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New estimates for the recursive low-rank truncation of block-structured matrices

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Abstract

The best approximation of a matrix by a low-rank matrix can be obtained by the singular value decomposition. For large-sized matrices this approach is too costly. Instead one may use a block decomposition. Approximating the small submatrices by low-rank matrices and agglomerating them into a new, coarser block decomposition, one obtains a recursive method. The required computation work is \(O(rnm)\) where \(r\) is the desired rank and \(n\times m\) is the size of the matrix. The paper discusses the new error estimates for \(A-B\) and \(M-A\) where \(A\) is the result of the recursive truncation applied to \(M\), while \(B\) is the best approximation.

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Notes

  1. The algorithm is completely different from spectral divide and conquer methods, as they are analysed, e.g., in [10] and the literature cited therein.

  2. The applications to hierarchical matrices are cheaper because of their special structures.

  3. In principle, the cost must depend on the underlying machine precision. The constants also depend on the implemented version of the algorithms etc. We introduce these explicit numbers to be able to indicate the cost of the later algorithm quantitatively. Otherwise, only statements using \(\mathcal {O}(\ldots )\) are possible.

  4. We use the term ‘orthogonal matrix’ also for rectangular matrices whose columns are pairwise orthonormal.

  5. The possible case of \({\text {rank}}(A)<r\) is included since we may choose \(\lambda _{i}=0\).

  6. The particular values are \(\mu =50\) and \(r=25.\) For \(\omega =0,\) the choice of \(2r=\mu \) yields the worse case \(q=(1+\sqrt{5})/2\). However, for larger values of \(\omega \), the results are rather independent of \(\mu \) and \(r\).

  7. For \(\omega \ge 12\), the worse case is again given by (4.6).

  8. \(\chi \circ \delta \) is the Hadamard product (entry-wise product) defined by \(\left( \chi \circ \delta \right) _{ij}=\chi _{ij} \delta _{ij}\).

  9. For subspaces \(V\subset \mathbb {R}^{I}\) and \(W\subset \mathbb {R}^{J},\) the tensor space \(V\otimes W\) is defined by \({\text {span}}\{vw^{\mathsf {T}}:v\in V,w\in W\}\) (cf. [7, Remark 1.3]).

References

  1. Bebendorf, M., Hackbusch, W.: Existence of \(\cal {H}\)-matrix approximants to the inverse FE-matrix of elliptic operators with \({L}^{\infty }\)-coefficients. Numer. Math. 95, 1–28 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Demmel, J., Gu, M., Eisenstat, S., Slapničar, I., Veselić, K., Drmač, Z.: Computing the singular value decomposition with high relative accuracy. Linear Alg. Appl 299, 21–80 (1999)

    Article  MATH  Google Scholar 

  3. Drmač, Z.: Accurate computation of the product-induced singular value decomposition with applications. SIAM J. Numer. Anal. 35, 1969–1994 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  5. Grasedyck, L., Hackbusch, W.: Construction and arithmetics of \(\cal {H}\)-matrices. Computing 70, 295–334 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hackbusch, W.: Hierarchische Matrizen—Algorithmen und Analysis, Springer, Berlin, 2009—Hierarchical Matrices—Algorithms and Analysis. Springer, Berlin (2015, to appear)

  7. Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. SCM. Springer, Berlin (2012)

    Book  Google Scholar 

  8. Hackbusch, W., Khoromskij, B., Kriemann, R.: Hierarchical matrices based on a weak admissibility criterion. Computing 73, 207–243 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kandler, U., Schröder, C.: Spectral error bounds for Hermitian inexact Krylov methods. Preprint 11–2014, Institute of Mathematics, Technische Universität Berlin (2014)

  10. Nakatsukasa, Y., Higham, N.J.: Stable and efficient divide and conquer algorithms for the symmetric eigenvalue decomposition and the SVD. SIAM J. Sci. Comput. 35, A1325–A1349 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Schmidt, E.: Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil: Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener. Math. Ann. 63, 433–476 (1907)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We thank Dr. R. Kriemann for providing the numerical tests.

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  1. Max-Planck-Institut Mathematik in den Naturwissenschaften, Inselstr. 22, 04103, Leipzig, Germany

    Wolfgang Hackbusch

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  1. Wolfgang Hackbusch
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Correspondence to Wolfgang Hackbusch.

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Hackbusch, W. New estimates for the recursive low-rank truncation of block-structured matrices. Numer. Math. 132, 303–328 (2016). https://doi.org/10.1007/s00211-015-0716-7

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  • DOI: https://doi.org/10.1007/s00211-015-0716-7

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