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# Algorithms for Rank and Tensor Structured Matrices

### Researcher:

• Thomas Mach
Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg,
Computational Methods in Systems and Control Theory,
Tel: +49 (0)391-6110-806

### Project Description

Large dense matrices can often not be handled with standard algorithms due to the limitation of storage. Special algorithms make use of the structure of the matrix. We investigate matrices with hierarchical rank structure (see [1]) and tensor-train structure (see [2]).

Often it is necessary to compute the eigenvalues of a matrix. The eigenvalues are important to determine the properties of the matrix and the problem behind. Sometimes it is not enough to know some eigenvalues, e.g. in the Anderson-model in particle physics.

The storage of hierarchical matrices (ℋ-matrices) needs a lower amount in compare to full matrices, so they are data-sparse matrices. Many matrix operation for ℋ-matrices only needs linear-polylogarithmic time.

One goal of this project is to find (if possible) all eigenvalues of an ℋ-matrix. These algorithms will be tested and evaluated at selected examples.

We also investigate the computation of some eigenvalues of matrices given in tensor-train matrix structure.

### Duration and Funding

• May 2008 - September 2010: Chemnitz University of Technology funded by a grant from the Free State of Saxony (Sächsisches Landesstipendium)
• October 2010 - June 2012: MPI Magdeburg

### Related Publications

 @Article{BenM12, author = {P. Benner and T. Mach}, title = {The Preconditioned Inverse Iteration for Hierarchical Matrices}, year = 2013, journal = {Numer. Lin. Alg. Appl.}, doi = {10.1002/nla.1830}, pages = {150--166}, volume=20, number=1 } Close BibTeX.185 The Preconditioned Inverse Iteration for Hierarchical Matrices; Benner, Peter; Mach, Thomas; Numer. Lin. Alg. Appl.  :  Vol. 20, No. 1, 150-166;2013. DOI: 10.1002/nla.1830also available as preprint MPIMD/11-01, 17 pages. @Article{BenM12b, author = {Peter Benner and Thomas Mach}, title = {Computing All or Some Eigenvalues of Symmetric $\mathcal{H}_{\ell}$-Matrices}, publisher = {SIAM}, year = {2012}, journal = {SIAM Journal on Scientific Computing}, volume = {34}, number = {1}, pages = {A485-A496}, keywords = {symmetric hierarchical matrices; eigenvalues; $\mathcal{H}_{\ell}$-matrices; slicing the spectrum}, url = {http://link.aip.org/link/?SCE/34/A485/1}, doi = {10.1137/100815323} } Close BibTeX.155 Computing all or some Eigenvalues of symmetric ℋℓ-Matrices; Benner, Peter; Mach, Thomas; SIAM Journal of Scientific Computing  :  Vol. 34, No. 1, A485-A496;2012. DOI: 10.1137/100815323also available as preprint MPIMD/10-01. @TECHREPORT{MPIMD12-05, author = {Peter Benner and Thomas Mach}, title = {The LR Cholesky Algorithm for Symmetric Hierarchical Matrices}, number = {MPIMD/12-05}, month = {February}, year = 2012, type = {MPI Magdeburg Preprint}, note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/}}, } Close BibTeX.227 The LR Cholesky Algorithm for Symmetric Hierarchical Matrices; Benner, Peter; Mach, Thomas ; 2012. available as preprint MPIMD/12-05, 16 pages. @TECHREPORT{MPIMD11-09, author = {Thomas Mach}, title = {Computing Inner Eigenvalues of Matrices in Tensor Train Matrix Format}, institution = {Max Planck Institute Magdeburg Preprints}, year = 2011, number = {MPIMD/11-09}, month = {December} } Close BibTeX.201 Computing Inner Eigenvalues of Matrices in Tensor Train Matrix Format; Mach, Thomas; in A. Cangiani, R.L. Davidchack, E.H. Georgoulis, A. Gorban, J. Levesley, M.V. Tretyakov: Numerical Mathematics and Advanced Applications 2011 - Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics, Leicester  :  Springer; 2012. accepted, available as preprint MPIMD/11-09, 11 pages. ï»¿@InProceedings{MacS12, author = {T. Mach and J. Saak}, title = {How Competitive is the ADI for Tensor Structured Equations?}, journal = {PAMM}, volume = {12}, number = {1}, publisher = {WILEY-VCH Verlag}, issn = {1617-7061}, year = {2011}, note = {DOI: 10.1002/pamm.201210306}, } Close BibTeX.233 How Competitive is the ADI for Tensor Structured Equations?; Mach, Thomas; Saak, Jens; Proceedings in Applied Mathematics and Mechanics  :  Wiley InterScience; 2012. Pages: 635â€“636, DOI: 10.1002/pamm.201210306. ï»¿@article {BenM11d, author = {P. Benner and T. Mach}, title = {Locally Optimal Block Preconditioned Conjugate Gradient Method for Hierarchical Matrices}, journal = {PAMM}, volume = {11}, number = {1}, publisher = {WILEY-VCH Verlag}, issn = {1617-7061}, url = {http://dx.doi.org/10.1002/pamm.201110360}, doi = {10.1002/pamm.201110360}, pages = {741--742}, year = {2011}, } Close BibTeX.192 Locally Optimal Block Preconditioned Conjugate Gradient Method for Hierarchical Matrices; Benner, Peter; Mach, Thomas; Proceedings in Applied Mathematics and Mechanics  :  Volume 11, Issue 1, pages 741â€“742, December 2011;Wiley InterScience; 2011. DOI: 10.1002/pamm.201110360. @TECHREPORT{MPIMD11-12, author = {Thomas Mach, Jens Saak}, title = {Towards an ADI iteration for Tensor Structured Equations}, institution = {Max Planck Institute Magdeburg Preprints}, year = 2011, number = {MPIMD/11-12}, month = {December} } Close BibTeX.204 Towards an ADI iteration for Tensor Structured Equations; Mach, Thomas; Saak, Jens; Max Planck Institute Magdeburg Preprints, MPIMD/11-12;2011. 16 pages. @ARTICLE{BenM10, author = {Peter Benner and Thomas Mach}, title = {{O}n the {QR} {D}ecomposition of $\mathcal{H}$-{M}atrices}, year = {2010}, journal = {Computing}, volume = {88}, number = {3--4}, pages = {111--129} } Close BibTeX.137 On the QR Decomposition of ℋ-Matrices; Benner, Peter; Mach, Thomas; Computing  :  Vol. 88, No. 3-4, pp. 111-129;Springer; 2010. DOI: 10.1007/s00607-010-0087-ysee also CSC preprint 09-04. 154 Computing the Eigenvalues of Hierarchical Matrices by LR-Cholesky Transformations; Benner, Peter; Mach, Thomas; Mathematisches Forschungsinstitut Oberwolfach, Report No. 37/2009, pp. 325-328;Mathematisches Forschungsinstitut Oberwolfach; 2009. DOI: 10.4171/OWR/2009/37.

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