396 research outputs found

    Inertia-based spectrum slicing for symmetric quadratic eigenvalue problems

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    [EN] In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos-based method for computing all real eigenvalues contained in a given interval of large-scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory-efficient by representing the computed pseudo-Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.Agencia Estatal de Investigacion, Grant/Award Number: TIN2016-75985-PCampos, C.; Román Moltó, JE. (2020). Inertia-based spectrum slicing for symmetric quadratic eigenvalue problems. Numerical Linear Algebra with Applications. 27(4):1-17. https://doi.org/10.1002/nla.2293S117274Tisseur, F., & Meerbergen, K. (2001). The Quadratic Eigenvalue Problem. SIAM Review, 43(2), 235-286. doi:10.1137/s0036144500381988Veselić, K. (2011). Damped Oscillations of Linear Systems. Lecture Notes in Mathematics. doi:10.1007/978-3-642-21335-9Grimes, R. G., Lewis, J. G., & Simon, H. D. (1994). A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems. SIAM Journal on Matrix Analysis and Applications, 15(1), 228-272. doi:10.1137/s0895479888151111Campos, C., & Roman, J. E. (2012). Strategies for spectrum slicing based on restarted Lanczos methods. Numerical Algorithms, 60(2), 279-295. doi:10.1007/s11075-012-9564-zLi, R., Xi, Y., Vecharynski, E., Yang, C., & Saad, Y. (2016). A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems. SIAM Journal on Scientific Computing, 38(4), A2512-A2534. doi:10.1137/15m1054493Guo, C.-H., Higham, N. J., & Tisseur, F. (2010). An Improved Arc Algorithm for Detecting Definite Hermitian Pairs. SIAM Journal on Matrix Analysis and Applications, 31(3), 1131-1151. doi:10.1137/08074218xNiendorf, V., & Voss, H. (2010). Detecting hyperbolic and definite matrix polynomials. Linear Algebra and its Applications, 432(4), 1017-1035. doi:10.1016/j.laa.2009.10.014NakatsukasaY NoferiniV. Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems;2017. Preprint arXiv:1711.00495.Parlett, B. N., & Chen, H. C. (1990). Use of indefinite pencils for computing damped natural modes. Linear Algebra and its Applications, 140, 53-88. doi:10.1016/0024-3795(90)90222-xCampos, C., & Roman, J. E. (2016). Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems. BIT Numerical Mathematics, 56(4), 1213-1236. doi:10.1007/s10543-016-0601-5Guo, C.-H., & Lancaster, P. (2005). Algorithms for hyperbolic quadratic eigenvalue problems. Mathematics of Computation, 74(252), 1777-1792. doi:10.1090/s0025-5718-05-01748-5Li, H., & Cai, Y. (2015). Solving the real eigenvalues of hermitian quadratic eigenvalue problems via bisection. The Electronic Journal of Linear Algebra, 30, 721-743. doi:10.13001/1081-3810.1979RomanJE CamposC RomeroE andTomasA. SLEPc users manual. DSIC‐II/24/02–Revision 3.9. D. Sistemes Informàtics i Computació Universitat Politècnica de València;2018.Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPc. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Guo, J.-S., Lin, W.-W., & Wang, C.-S. (1995). Numerical solutions for large sparse quadratic eigenvalue problems. Linear Algebra and its Applications, 225, 57-89. doi:10.1016/0024-3795(93)00318-tSleijpen, G. L. G., Booten, A. G. L., Fokkema, D. R., & van der Vorst, H. A. (1996). Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT Numerical Mathematics, 36(3), 595-633. doi:10.1007/bf01731936Bai, Z., & Su, Y. (2005). SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem. SIAM Journal on Matrix Analysis and Applications, 26(3), 640-659. doi:10.1137/s0895479803438523Güttel, S., & Tisseur, F. (2017). The nonlinear eigenvalue problem. Acta Numerica, 26, 1-94. doi:10.1017/s0962492917000034Yang, L., Sun, Y., & Gong, F. (2018). The inexact residual iteration method for quadratic eigenvalue problem and the analysis of convergence. Journal of Computational and Applied Mathematics, 332, 45-55. doi:10.1016/j.cam.2017.10.003Keçeli, M., Corsetti, F., Campos, C., Roman, J. E., Zhang, H., Vázquez-Mayagoitia, Á., … Wagner, A. F. (2018). SIESTA-SIPs: Massively parallel spectrum-slicing eigensolver for an ab initio molecular dynamics package. Journal of Computational Chemistry, 39(22), 1806-1814. doi:10.1002/jcc.25350Voss, H., Werner, B., & Hadeler, K. P. (1982). A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems. Mathematical Methods in the Applied Sciences, 4(1), 415-424. doi:10.1002/mma.1670040126Higham, N. J., Mackey, D. S., & Tisseur, F. (2009). Definite Matrix Polynomials and their Linearization by Definite Pencils. SIAM Journal on Matrix Analysis and Applications, 31(2), 478-502. doi:10.1137/080721406Al-Ammari, M., & Tisseur, F. (2012). Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification. Linear Algebra and its Applications, 436(10), 3954-3973. doi:10.1016/j.laa.2010.08.035Gohberg, I., Lancaster, P., & Rodman, L. (1980). Spectral Analysis of Selfadjoint Matrix Polynomials. The Annals of Mathematics, 112(1), 33. doi:10.2307/1971320RozložnÍk, M., Okulicka-DŁużewska, F., & Smoktunowicz, A. (2015). Cholesky-Like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms. SIAM Journal on Matrix Analysis and Applications, 36(2), 727-751. doi:10.1137/130947003Lu, D., Su, Y., & Bai, Z. (2016). Stability Analysis of the Two-level Orthogonal Arnoldi Procedure. SIAM Journal on Matrix Analysis and Applications, 37(1), 195-214. doi:10.1137/151005142Campos, C., & Roman, J. E. (2016). Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc. SIAM Journal on Scientific Computing, 38(5), S385-S411. doi:10.1137/15m1022458Higham, N. J., Mackey, D. S., Mackey, N., & Tisseur, F. (2007). Symmetric Linearizations for Matrix Polynomials. SIAM Journal on Matrix Analysis and Applications, 29(1), 143-159. doi:10.1137/050646202BalayS AbhyankarS AdamsM et al. PETSc users manual. ANL‐95/11 ‐ Revision 3.10. Argonne National Laboratory;2018.Betcke, T., Higham, N. J., Mehrmann, V., Schröder, C., & Tisseur, F. (2013). NLEVP. 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    Graph Clustering by Flow Simulation

