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Fast algorithms for spectral collocation with non-periodic boundary conditions

W. Lyons, H.D. Ceniceros, S. Chandrasekaran, M. Gu
2005 Journal of Computational Physics  
The underlying ideas are introduced and developed in the context of linearly implicit methods for stiff equations.  ...  By employing a structured representation of linear operators we are able to use fast algorithms without being restricted to periodic boundary conditions.  ...  The off-diagonal blocks in this partitioning are shown to have low rank.  ... 
doi:10.1016/j.jcp.2005.01.007 fatcat:xefok3i3fvfdtis4enyqfzxms4

Eigenvalue computation for unitary rank structured matrices

Steven Delvaux, Marc Van Barel
2008 Journal of Computational and Applied Mathematics  
In this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps.  ...  First we perform a reduction of the given matrix into Hessenberg form, next we compute the eigenvalues of this resulting Hessenberg matrix via an implicit QR-algorithm.  ...  low rank blocks to the main diagonal, and where n denotes the matrix size.  ... 
doi:10.1016/j.cam.2007.01.006 fatcat:ibaihjs22rhfveeac4ylwg7wca

Block computation and representation of a sparse nullspace basis of a rectangular matrix

Sabine Le Borne
2008 Linear Algebra and its Applications  
We will employ this observation to design an efficient block algorithm that computes a sparse representation of the nullspace basis in almost optimal complexity.  ...  We assume that B has full rank, i.e., rank(B) = m. It is well known that the last n − m columns of the orthogonal matrix Q in a QR factorization B = QR form such a desired null basis.  ...  Efficient realization of the implicit block representation by ordering and low rank approximation In this section, we propose some strategies which will greatly decrease the computational costs of computing  ... 
doi:10.1016/j.laa.2007.11.025 fatcat:ykzck3flqrgxnldjii2by6kq5q

Fast QR iterations for unitary plus low rank matrices [article]

Roberto Bevilacqua and Gianna M. Del Corso and Luca Gemignani
2019 arXiv   pre-print
Some fast algorithms for computing the eigenvalues of a block companion matrix A = U + XY^H, where U∈C^n× n is unitary block circulant and X, Y ∈C^n × k, have recently appeared in the literature.  ...  A remarkable case is U unitary diagonal which makes possible to deal with interpolation techniques for rootfinding problems and nonlinear eigenvalue problems.  ...  Matrix Q (1) is still block-diagonal with the leading block k × k unitary diagonal, and the tailing block Hessenberg.  ... 
arXiv:1810.02708v2 fatcat:oykg7yfnzfclvkata4m5nv7coe

hm-toolbox: Matlab software for HODLR and HSS matrices [article]

Stefano Massei, Leonardo Robol, Daniel Kressner
2020 arXiv   pre-print
Matrices with hierarchical low-rank structure, including HODLR and HSS matrices, constitute a versatile tool to develop fast algorithms for addressing large-scale problems.  ...  Nevertheless, it maintains the favorable complexity of hierarchical low-rank matrices and offers, at the same time, a convenient way of prototyping and experimenting with algorithms.  ...  An HSS matrix is converted into a HODLR matrix by simply building explicit low-rank factorizations of the off-diagonal blocks from their implicit nested representation in the HSS format.  ... 
arXiv:1909.07909v3 fatcat:blvtvqamh5dxlaktli3fpugvoe

Understanding Dimensional Collapse in Contrastive Self-supervised Learning [article]

Li Jing, Pascal Vincent, Yann LeCun, Yuandong Tian
2022 arXiv   pre-print
Self-supervised visual representation learning aims to learn useful representations without relying on human annotations.  ...  Inspired by our theory, we propose a novel contrastive learning method, called DirectCLR, which directly optimizes the representation space without relying on an explicit trainable projector.  ...  Projector diagonal low-rank Top-1 Accuracy no projector 51.5 orthogonal projector 52.2 trainable projector 61.1 trainable diagonal projector 60.2 fixed low-rank projector 62.3 fixed low-rank diagonal projector  ... 
arXiv:2110.09348v3 fatcat:3etdtubetvcl5blmcrl24bscuy

Efficient Reduction of Compressed Unitary plus Low-rank Matrices to Hessenberg form [article]

Roberto Bevilacqua, Gianna M. Del Corso, Luca Gemignani
2019 arXiv   pre-print
We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix A=G+U V^H, where G∈C^n× n is a unitary matrix represented in some compressed format using O(nk  ...  Small Rank Modifications of Unitary Block Diagonal Matrices.  ...  In particular, the representation is suited for the fast eigensolver for unitary plus low rank matrices developed in [8] . Our derivation is based on three key ingredients or building blocks: 1.  ... 
arXiv:1901.08411v2 fatcat:5p6pwym6u5cj3ogmuzdengbmci

Efficient construction of an HSS preconditioner for symmetric positive definite ℋ^2 matrices [article]

