2,241 research outputs found

    The symmetric-Toeplitz linear system problem in parallel

    Full text link
    [EN] Many algorithms exist that exploit the special structure of Toeplitz matrices for solving linear systems. Nevertheless, these algorithms are difficult to parallelize due to its lower computational cost and the great dependency of the operations involved that produces a great communication cost. The foundation of the parallel algorithm presented in this paper consists of transforming the Toeplitz matrix into a another structured matrix called Cauchy¿like. The particular properties of Cauchy¿like matrices are exploited in order to obtain two levels of parallelism that makes possible to highly reduce the execution time. The experimental results were obtained in a cluster of PC¿s.Supported by Spanish MCYT and FEDER under Grant TIC 2003-08238-C02-02Alonso-Jordá, P.; Vidal Maciá, AM. (2005). The symmetric-Toeplitz linear system problem in parallel. Computational Science -- ICCS 2005,Pt 1, Proceedings. 3514:220-228. https://doi.org/10.1007/11428831_28S2202283514Sweet, D.R.: The use of linear-time systolic algorithms for the solution of toeplitz problems. k Technical Report JCU-CS-91/1, Department of Computer Science, James Cook University, Tue, 23 April 1996 15, 17, 55 GMT (1991)Evans, D.J., Oka, G.: Parallel solution of symmetric positive definite Toeplitz systems. Parallel Algorithms and Applications 12, 297–303 (1998)Gohberg, I., Koltracht, I., Averbuch, A., Shoham, B.: Timing analysis of a parallel algorithm for Toeplitz matrices on a MIMD parallel machine. Parallel Computing 17, 563–577 (1991)Gallivan, K., Thirumalai, S., Dooren, P.V.: On solving block toeplitz systems using a block schur algorithm. In: Proceedings of the 23rd International Conference on Parallel Processing, Boca Raton, FL, USA, vol. 3, pp. 274–281. CRC Press, Boca Raton (1994)Thirumalai, S.: High performance algorithms to solve Toeplitz and block Toeplitz systems. Ph.d. th., Grad. College of the U. of Illinois at Urbana–Champaign (1996)Alonso, P., Badía, J.M., Vidal, A.M.: Parallel algorithms for the solution of toeplitz systems of linear equations. In: Wyrzykowski, R., Dongarra, J., Paprzycki, M., Waśniewski, J. (eds.) PPAM 2004. LNCS, vol. 3019, pp. 969–976. Springer, Heidelberg (2004)Anderson, E., et al.: LAPACK Users’ Guide. SIAM, Philadelphia (1995)Blackford, L., et al.: ScaLAPACK Users’ Guide. SIAM, Philadelphia (1997)Alonso, P., Badía, J.M., González, A., Vidal, A.M.: Parallel design of multichannel inverse filters for audio reproduction. In: Parallel and Distributed Computing and Systems, IASTED, Marina del Rey, CA, USA, vol. II, pp. 719–724 (2003)Loan, C.V.: Computational Frameworks for the Fast Fourier Transform. SIAM Press, Philadelphia (1992)Heinig, G.: Inversion of generalized Cauchy matrices and other classes of structured matrices. Linear Algebra and Signal Proc., IMA, Math. Appl. 69, 95–114 (1994)Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Mathematics of Computation 64, 1557–1576 (1995)Alonso, P., Vidal, A.M.: An efficient and stable parallel solution for symmetric toeplitz linear systems. TR DSIC-II/2005, DSIC–Univ. Polit. Valencia (2005)Kailath, T., Sayed, A.H.: Displacement structure: Theory and applications. SIAM Review 37, 297–386 (1995

    Applications and accuracy of the parallel diagonal dominant algorithm

    Get PDF
    The Parallel Diagonal Dominant (PDD) algorithm is a highly efficient, ideally scalable tridiagonal solver. In this paper, a detailed study of the PDD algorithm is given. First the PDD algorithm is introduced. Then the algorithm is extended to solve periodic tridiagonal systems. A variant, the reduced PDD algorithm, is also proposed. Accuracy analysis is provided for a class of tridiagonal systems, the symmetric, and anti-symmetric Toeplitz tridiagonal systems. Implementation results show that the analysis gives a good bound on the relative error, and the algorithm is a good candidate for the emerging massively parallel machines

    Multigrid waveform relaxation for the time-fractional heat equation

    Get PDF
    In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of O(NMlog(M))O(N M \log(M)) operations, where MM is the number of time steps and NN is the number of spatial grid points. A semi-algebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with non-smooth solutions and a nonlinear problem with applications in porous media, are presented
    corecore