7,474 research outputs found
Numerical Methods for the QCD Overlap Operator:III. Nested Iterations
The numerical and computational aspects of chiral fermions in lattice quantum
chromodynamics are extremely demanding. In the overlap framework, the
computation of the fermion propagator leads to a nested iteration where the
matrix vector multiplications in each step of an outer iteration have to be
accomplished by an inner iteration; the latter approximates the product of the
sign function of the hermitian Wilson fermion matrix with a vector. In this
paper we investigate aspects of this nested paradigm. We examine several Krylov
subspace methods to be used as an outer iteration for both propagator
computations and the Hybrid Monte-Carlo scheme. We establish criteria on the
accuracy of the inner iteration which allow to preserve an a priori given
precision for the overall computation. It will turn out that the accuracy of
the sign function can be relaxed as the outer iteration proceeds. Furthermore,
we consider preconditioning strategies, where the preconditioner is built upon
an inaccurate approximation to the sign function. Relaxation combined with
preconditioning allows for considerable savings in computational efforts up to
a factor of 4 as our numerical experiments illustrate. We also discuss the
possibility of projecting the squared overlap operator into one chiral sector.Comment: 33 Pages; citations adde
Solving Factored MDPs with Hybrid State and Action Variables
Efficient representations and solutions for large decision problems with
continuous and discrete variables are among the most important challenges faced
by the designers of automated decision support systems. In this paper, we
describe a novel hybrid factored Markov decision process (MDP) model that
allows for a compact representation of these problems, and a new hybrid
approximate linear programming (HALP) framework that permits their efficient
solutions. The central idea of HALP is to approximate the optimal value
function by a linear combination of basis functions and optimize its weights by
linear programming. We analyze both theoretical and computational aspects of
this approach, and demonstrate its scale-up potential on several hybrid
optimization problems
A Continuation Multilevel Monte Carlo algorithm
We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for
weak approximation of stochastic models. The CMLMC algorithm solves the given
approximation problem for a sequence of decreasing tolerances, ending when the
required error tolerance is satisfied. CMLMC assumes discretization hierarchies
that are defined a priori for each level and are geometrically refined across
levels. The actual choice of computational work across levels is based on
parametric models for the average cost per sample and the corresponding weak
and strong errors. These parameters are calibrated using Bayesian estimation,
taking particular notice of the deepest levels of the discretization hierarchy,
where only few realizations are available to produce the estimates. The
resulting CMLMC estimator exhibits a non-trivial splitting between bias and
statistical contributions. We also show the asymptotic normality of the
statistical error in the MLMC estimator and justify in this way our error
estimate that allows prescribing both required accuracy and confidence in the
final result. Numerical results substantiate the above results and illustrate
the corresponding computational savings in examples that are described in terms
of differential equations either driven by random measures or with random
coefficients
Particle conservation in numerical models of the tokamak plasma edge
The test particle Monte-Carlo models for neutral particles are often used in
the tokamak edge modelling codes. The drawback of this approach is that the
self-consistent solution suffers from random error introduced by the
statistical method. A particular case where the onset of nonphysical solutions
can be clearly identified is violation of the global particle balance due to
non-converged residuals. There are techniques which can reduce the residuals -
such as internal iterations in the code B2-EIRENE - but they may pose severe
restrictions on the time-step and slow down the computations. Numerical
diagnostics described in the paper can be used to unambiguously identify when
the too large error in the global particle balance is due to finite-volume
residuals, and their reduction is absolutely necessary. Algorithms which reduce
the error while allowing large time-step are also discussed.Comment: Link to journal publication:
http://aip.scitation.org/doi/full/10.1063/1.498085
Multigrid Methods in Lattice Field Computations
The multigrid methodology is reviewed. By integrating numerical processes at
all scales of a problem, it seeks to perform various computational tasks at a
cost that rises as slowly as possible as a function of , the number of
degrees of freedom in the problem. Current and potential benefits for lattice
field computations are outlined. They include: solution of Dirac
equations; just operations in updating the solution (upon any local
change of data, including the gauge field); similar efficiency in gauge fixing
and updating; operations in updating the inverse matrix and in
calculating the change in the logarithm of its determinant; operations
per producing each independent configuration in statistical simulations
(eliminating CSD), and, more important, effectively just operations per
each independent measurement (eliminating the volume factor as well). These
potential capabilities have been demonstrated on simple model problems.
Extensions to real life are explored.Comment: 4
Exponential Runge-Kutta methods for stiff kinetic equations
We introduce a class of exponential Runge-Kutta integration methods for
kinetic equations. The methods are based on a decomposition of the collision
operator into an equilibrium and a non equilibrium part and are exact for
relaxation operators of BGK type. For Boltzmann type kinetic equations they
work uniformly for a wide range of relaxation times and avoid the solution of
nonlinear systems of equations even in stiff regimes. We give sufficient
conditions in order that such methods are unconditionally asymptotically stable
and asymptotic preserving. Such stability properties are essential to guarantee
the correct asymptotic behavior for small relaxation times. The methods also
offer favorable properties such as nonnegativity of the solution and entropy
inequality. For this reason, as we will show, the methods are suitable both for
deterministic as well as probabilistic numerical techniques
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