Xin Xing, Hua Huang, Edmond Chow
2021 arXiv   pre-print
Fast (linear-scaling) matrix-vector products are available by expressing the kernel matrix in an ℋ^2 representation or an equivalent fast multipole method representation.  ...  Preconditioning such matrices, however, requires a structured matrix approximation that is more regular than the ℋ^2 representation, such as the hierarchically semiseparable (HSS) matrix representation  ...  The HSS representation generally requires more of its off-diagonal blocks to be compressed into low-rank form than the H 2 representation.  ... 
arXiv:2011.07632v2 fatcat:w2l7e45cvzeitj7lg7rmwpzjuu

Implicit double shift QR-algorithm for companion matrices

Marc Van Barel, Raf Vandebril, Paul Van Dooren, Katrijn Frederix
2010 Numerische Mathematik  
Moreover, no compression is needed as the specific representation of the involved matrices is maintained. Finally, also a double shift version of the implicit method is presented.  ...  Moreover, no compression is needed as the specific representation of the involved matrices is maintained. Finally, also a double shift version of the implicit method is presented.  ...  The representation designed in this section is the one that will be used for developing an implicit QR-method for unitary plus low rank matrices.  ... 
doi:10.1007/s00211-010-0302-y fatcat:boutu3uz4vhvvowfkwvdcoty3y

Unitary rank structured matrices

Steven Delvaux, Marc Van Barel
2008 Journal of Computational and Applied Mathematics  
pull-through process of the two branches of the representation.  ...  We also provide some basic operations for manipulating the representation, such as the transition to zero-creating form, the transition to a unitary/Givens-weight representation, as well as an internal  ...  It reveals then that for each pure structure block of a unitary matrix, the complementary submatrix must have low rank as well.  ... 
doi:10.1016/j.cam.2007.03.020 fatcat:jmgli3tg6ngufiej3jtjrrx4mi

A bibliography on semiseparable matrices*

R. Vandebril, M. Van Barel, G. Golub, N. Mastronardi
2005 Calcolo  
matrices, while in other cases they were defined as matrices having low rank blocks below the diagonal.  ...  matrices, while in other cases they were defined as matrices having low rank blocks below the diagonal.  ...  -One can generalize all the above representations to come to block forms. Such as recursively semiseparable, sequentially semiseparable matrices or low Hankel rank matrices.  ... 
doi:10.1007/s10092-005-0107-z fatcat:dj3ema4whveg3l725gofhrpir4

Fast algorithms for hierarchically semiseparable matrices

Jianlin Xia, Shivkumar Chandrasekaran, Ming Gu, Xiaoye S. Li
2010 Numerical Linear Algebra with Applications  
Moreover, we provide a new linear complexity explicit U LV factorization algorithm for symmetric positive definite HSS matrices with a low-rank property.  ...  In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices.  ...  The low-rank property is concerned with the ranks or numerical ranks of certain types of off-diagonal blocks.  ... 
doi:10.1002/nla.691 fatcat:neof3jzipfdhxevk53ha5jwb2u

A preconditioned GMRES for complex dense linear systems from electromagnetic wave scattering problems

Angelika Bunse-Gerstner, Ignacio Gutiérrez-Cañas
2006 Linear Algebra and its Applications  
The novelty of this work is the construction of an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups  ...  In the light of this experience we propose a multilevel near-field matrix and its corresponding HSS representation as a hierarchical preconditioner in order to substantially reduce the number of iterations  ...  HSS matrices are characterized by a hierarchical low-rank structure in the off-diagonal blocks.  ... 
doi:10.1016/j.laa.2005.11.022 fatcat:nbnnnabtfndn7fgoakcqzoboci

Efficient Reduction of Compressed Unitary Plus Low Rank Matrices to Hessenberg Form

Roberto Bevilacqua, Gianna M. Del Corso, Luca Gemignani
2020 SIAM Journal on Matrix Analysis and Applications  
We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low rank matrix A = G + U V H , where G \in \BbbC n\times n is a unitary matrix represented in some compressed  ...  Small rank modifications of unitary block diagonal matrices.  ...  However the Hessenberg matrix also obtained with that algorithm, which is based on the Givens-vector representation of the low rank part, is not directly exploitable by available fast eigensolvers.  ... 
doi:10.1137/19m1280363 fatcat:6wnrhmeuqzc2hmxwvg6orizbn4

Discriminative Block-Diagonal Representation Learning for Image Recognition

Zheng Zhang, Yong Xu, Ling Shao, Jian Yang
2018 IEEE Transactions on Neural Networks and Learning Systems  
In this paper, we propose a discriminative block-diagonal low-rank representation (BDLRR) method for recognition.  ...  representation under the semi-supervised framework of low-rank representation.  ...  For CBDS, it locally enforces the classwise diagonal structure on the low-rank criterion, whereas our BDLRR method globally imposes the block-diagonal constraint on the low-rank criterion by directly minimizing  ... 
doi:10.1109/tnnls.2017.2712801 pmid:28692990 fatcat:53nnhax2tvh4finsqw2tevxaaq
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