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Efficient, sparse representation of manifold distance matrices for classical scaling [article]

Javier S. Turek, Alexander Huth
2018 arXiv   pre-print
Thus for large point sets it is common to use a low-rank approximation to the distance matrix, which fits in memory and can be efficiently analyzed using methods such as multidimensional scaling (MDS).  ...  In this paper we present a novel sparse method for efficiently representing geodesic distance matrices using biharmonic interpolation.  ...  In the low-rank approximation step we draw a set of landmark points and then compute the interpolation operator P and the matrix W.  ... 
arXiv:1705.10887v2 fatcat:wmbf4ynkyvhstbifket3gzzgu4

Efficient, Sparse Representation of Manifold Distance Matrices for Classical Scaling

Javier S. Turek, Alexander G. Huth
2018 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition  
Thus for large point sets it is common to use a low-rank approximation to the distance matrix, which fits in memory and can be efficiently analyzed using methods such as multidimensional scaling (MDS).  ...  In this paper we present a novel sparse method for efficiently representing geodesic distance matrices using biharmonic interpolation.  ...  In the low-rank approximation step we draw a set of landmark points and then compute the interpolation operator P and the matrix W.  ... 
doi:10.1109/cvpr.2018.00301 dblp:conf/cvpr/TurekH18 fatcat:yzkw3gby6fhgxh6f6dqlnma22m

Efficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole method

Pieter Coulier, Eric Darve
2016 Computer Methods in Applied Mechanics and Engineering  
The linear complexity is achieved by transforming the dense system into an extended sparse system, along with the compression of certain matrix blocks into low-rank factorizations.  ...  The solver is inexact, although the error can be controlled and made as small as needed; a low accuracy solver can hence be used as an efficient preconditioner in an iterative scheme.  ...  The first author is a post-doctoral fellow of the Research Foundation Flanders (FWO) and a Francqui Foundation fellow of the Belgian American Educational Foundation (BAEF).  ... 
doi:10.1016/j.cma.2016.05.029 fatcat:e26hfcxgwzbejk2a32m5ws43ey

Layer-oriented multigrid wavefront reconstruction algorithms for multiconjugate adaptive optics

Luc Gilles, Brent L. Ellerbroek, Curtis R. Vogel, Peter L. Wizinowich, Domenico Bonaccini
2003 Adaptive Optical System Technologies II  
Using standard matrix methods to compute, optimize, and implement wavefront control algorithms for these systems is impractical, since the number of calculations required to compute and apply the reconstruction  ...  matrix scales respectively with the cube and the square of the number of AO degrees of freedom.  ...  ACKNOWLEDGMENTS We acknowledge the support of the US Air Force Office of Scientific Research and the Gemini Observatory  ... 
doi:10.1117/12.459347 fatcat:62p2yqkqzffqvehu73543rygom

Fast Direct Methods for Gaussian Processes [article]

Sivaram Ambikasaran, Daniel Foreman-Mackey, Leslie Greengard, David W. Hogg, Michael O'Neil
2015 arXiv   pre-print
Here, we show that for the most commonly used covariance functions, the matrix C can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an O (n^2 n) algorithm  ...  A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution.  ...  ACKNOWLEDGMENT The authors would like to thank Iain Murray for several useful and detailed discussions.  ... 
arXiv:1403.6015v2 fatcat:qt6pddpoyrahhfyyxbcbasro6q

Knot Optimization for Biharmonic B-splines on Manifold Triangle Meshes

Fei Hou, Ying He, Hong Qin, Aimin Hao
2017 IEEE Transactions on Visualization and Computer Graphics  
We develop algorithms for spline evaluation, data interpolation and hierarchical data decomposition.  ...  Biharmonic B-splines, proposed by Feng and Warren, are an elegant generalization of univariate B-splines to planar and curved domains with fully irregular knot configuration.  ...  Acknowledgements: We thank the reviewers for their detailed and constructive comments, which help us to improve the quality of the paper significantly.  ... 
doi:10.1109/tvcg.2016.2605092 pmid:27608469 fatcat:2jeowy73y5b77ijn7cxnhi73ge

Book Review: Lectures on finite precision computations

Hans J. Stetter, Charles Van Loan, Michael Holst, Frank Stenger, Chi-Wang Shu, R. Mattheij, Stephen J. Wright, Thomas F. Coleman, Lars B. Wahlbin
1997 Mathematics of Computation  
In this setting "proximity" often means the defining matrices differ by a few rows or columns or, more generally, by a low-rank matrix.  ...  The mechanism of finding the interpolation or differentiation matrices is discussed, and examples given for these matrices for different node distributions.  ... 
doi:10.1090/s0025-5718-97-00910-1 fatcat:kiiohmhwnrh5rkjl672eqoqnre

Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions [article]

