3,140 research outputs found
Preconditioning complex symmetric linear systems
A new polynomial preconditioner for symmetric complex linear systems based on
Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear
systems is herein presented. It applies to Conjugate Orthogonal Conjugate
Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative
solvers and does not require any estimation of the spectrum of the coefficient
matrix. An upper bound of the condition number of the preconditioned linear
system is provided. Moreover, to reduce the computational cost, an inexact
variant based on incomplete Cholesky decomposition or orthogonal polynomials is
proposed. Numerical results show that the present preconditioner and its
inexact variant are efficient and robust solvers for this class of linear
systems. A stability analysis of the method completes the description of the
preconditioner.Comment: 26 pages, 4 figures, 4 table
Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
The numerical solution of partial differential equations on high-dimensional
domains gives rise to computationally challenging linear systems. When using
standard discretization techniques, the size of the linear system grows
exponentially with the number of dimensions, making the use of classic
iterative solvers infeasible. During the last few years, low-rank tensor
approaches have been developed that allow to mitigate this curse of
dimensionality by exploiting the underlying structure of the linear operator.
In this work, we focus on tensors represented in the Tucker and tensor train
formats. We propose two preconditioned gradient methods on the corresponding
low-rank tensor manifolds: A Riemannian version of the preconditioned
Richardson method as well as an approximate Newton scheme based on the
Riemannian Hessian. For the latter, considerable attention is given to the
efficient solution of the resulting Newton equation. In numerical experiments,
we compare the efficiency of our Riemannian algorithms with other established
tensor-based approaches such as a truncated preconditioned Richardson method
and the alternating linear scheme. The results show that our approximate
Riemannian Newton scheme is significantly faster in cases when the application
of the linear operator is expensive.Comment: 24 pages, 8 figure
Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems
In this paper we solve the Helmholtz equation with multigrid preconditioned
Krylov subspace methods. The class of Shifted Laplacian preconditioners are
known to significantly speed-up Krylov convergence. However, these
preconditioners have a parameter beta, a measure of the complex shift. Due to
contradictory requirements for the multigrid and Krylov convergence, the choice
of this shift parameter can be a bottleneck in applying the method. In this
paper, we propose a wavenumber-dependent minimal complex shift parameter which
is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid
scheme. We claim that, given any (regionally constant) wavenumber, this minimal
complex shift parameter provides the reader with a parameter choice that leads
to efficient Krylov convergence. Numerical experiments in one and two spatial
dimensions validate the theoretical results. It appears that the proposed
complex shift is both the minimal requirement for a multigrid V-cycle to
converge, as well as being near-optimal in terms of Krylov iteration count.Comment: 20 page
On Convergence of the Inexact Rayleigh Quotient Iteration with the Lanczos Method Used for Solving Linear Systems
For the Hermitian inexact Rayleigh quotient iteration (RQI), the author has
established new local general convergence results, independent of iterative
solvers for inner linear systems. The theory shows that the method locally
converges quadratically under a new condition, called the uniform positiveness
condition. In this paper we first consider the local convergence of the inexact
RQI with the unpreconditioned Lanczos method for the linear systems. Some
attractive properties are derived for the residuals, whose norms are
's, of the linear systems obtained by the Lanczos method. Based on
them and the new general convergence results, we make a refined analysis and
establish new local convergence results. It is proved that the inexact RQI with
Lanczos converges quadratically provided that with a
constant . The method is guaranteed to converge linearly provided
that is bounded by a small multiple of the reciprocal of the
residual norm of the current approximate eigenpair. The results are
fundamentally different from the existing convergence results that always
require , and they have a strong impact on effective
implementations of the method. We extend the new theory to the inexact RQI with
a tuned preconditioned Lanczos for the linear systems. Based on the new theory,
we can design practical criteria to control to achieve quadratic
convergence and implement the method more effectively than ever before.
Numerical experiments confirm our theory.Comment: 20 pages, 8 figures. arXiv admin note: text overlap with
arXiv:0906.223
A Numerical Approach to Space-Time Finite Elements for the Wave Equation
We study a space-time finite element approach for the nonhomogeneous wave
equation using a continuous time Galerkin method. We present fully implicit
examples in 1+1, 2+1, and 3+1 dimensions using linear quadrilateral,
hexahedral, and tesseractic elements. Krylov solvers with additive Schwarz
preconditioning are used for solving the linear system. We introduce a time
decomposition strategy in preconditioning which significantly improves
performance when compared with unpreconditioned cases.Comment: 9 pages, 5 figures, 5 table
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