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    Analysis of structured polynomial eigenvalue problems

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    This thesis considers Hermitian/symmetric, alternating and palindromic matrix polynomials which all arise frequently in a variety of applications, such as vibration analysis of dynamical systems and optimal control problems. A classification of Hermitian matrix polynomials whose eigenvalues belong to the extended real line, with each eigenvalue being of definite type, is provided first. We call such polynomials quasidefinite. Definite pencils, definitizable pencils, overdamped quadratics, gyroscopically stabilized quadratics, (quasi)hyperbolic and definite matrix polynomials are all quasidefinite. We show, using homogeneous rotations, special Hermitian linearizations and a new characterization of hyperbolic matrix polynomials, that the main common thread between these many subclasses is the distribution of their eigenvalue types. We also identify, amongst all quasihyperbolic matrix polynomials, those that can be diagonalized by a congruence transformation applied to a Hermitian linearization of the matrix polynomial while maintaining the structure of the linearization. Secondly, we generalize the notion of self-adjoint standard triples associated with Hermitian matrix polynomials in Gohberg, Lancaster and Rodman's theory of matrix polynomials to present spectral decompositions of structured matrix polynomials in terms of standard pairs (X,T), which are either real or complex, plus a parameter matrix S that acquires particular properties depending on the structure under investigation. These decompositions are mainly an extension of the Jordan canonical form for a matrix over the real or complex field so we investigate the important special case of structured Jordan triples. Finally, we use the concept of structured Jordan triples to solve a structured inverse polynomial eigenvalue problem. As a consequence, we can enlarge the collection of nonlinear eigenvalue problems [NLEVP, 2010] by generating quadratic and cubic quasidefinite matrix polynomials in different subclasses from some given spectral data by solving an appropriate inverse eigenvalue problem. For the quadratic case, we employ available algorithms to provide tridiagonal definite matrix polynomials.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

    Cortical spatio-temporal dimensionality reduction for visual grouping

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    The visual systems of many mammals, including humans, is able to integrate the geometric information of visual stimuli and to perform cognitive tasks already at the first stages of the cortical processing. This is thought to be the result of a combination of mechanisms, which include feature extraction at single cell level and geometric processing by means of cells connectivity. We present a geometric model of such connectivities in the space of detected features associated to spatio-temporal visual stimuli, and show how they can be used to obtain low-level object segmentation. The main idea is that of defining a spectral clustering procedure with anisotropic affinities over datasets consisting of embeddings of the visual stimuli into higher dimensional spaces. Neural plausibility of the proposed arguments will be discussed

    Application of the Non-Hermitian Singular Spectrum Analysis to the exponential retrieval problem

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    We present a new approach to solve the exponential retrieval problem. We derive a stable technique, based on the singular value decomposition (SVD) of lag-covariance and crosscovariance matrices consisting of covariance coefficients computed for index translated copies of an initial time series. For these matrices a generalized eigenvalue problem is solved. The initial signal is mapped into the basis of the generalized eigenvectors and phase portraits are consequently analyzed. Pattern recognition techniques could be applied to distinguish phase portraits related to the exponentials and noise. Each frequency is evaluated by unwrapping phases of the corresponding portrait, detecting potential wrapping events and estimation of the phase slope. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise
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