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Journal articles on the topic 'Randomized Numerical Linear Algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Kannan, Ravindran, and Santosh Vempala. "Randomized algorithms in numerical linear algebra." Acta Numerica 26 (May 1, 2017): 95–135. http://dx.doi.org/10.1017/s0962492917000058.

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This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A central unifying idea is sampling the columns (or rows) of a matrix according to their squared lengths.

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2

Martinsson, Per-Gunnar, and Joel A. Tropp. "Randomized numerical linear algebra: Foundations and algorithms." Acta Numerica 29 (May 2020): 403–572. http://dx.doi.org/10.1017/s0962492920000021.

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This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues.Topics include norm estimation, matrix approximation by sampling, structured and unstructured random embeddings, linear regression problems, low-rank approximation, subspace iteration and Krylov methods, error estimation and adaptivity, interpolatory and CUR factorizations, Nyst

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3

Dereziński, Michał, and Michael W. Mahoney. "Determinantal Point Processes in Randomized Numerical Linear Algebra." Notices of the American Mathematical Society 68, no. 01 (2021): 1. http://dx.doi.org/10.1090/noti2202.

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4

Lim, Lek-Heng, and Jonathan Weare. "Fast Randomized Iteration: Diffusion Monte Carlo through the Lens of Numerical Linear Algebra." SIAM Review 59, no. 3 (2017): 547–87. http://dx.doi.org/10.1137/15m1040827.

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5

Novak, Erich. "Optimal linear randomized methods for linear operators in Hilbert spaces." Journal of Complexity 8, no. 1 (1992): 22–36. http://dx.doi.org/10.1016/0885-064x(92)90032-7.

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6

Pan, Victor Y., and Guoliang Qian. "Randomized Preprocessing of Homogeneous Linear Systems of Equations." Linear Algebra and its Applications 432, no. 12 (2010): 3272–318. http://dx.doi.org/10.1016/j.laa.2010.01.023.

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7

Oymak, Samet, and Joel A. Tropp. "Universality laws for randomized dimension reduction, with applications." Information and Inference: A Journal of the IMA 7, no. 3 (2017): 337–446. http://dx.doi.org/10.1093/imaiai/iax011.

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Abstract Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability. This article studies a natural family of randomized dimension reduction maps and a large class of data s

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8

Liu, Yong, and Chuan-Qing Gu. "On greedy randomized block Kaczmarz method for consistent linear systems." Linear Algebra and its Applications 616 (May 2021): 178–200. http://dx.doi.org/10.1016/j.laa.2021.01.024.

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9

Rudi, Alessandro, Leonard Wossnig, Carlo Ciliberto, Andrea Rocchetto, Massimiliano Pontil, and Simone Severini. "Approximating Hamiltonian dynamics with the Nyström method." Quantum 4 (February 20, 2020): 234. http://dx.doi.org/10.22331/q-2020-02-20-234.

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Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simu

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10

Novak, Erich, and Henryk Woźniakowski. "Lower bounds for the complexity of linear functionals in the randomized setting." Journal of Complexity 27, no. 1 (2011): 1–22. http://dx.doi.org/10.1016/j.jco.2010.08.002.

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11

Bai, Zhong-Zhi, and Wen-Ting Wu. "On partially randomized extended Kaczmarz method for solving large sparse overdetermined inconsistent linear systems." Linear Algebra and its Applications 578 (October 2019): 225–50. http://dx.doi.org/10.1016/j.laa.2019.05.005.

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12

Pelling, Art J. R., and Ennes Sarradj. "Efficient Forced Response Computations of Acoustical Systems with a State-Space Approach." Acoustics 3, no. 3 (2021): 581–94. http://dx.doi.org/10.3390/acoustics3030037.

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State-space models have been successfully employed for model order reduction and control purposes in acoustics in the past. However, due to the cubic complexity of the singular value decomposition, which makes up the core of many subspace system identification (SSID) methods, the construction of large scale state-space models from high-dimensional measurement data has been problematic in the past. Recent advances of numerical linear algebra have brought forth computationally efficient randomized rank-revealing matrix factorizations and it has been shown that these factorizations can be used to

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13

Gentle, James. "Matrix Analysis and Applied Linear Algebra, Numerical Linear Algebra, and Applied Numerical Linear Algebra." Journal of the American Statistical Association 96, no. 455 (2001): 1136–37. http://dx.doi.org/10.1198/jasa.2001.s412.

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14

Demmel, James W., Michael T. Heath, and Henk A. van der Vorst. "Parallel numerical linear algebra." Acta Numerica 2 (January 1993): 111–97. http://dx.doi.org/10.1017/s096249290000235x.

