1,399 research outputs found
Nova rješenja u φ 4 -teoriji s gušenjem
Two-kink and kink-antikink states in φ 4 -theory with damping are constructed.U okviru φ 4 teorije konstruirana su rješenja za dvojni prijelom i prijelomantiprijelom s gušenjem
Gradient flow approach to geometric convergence analysis of preconditioned eigensolvers
Preconditioned eigenvalue solvers (eigensolvers) are gaining popularity, but
their convergence theory remains sparse and complex. We consider the simplest
preconditioned eigensolver--the gradient iterative method with a fixed step
size--for symmetric generalized eigenvalue problems, where we use the gradient
of the Rayleigh quotient as an optimization direction. A sharp convergence rate
bound for this method has been obtained in 2001--2003. It still remains the
only known such bound for any of the methods in this class. While the bound is
short and simple, its proof is not. We extend the bound to Hermitian matrices
in the complex space and present a new self-contained and significantly shorter
proof using novel geometric ideas.Comment: 8 pages, 2 figures. Accepted to SIAM J. Matrix Anal. (SIMAX
Evaluation of Directive-Based GPU Programming Models on a Block Eigensolver with Consideration of Large Sparse Matrices
Achieving high performance and performance portability for large-scale scientific applications is a major challenge on heterogeneous computing systems such as many-core CPUs and accelerators like GPUs. In this work, we implement a widely used block eigensolver, Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG), using two popular directive based programming models (OpenMP and OpenACC) for GPU-accelerated systems. Our work differs from existing work in that it adopts a holistic approach that optimizes the full solver performance rather than narrowing the problem into small kernels (e.g., SpMM, SpMV). Our LOPBCG GPU implementation achieves a 2.8–4.3 speedup over an optimized CPU implementation when tested with four different input matrices. The evaluated configuration compared one Skylake CPU to one Skylake CPU and one NVIDIA V100 GPU. Our OpenMP and OpenACC LOBPCG GPU implementations gave nearly identical performance. We also consider how to create an efficient LOBPCG solver that can solve problems larger than GPU memory capacity. To this end, we create microbenchmarks representing the two dominant kernels (inner product and SpMM kernel) in LOBPCG and then evaluate performance when using two different programming approaches: tiling the kernels, and using Unified Memory with the original kernels. Our tiled SpMM implementation achieves a 2.9 and 48.2 speedup over the Unified Memory implementation on supercomputers with PCIe Gen3 and NVLink 2.0 CPU to GPU interconnects, respectively
Absolute value preconditioning for symmetric indefinite linear systems
We introduce a novel strategy for constructing symmetric positive definite
(SPD) preconditioners for linear systems with symmetric indefinite matrices.
The strategy, called absolute value preconditioning, is motivated by the
observation that the preconditioned minimal residual method with the inverse of
the absolute value of the matrix as a preconditioner converges to the exact
solution of the system in at most two steps. Neither the exact absolute value
of the matrix nor its exact inverse are computationally feasible to construct
in general. However, we provide a practical example of an SPD preconditioner
that is based on the suggested approach. In this example we consider a model
problem with a shifted discrete negative Laplacian, and suggest a geometric
multigrid (MG) preconditioner, where the inverse of the matrix absolute value
appears only on the coarse grid, while operations on finer grids are based on
the Laplacian. Our numerical tests demonstrate practical effectiveness of the
new MG preconditioner, which leads to a robust iterative scheme with minimalist
memory requirements
Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix
The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a
matrix with columns that form an orthonormal basis for a subspace \X, and
a Hermitian matrix , the eigenvalues of are called Ritz values of
with respect to \X. If the subspace \X is -invariant then the Ritz
values are some of the eigenvalues of . If the -invariant subspace \X
is perturbed to give rise to another subspace \Y, then the vector of absolute
values of changes in Ritz values of represents the absolute eigenvalue
approximation error using \Y. We bound the error in terms of principal angles
between \X and \Y. We capitalize on ideas from a recent paper [DOI:
10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute
values of differences between Ritz values for subspaces \X and \Y was
weakly (sub-)majorized by a constant times the sine of the vector of principal
angles between \X and \Y, the constant being the spread of the spectrum of
. In that result no assumption was made on either subspace being
-invariant. It was conjectured there that if one of the trial subspaces is
-invariant then an analogous weak majorization bound should only involve
terms of the order of sine squared. Here we confirm this conjecture.
Specifically we prove that the absolute eigenvalue error is weakly majorized by
a constant times the sine squared of the vector of principal angles between the
subspaces \X and \Y, where the constant is proportional to the spread of
the spectrum of . For many practical cases we show that the proportionality
factor is simply one, and that this bound is sharp. For the general case we can
only prove the result with a slightly larger constant, which we believe is
artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and
Applications (SIMAX
Theoretical investigation of TbNi_{5-x}Cu_x optical properties
In this paper we present theoretical investigation of optical conductivity
for intermetallic TbNi_{5-x}Cu_x series. In the frame of LSDA+U calculations
electronic structure for x=0,1,2 and on top of that optical conductivities were
calculated. Disorder effects of Ni for Cu substitution on a level of LSDA+U
densities of states (DOS) were taken into account via averaging over all
possible Cu ion positions for given doping level x. Gradual suppression and
loosing of structure of optical conductivity at 2 eV together with simultaneous
intensity growth at 4 eV correspond to increase of Cu and decrease of Ni
content. As reported before [Knyazev et al., Optics and Spectroscopy 104, 360
(2008)] plasma frequency has non monotonic doping behaviour with maximum at
x=1. This behaviour is explained as competition between lowering of total
density of states on the Fermi level N(E_F) and growing of number of carriers.
Our theoretical results agree well with variety of recent experiments.Comment: 4 pages, 3 figure
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