Arvind K. Saibaba, Rachel Minster, Misha E. Kilmer
2021 arXiv   pre-print
and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes.  ...  We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points; the resulting low-rank  ...  Since the resulting matrix is symmetric and positive definite, we would like to preserve the symmetry in the low-rank approximation.  ... 
arXiv:2107.13107v1 fatcat:ondanxg3i5eb3c2rmq62k2m6km

Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators

W. Hackbusch, B. N. Khoromskij
2005 Computing  
We focus on the approximation of the operator-valued functions A −σ , σ > 0, and sign(A) for a class of finite difference discretisations A ∈ R N×N .  ...  The asymptotic complexity of our data-sparse representations can be estimated by O(n p log q n), p = 1, 2, with q independent of d, where n = N 1/d is the dimension of the discrete problem in one space  ...  Applying Lemmata 3.1 and 3.2 proves the existence of a low Kronecker rank HKT approximation to the class of multi-dimensional integral operators.  ... 
doi:10.1007/s00607-005-0145-z fatcat:p33xiircrzcb7cimg2m7pmu7hu

Efficient Block Preconditioning for a $C^1$ Finite Element Discretization of the Dirichlet Biharmonic Problem

J. Pestana, R. Muddle, M. Heil, F. Tisseur, M. Mihajlović
2016 SIAM Journal on Scientific Computing  
The eigenvalue analysis is based on the fact that the proposed preconditioner, like the coefficient matrix itself, is symmetric positive definite (SPD) and assembled from element matrices.  ...  Finally, we study robustness of this preconditioner with respect to element stretching, domain distortion, and nonconvex domains.  ...  The authors would like to thank Andy Wathen for fruitful discussions, and the referees for helpful comments and references.  ... 
doi:10.1137/15m1014887 fatcat:rll6oyshzzcfxif4llrkxulnfe

Sparse Inpainting with Smoothed Particle Hydrodynamics [article]

Viktor Daropoulos, Matthias Augustin, Joachim Weickert
2021 arXiv   pre-print
As, in its naive formulation, the SPH technique is not even capable of reproducing constant functions, we modify the approach to obtain an approximation which can reproduce constant and linear functions  ...  The main goal of this work is to perform the image inpainting process from a set of sparsely distributed image samples with the Smoothed Particle Hydrodynamics (SPH) technique.  ...  We thank Vassillen Chizhov for his support regarding programs with spatial and tonal optimization for harmonic and biharmonic inpainting.  ... 
arXiv:2011.11289v4 fatcat:koabmy3z4feybn2jimk4x7uqfa

Fast Numerical Methods for Non-local Operators

Wolfgang Hackbusch, Stefan Sauter, Christoph Schwab
2004 Oberwolfach Reports  
It was Fast Numerical Methods for Non-local Operators 1745 shown that dense matrices arising in BETI could be avoided by using the Fast Multipole Method. • H-matrix arithmetics H-matrices allow the sparse  ...  some of these methods: (1) Cluster methods for the sparse representation of classical Fredholm integral operators. (2) Multipole methods for the fast evaluation of Coulomb-type potentials. (3) H-matrices  ...  of the matrix K can be approximated by matrices of low rank.  ... 
doi:10.4171/owr/2004/33 fatcat:y2fti3akzrbrxjf7i3h7nnv7oa

Mesh deformation based on radial basis function interpolation

A. de Boer, M.S. van der Schoot, H. Bijl
2007 Computers & structures  
The method can handle large mesh deformations caused by translations, rotations and deformations of the boudary of the domain. However, the performance depends on the used RBF.  ...  The method is based on using radial basis functions (RBF's) to interpolate the displacements of the boundary nodes to the whole flow mesh.  ...  The systems encounterd there are approximately as large as n in × n in , with n in the total number of mesh points.  ... 
doi:10.1016/j.compstruc.2007.01.013 fatcat:e42v5vkrarddnjekjmcgfyndva

Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations

E.J. Kansa, Y.C. Hon
2000 Computers and Mathematics with Applications  
Full matrices tend to become progressively more ill-conditioned as the rank increases.  ...  The hybrid combination of these methods contribute to very accurate solutions. Even though FEM gives rise to sparse coefficient matrices, these matrices in practice can become very ill-conditioned.  ...  The work described in this paper was partially supported by a grant from the the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 9040428) and a grant from the  ... 
doi:10.1016/s0898-1221(00)00071-7 fatcat:nf6rxfndfngfjgxbbowm2vjgvi

Dynamic harmonic fields for surface processing

Kai Xu, Hao Zhang, Daniel Cohen-Or, Yueshan Xiong
2009 Computers & graphics  
It maintains the symmetry of the Laplacian system and takes advantage of fast multi-rank updating and downdating of Cholesky factorization, achieving both speed and numerical stability.  ...  In this paper, we propose a method for fast updating of harmonic fields defined on polygonal meshes, enabling real-time insertion and deletion of constraints.  ...  Exploiting supernodal structures improves the locality for large sparse matrices, leading to higher performance in computation and memory access.  ... 
doi:10.1016/j.cag.2009.03.022 fatcat:gbh4qstp7na2jg4tvhznyos3he
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