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We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of paralled processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymm

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15

Stewart, G. W., Tom King, Yves Achdou, and Frank Stenger. "Book Review: Numerical linear algebra." Mathematics of Computation 68, no. 225 (1999): 453–60. http://dx.doi.org/10.1090/s0025-5718-99-01069-8.

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16

Ballard, Grey, James Demmel, Olga Holtz, and Oded Schwartz. "Minimizing Communication in Numerical Linear Algebra." SIAM Journal on Matrix Analysis and Applications 32, no. 3 (2011): 866–901. http://dx.doi.org/10.1137/090769156.

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17

Dongarra, Jack J., and Victor Eijkhout. "Numerical linear algebra algorithms and software." Journal of Computational and Applied Mathematics 123, no. 1-2 (2000): 489–514. http://dx.doi.org/10.1016/s0377-0427(00)00400-3.

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18

Eldén, Lars. "Numerical linear algebra in data mining." Acta Numerica 15 (May 2006): 327–84. http://dx.doi.org/10.1017/s0962492906240017.

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Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.

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19

Arveson, W. "C*-Algebras and Numerical Linear Algebra." Journal of Functional Analysis 122, no. 2 (1994): 333–60. http://dx.doi.org/10.1006/jfan.1994.1072.

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20

Mastronardi, Nicola, and Sabine Van Huffel. "Numerical linear algebra and its applications." Numerical Linear Algebra with Applications 12, no. 8 (2005): 683. http://dx.doi.org/10.1002/nla.443.

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21

Gonzaga, Clovis Caesar, and Jin Yun Yuan. "Foz2006 numerical linear algebra and optimization." Numerical Linear Algebra with Applications 15, no. 10 (2008): 887–89. http://dx.doi.org/10.1002/nla.601.

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22

Bru, Rafael, and Juan Manuel Peña. "Preface: numerical and applied linear algebra." Advances in Computational Mathematics 35, no. 2-4 (2011): 99–102. http://dx.doi.org/10.1007/s10444-011-9170-y.

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23

Kostlan, Eric. "Complexity theory of numerical linear algebra." Journal of Computational and Applied Mathematics 22, no. 2-3 (1988): 219–30. http://dx.doi.org/10.1016/0377-0427(88)90402-5.

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24

Ziegel, Eric R., and James E. Gentle. "Numerical Linear Algebra for Applications in Statistics." Technometrics 41, no. 3 (1999): 272. http://dx.doi.org/10.2307/1270592.

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25

Appa, Gautam, Allan Findlay, Philip E. Gill, Walter Murray, and Margaret H. Wright. "Numerical Linear Algebra and Optimization: Volume 1." Journal of the Operational Research Society 43, no. 1 (1992): 74. http://dx.doi.org/10.2307/2583704.

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26

Cowles, Mary Kathryn, James E. Gentle, and Kenneth Lange. "Numerical Linear Algebra for Applications in Statistics." Journal of the American Statistical Association 95, no. 450 (2000): 675. http://dx.doi.org/10.2307/2669416.

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27

Appa, Gautam, and Allan Findlay. "Numerical Linear Algebra and Optimization: Volume 1." Journal of the Operational Research Society 43, no. 1 (1992): 74–75. http://dx.doi.org/10.1057/jors.1992.12.

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28

G., W., and Philippe G. Ciarlet. "Introduction to Numerical Linear Algebra and Optimisation." Mathematics of Computation 55, no. 191 (1990): 395. http://dx.doi.org/10.2307/2008817.

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29

Codenotti, B., M. Leoncini, and G. Resta. "Oracle computations in parallel numerical linear algebra." Theoretical Computer Science 127, no. 1 (1994): 99–121. http://dx.doi.org/10.1016/0304-3975(94)90102-3.

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30

Sucharov, L. "Numerical linear algebra and optimization, volume 1." Advances in Engineering Software 14, no. 3 (1992): 237. http://dx.doi.org/10.1016/0965-9978(92)90031-a.

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31

Nachaoui, Abdeljalil. "Numerical linear algebra for reconstruction inverse problems." Journal of Computational and Applied Mathematics 162, no. 1 (2004): 147–64. http://dx.doi.org/10.1016/j.cam.2003.08.009.

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32

Shi, Xiaofei, and David P. Woodruff. "Sublinear Time Numerical Linear Algebra for Structured Matrices." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 4918–25. http://dx.doi.org/10.1609/aaai.v33i01.33014918.

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We show how to solve a number of problems in numerical linear algebra, such as least squares regression, lp-regression for any p ≥ 1, low rank approximation, and kernel regression, in time T(A)poly(log(nd)), where for a given input matrix A ∈ Rn×d, T(A) is the time needed to compute A · y for an arbitrary vector y ∈ Rd. Since T(A) ≤ O(nnz(A)), where nnz(A) denotes the number of non-zero entries of A, the time is no worse, up to polylogarithmic factors, as all of the recent advances for such problems that run in input-sparsity time. However, for many applications, T(A) can be much smaller than

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33

Sturmfels, Bernd, Sascha Timme, and Piotr Zwiernik. "Estimating linear covariance models with numerical nonlinear algebra." Algebraic Statistics 11, no. 1 (2020): 31–52. http://dx.doi.org/10.2140/astat.2020.11.31.

