539 research outputs found
Fast and accurate con-eigenvalue algorithm for optimal rational approximations
The need to compute small con-eigenvalues and the associated con-eigenvectors
of positive-definite Cauchy matrices naturally arises when constructing
rational approximations with a (near) optimally small error.
Specifically, given a rational function with poles in the unit disk, a
rational approximation with poles in the unit disk may be obtained
from the th con-eigenvector of an Cauchy matrix, where the
associated con-eigenvalue gives the approximation error in the
norm. Unfortunately, standard algorithms do not accurately compute
small con-eigenvalues (and the associated con-eigenvectors) and, in particular,
yield few or no correct digits for con-eigenvalues smaller than the machine
roundoff. We develop a fast and accurate algorithm for computing
con-eigenvalues and con-eigenvectors of positive-definite Cauchy matrices,
yielding even the tiniest con-eigenvalues with high relative accuracy. The
algorithm computes the th con-eigenvalue in operations
and, since the con-eigenvalues of positive-definite Cauchy matrices decay
exponentially fast, we obtain (near) optimal rational approximations in
operations, where is the
approximation error in the norm. We derive error bounds
demonstrating high relative accuracy of the computed con-eigenvalues and the
high accuracy of the unit con-eigenvectors. We also provide examples of using
the algorithm to compute (near) optimal rational approximations of functions
with singularities and sharp transitions, where approximation errors close to
machine precision are obtained. Finally, we present numerical tests on random
(complex-valued) Cauchy matrices to show that the algorithm computes all the
con-eigenvalues and con-eigenvectors with nearly full precision
Parareal Convergence for Oscillatory PDEs with Finite Time-scale Separation
This is the final version. Available on open access from the Society for Industrial and Applied Mathematics via the DOI in this recordA variant of the Parareal method for highly oscillatory systems of PDEs was proposed by Haut and Wingate (2014). In that work they proved superlinear conver- gence of the method in the limit of infinite time scale separation. Their coarse solver features a coordinate transformation and a fast-wave averag- ing method inspired by analysis of multiple scales PDEs and is integrated using an HMM-type method. However, for many physical applications the timescale separation is finite, not infinite. In this paper we prove con- vergence for finite timescale separaration by extending the error bound on the coarse propagator to this case. We show that convergence requires the solution of an optimization problem that involves the averaging win- dow interval, the time step, and the parameters in the problem. We also propose a method for choosing the averaging window relative to the time step based as a function of the finite frequencies inherent in the problem.University of Exete
The effect of two distinct fast time scales in the rotating, stratified Boussinesq equations: variations from quasi-geostrophy
This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Inspired by the use of fast singular limits in time-parallel numerical methods for a single fast frequency, we consider the limiting, nonlinear dynamics for a system of partial differential equations when two fast, distinct time scales are
present. First order slow equations are derived via the method of multiple time scales when the two small parameters are related by a rational power. We find that the resultant system depends only on the relationship of the two fast time-scales, i.e. which fast time is fastest? Using the theory of cancellation of fast oscillations, we show that with the appropriate assumptions on the nonlinear operator of the full system, this reduced slow system is exactly that which the solution will converge to if each asymptotic limit is considered sequentially. The same result is also obtained via the method of renormalization. The specific example of the rotating, stratified Boussinesq equations is explored in detail, indicating that the most common distinguished limit of this system – quasi-geostrophy, is not the only limiting asymptotic system.We wish to thank the 2 anonymous referees whose comments significantly enhanced the presentation and scope of this article. J. P. W. would like to thank A.Larios, K. Julien, G. Chini, and A. Farhat for various discussions that prompted and motivated this work as well as generous support from the Mathematics Department of Brigham Young University. All of the authors wish to acknowledge the DOE LANL/LDRD program for support, as well as the hospitality of the Courant Institute of New York University where some of this work was completed. Wingate also wishes to thank the University of Exeter for support during the completion of this manuscript
A high-order scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator
ArticleThis is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record IMA J Numer Anal (2015) is available online at http://imajna.oxfordjournals.org/content/early/2015/06/16/imanum.drv021The manuscript presents a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u/∂t=Lu∂u/∂t=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(τL)exp(τL) for a relatively large time-step ττ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge–Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials
Beyond spatial scalability limitations with a massively parallel method for linear oscillatory problems
This is the author accepted manuscript. The final version is available from SAGE Publications via the DOI in this record.This paper presents, discusses and analyses a massively parallel-in-time solver for linear oscillatory PDEs, which
is a key numerical component for evolving weather, ocean, climate and seismic models. The time parallelization in
this solver allows us to significantly exceed the computing resources used by parallelization-in-space methods and
results in a correspondingly significantly reduced wall-clock time. One of the major difficulties of achieving Exascale
performance for weather prediction is that the strong scaling limit – the parallel performance for a fixed problem size
with an increasing number of processors – saturates. A main avenue to circumvent this problem is to introduce new
numerical techniques that take advantage of time parallelism. In this paper we use a time-parallel approximation that
retains the frequency information of oscillatory problems. This approximation is based on (a) reformulating the original
problem into a large set of independent terms and (b) solving each of these terms independently of each other which
can now be accomplished on a large number of HPC resources. Our results are conducted on up to 3586 cores for
problem sizes with the parallelization-in-space scalability limited already on a single node. We gain significant reductions
in the time-to-solution of 118.3 for spectral methods and 1503.0 for finite-difference methods with the parallelizationin-time
approach. A developed and calibrated performance model gives the scalability limitations a-priory for this new
approach and allows us to extrapolate the performance method towards large-scale system. This work has the potential
to contribute as a basic building block of parallelization-in-time approaches, with possible major implications in applied
areas modelling oscillatory dominated problems.The authors gratefully acknowledge the Gauss Centre for Supercomputing
e.V. (www.gauss-centre.eu) for funding this
project by providing computing time on the GCS Supercomputer
SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.
de). We also acknowledge use of Hartree Centre resources in this
work on which the early evaluation of the parallelization concepts
were done
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