77,623 research outputs found
Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential
The overlap operator in lattice QCD requires the computation of the sign
function of a matrix, which is non-Hermitian in the presence of a quark
chemical potential. In previous work we introduced an Arnoldi-based Krylov
subspace approximation, which uses long recurrences. Even after the deflation
of critical eigenvalues, the low efficiency of the method restricts its
application to small lattices. Here we propose new short-recurrence methods
which strongly enhance the efficiency of the computational method. Using
rational approximations to the sign function we introduce two variants, based
on the restarted Arnoldi process and on the two-sided Lanczos method,
respectively, which become very efficient when combined with multishift
solvers. Alternatively, in the variant based on the two-sided Lanczos method
the sign function can be evaluated directly. We present numerical results which
compare the efficiencies of a restarted Arnoldi-based method and the direct
two-sided Lanczos approximation for various lattice sizes. We also show that
our new methods gain substantially when combined with deflation.Comment: 14 pages, 4 figures; as published in Comput. Phys. Commun., modified
data in Figs. 2,3 and 4 for improved implementation of FOM algorithm,
extended discussion of the algorithmic cos
Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection
Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi method, but we also discuss the nonorthogonal generalized Leja point (or PAIN) method. The issue of optimal pole selection for rational Krylov methods applied for approximating the resolvent and exponential function, and functions of Markov type, is treated in some detail
Fast Color Space Transformations Using Minimax Approximations
Color space transformations are frequently used in image processing,
graphics, and visualization applications. In many cases, these transformations
are complex nonlinear functions, which prohibits their use in time-critical
applications. In this paper, we present a new approach called Minimax
Approximations for Color-space Transformations (MACT).We demonstrate MACT on
three commonly used color space transformations. Extensive experiments on a
large and diverse image set and comparisons with well-known multidimensional
lookup table interpolation methods show that MACT achieves an excellent balance
among four criteria: ease of implementation, memory usage, accuracy, and
computational speed
Spectrum optimization in multi-user multi-carrier systems with iterative convex and nonconvex approximation methods
Several practical multi-user multi-carrier communication systems are
characterized by a multi-carrier interference channel system model where the
interference is treated as noise. For these systems, spectrum optimization is a
promising means to mitigate interference. This however corresponds to a
challenging nonconvex optimization problem. Existing iterative convex
approximation (ICA) methods consist in solving a series of improving convex
approximations and are typically implemented in a per-user iterative approach.
However they do not take this typical iterative implementation into account in
their design. This paper proposes a novel class of iterative approximation
methods that focuses explicitly on the per-user iterative implementation, which
allows to relax the problem significantly, dropping joint convexity and even
convexity requirements for the approximations. A systematic design framework is
proposed to construct instances of this novel class, where several new
iterative approximation methods are developed with improved per-user convex and
nonconvex approximations that are both tighter and simpler to solve (in
closed-form). As a result, these novel methods display a much faster
convergence speed and require a significantly lower computational cost.
Furthermore, a majority of the proposed methods can tackle the issue of getting
stuck in bad locally optimal solutions, and hence improve solution quality
compared to existing ICA methods.Comment: 33 pages, 7 figures. This work has been submitted for possible
publicatio
New Laplace and Helmholtz solvers
New numerical algorithms based on rational functions are introduced that can
solve certain Laplace and Helmholtz problems on two-dimensional domains with
corners faster and more accurately than the standard methods of finite elements
and integral equations. The new algorithms point to a reconsideration of the
assumptions underlying existing numerical analysis for partial differential
equations
High-Performance Passive Macromodeling Algorithms for Parallel Computing Platforms
This paper presents a comprehensive strategy for fast generation of passive macromodels of linear devices and interconnects on parallel computing hardware. Starting from a raw characterization of the structure in terms of frequency-domain tabulated scattering responses, we perform a rational curve fitting and a postprocessing passivity enforcement. Both algorithms are parallelized and cast in a form that is suitable for deployment on shared-memory multicore platforms. Particular emphasis is placed on the passivity characterization step, which is performed using two complementary strategies. The first uses an iterative restarted and deflated rational Arnoldi process to extract the imaginary Hamiltonian eigenvalues associated with the model. The second is based on an accuracy-controlled adaptive sampling. Various parallelization strategies are discussed for both schemes, with particular care on load balancing between different computing threads and memory occupation. The resulting parallel macromodeling flow is demonstrated on a number of medium- and large-scale structures, showing good scalability up to 16 computational core
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