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34

Clements, Dick. "Teaching topics in numerical linear algebra using spreadsheets." International Journal of Mathematical Education in Science and Technology 22, no. 6 (1991): 1003–17. http://dx.doi.org/10.1080/0020739910220619.

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35

Oancea, Bogdan, Ion Gh Rosca, Tudorel Andrei, and Andreea Iluzia Iacob. "Evaluating Java performance for linear algebra numerical computations." Procedia Computer Science 3 (2011): 474–78. http://dx.doi.org/10.1016/j.procs.2010.12.080.

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36

Laub, A. "Numerical linear algebra aspects of control design computations." IEEE Transactions on Automatic Control 30, no. 2 (1985): 97–108. http://dx.doi.org/10.1109/tac.1985.1103900.

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37

Anzt, Hartwig, Terry Cojean, Yen-Chen Chen, et al. "Ginkgo: A high performance numerical linear algebra library." Journal of Open Source Software 5, no. 52 (2020): 2260. http://dx.doi.org/10.21105/joss.02260.

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38

Hansen, Anders C. "Infinite-dimensional numerical linear algebra: theory and applications." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2124 (2010): 3539–59. http://dx.doi.org/10.1098/rspa.2009.0617.

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We present a new method for computing spectra and pseudospectra of bounded operators on separable Hilbert spaces. The core in this theory is a generalization of the pseudospectrum called the n -pseudospectrum.

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39

Pennestrì, E., and R. Stefanelli. "Linear algebra and numerical algorithms using dual numbers." Multibody System Dynamics 18, no. 3 (2007): 323–44. http://dx.doi.org/10.1007/s11044-007-9088-9.

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40

Datta, Biswa Nath. "Linear and numerical linear algebra in control theory: some research problems." Linear Algebra and its Applications 197-198 (January 1994): 755–90. http://dx.doi.org/10.1016/0024-3795(94)90512-6.

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41

Manteuffel, Thomas A., and Linda R. Petzold. "Special Issue on Iterative Methods in Numerical Linear Algebra." SIAM Journal on Scientific and Statistical Computing 13, no. 1 (1992): vii—viii. http://dx.doi.org/10.1137/0913intro1.

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42

Manteuffel, Tom, and Steve McCormick. "Special Section on Iterative Methods in Numerical Linear Algebra." SIAM Journal on Scientific Computing 15, no. 2 (1994): 295–96. http://dx.doi.org/10.1137/0915intro1.

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43

Manteuffel, Tom, and Steve McCormick. "Special Section on Iterative Methods in Numerical Linear Algebra." SIAM Journal on Scientific Computing 15, no. 3 (1994): 545–46. http://dx.doi.org/10.1137/0915intro2.

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44

Manteuffel, Tom. "Special Issue on Iterative Methods in Numerical Linear Algebra." SIAM Journal on Scientific Computing 17, no. 1 (1996): vii—viii. http://dx.doi.org/10.1137/0917001.

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45

Watson, Layne T. "Numerical Linear Algebra Aspects of Globally Convergent Homotopy Methods." SIAM Review 28, no. 4 (1986): 529–45. http://dx.doi.org/10.1137/1028157.

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46

Hajarian, Masoud, Feng Ding, and Jein-Shan Chen. "Applications of Methods of Numerical Linear Algebra in Engineering." Mathematical Problems in Engineering 2014 (2014): 1–2. http://dx.doi.org/10.1155/2014/718560.

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47

Krüger, Jens, and Rüdiger Westermann. "Linear algebra operators for GPU implementation of numerical algorithms." ACM Transactions on Graphics 22, no. 3 (2003): 908–16. http://dx.doi.org/10.1145/882262.882363.

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48

Dekker, T. J., W. Hoffmann, and P. P. M. De Rijk. "Algorithms for solving numerical linear algebra problems on supercomputers." Future Generation Computer Systems 4, no. 4 (1989): 255–63. http://dx.doi.org/10.1016/0167-739x(89)90001-0.

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49

Bertram, Wolfgang. "From linear algebra via affine algebra to projective algebra." Linear Algebra and its Applications 378 (February 2004): 109–34. http://dx.doi.org/10.1016/j.laa.2003.06.021.

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50

Butkovič, Peter. "Max-algebra: the linear algebra of combinatorics?" Linear Algebra and its Applications 367 (July 2003): 313–35. http://dx.doi.org/10.1016/s0024-3795(02)00655-9.

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