Honghua Chen ◽  
Weiming Wang ◽  
Haoran Xie ◽  
Gary Cheng ◽  
...  
Keyword(s):  
Point Cloud ◽  
Low Rank ◽  
3D Measurement ◽  
Non Local

Non-convex low-rank representation combined with rank-one matrix sum for subspace clustering

2020 ◽  
Vol 24 (20) ◽  
pp. 15317-15326
Author(s):  
Xiaofang Liu ◽  
Jun Wang ◽  
Dansong Cheng ◽  
Daming Shi ◽  
Yongqiang Zhang
Keyword(s):  
Subspace Clustering ◽  
Low Rank ◽  
Rank One ◽  
Low Rank Representation

scholarly journals Scattering on a square lattice from a crack with a damage zone

2020 ◽  
Vol 476 (2235) ◽  
pp. 20190686 ◽  
Author(s):  
Basant Lal Sharma ◽  
Gennady Mishuris
Keyword(s):  
Asymptotic Approximation ◽  
Damage Zone ◽  
Crack Tip ◽  
Square Lattice ◽  
Open Crack ◽  
Matrix Kernel ◽  
Link Type ◽  
Damaged Zone ◽  
Sharp Crack ◽  
Appl Math

A semi-infinite crack in an infinite square lattice is subjected to a wave coming from infinity, thereby leading to its scattering by the crack surfaces. A partially damaged zone ahead of the crack tip is modelled by an arbitrarily distributed stiffness of the damaged links. While an open crack, with an atomically sharp crack tip, in the lattice has been solved in closed form with the help of the scalar Wiener–Hopf formulation (Sharma 2015 SIAM J. Appl. Math. , 75 , 1171–1192 ( doi:10.1137/140985093 ); Sharma 2015 SIAM J. Appl. Math. 75 , 1915–1940. ( doi:10.1137/15M1010646 )), the problem considered here becomes very intricate depending on the nature of the damaged links. For instance, in the case of a partially bridged finite zone it involves a 2 × 2 matrix kernel of formidable class. But using an original technique, the problem, including the general case of arbitrarily damaged links, is reduced to a scalar one with the exception that it involves solving an auxiliary linear system of N  ×  N equations, where N defines the length of the damage zone. The proposed method does allow, effectively, the construction of an exact solution. Numerical examples and the asymptotic approximation of the scattered field far away from the crack tip are also presented.

scholarly journals SLRL4D: Joint Restoration of Subspace Low-Rank Learning and Non-Local 4-D Transform Filtering for Hyperspectral Image

2020 ◽  
Vol 12 (18) ◽  
pp. 2979
Author(s):  
Le Sun ◽  
Chengxun He ◽  
Yuhui Zheng ◽  
Songze Tang
Keyword(s):  
Hyperspectral Image ◽  
Dimensional Subspace ◽  
Low Rank ◽  
Spatial Correlations ◽  
Self Similarity ◽  
Multiple Datasets ◽  
Alternating Direction ◽  
Non Local ◽  
Low Dimensional ◽  
Spatial Redundancy

During the process of signal sampling and digital imaging, hyperspectral images (HSI) inevitably suffer from the contamination of mixed noises. The fidelity and efficiency of subsequent applications are considerably reduced along with this degradation. Recently, as a formidable implement for image processing, low-rank regularization has been widely extended to the restoration of HSI. Meanwhile, further exploration of the non-local self-similarity of low-rank images are proven useful in exploiting the spatial redundancy of HSI. Better preservation of spatial-spectral features is achieved under both low-rank and non-local regularizations. However, existing methods generally regularize the original space of HSI, the exploration of the intrinsic properties in subspace, which leads to better denoising performance, is relatively rare. To address these challenges, a joint method of subspace low-rank learning and non-local 4-d transform filtering, named SLRL4D, is put forward for HSI restoration. Technically, the original HSI is projected into a low-dimensional subspace. Then, both spectral and spatial correlations are explored simultaneously by imposing low-rank learning and non-local 4-d transform filtering on the subspace. The alternating direction method of multipliers-based algorithm is designed to solve the formulated convex signal-noise isolation problem. Finally, experiments on multiple datasets are conducted to illustrate the accuracy and efficiency of SLRL4D.

Parallel Iterative Methods for the Helmholtz Equation With Exact Nonreflecting Boundaries

2002 ◽  
Author(s):  
Cristian Ianculescu ◽  
Lonny L. Thompson
Keyword(s):  
Helmholtz Equation ◽  
Iterative Methods ◽  
Scale Up ◽  
Low Rank ◽  
Interprocessor Communication ◽  
Parallel Iterative Methods ◽  
Local Boundary Condition ◽  
Non Local ◽  
Parallel Iterative ◽  
Low Rank Update

Parallel iterative methods for fast solution of large-scale acoustic radiation and scattering problems are developed using exact Dirichlet-to-Neumann (DtN), nonreflecting boundaries. A separable elliptic nonreflecting boundary is used to efficiently model unbounded regions surrounding elongated structures. We exploit the special structure of the non-local DtN map as a low-rank update of the system matrix to efficiently compute the matrix-by-vector products found in Krylov subspace based iterative methods. For the complex non-hermitian matrices resulting from the Helmholtz equation, we use a distributed-memory parallel BICG-STAB iterative method in conjunction with a parallel Jacobi preconditioner. Domain decomposition with interface minimization was performed to ensure optimal interprocessor communication. For the architectures tested, and using the MPICH version of MPI, we show that when implemented as a low-rank update, the non-local character of the DtN map does not signicantly decrease the scale up and parallel eciency versus a purely approximate local boundary condition.

scholarly journals A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 100 ◽  
Author(s):  
Luca Bergamaschi
Keyword(s):  
Linear Systems ◽  
Eigenvalue Problems ◽  
Real Life ◽  
Low Rank ◽  
Preconditioned Conjugate Gradient ◽  
Krylov Methods ◽  
Rank Matrix ◽  
Recent Developments ◽  
Speed Up ◽  
Low Rank Matrix

The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems A k x k = b k , k = 1 , … arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

Low-rank tensor completion via combined non-local self-similarity and low-rank regularization

2019 ◽  
Vol 367 ◽  
pp. 1-12 ◽  
Author(s):  
Xiao-Tong Li ◽  
Xi-Le Zhao ◽  
Tai-Xiang Jiang ◽  
Yu-Bang Zheng ◽  
Teng-Yu Ji ◽  
...  
Keyword(s):  
Low Rank ◽  
Tensor Completion ◽  
Self Similarity ◽  
Rank Tensor ◽  
Non Local
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