Comparison of Various Improved-Partition Fuzzy c-Means Clustering Algorithms in Fast Color Reduction

This paper provides a comparative study of sev- eral enhanced versions of the fuzzy c -means clustering al- gorithm in an application of histogram-based image color reduction. A common preprocessing is performed before clus- tering, consisting of a preliminary color quantization, histogram extraction and selection of frequently occurring colors of the image. These selected colors will be clustered by tested c -means algorithms. Clustering is followed by another common step, which creates the output image. Besides conventional hard (HCM) and fuzzy c -means (FCM) clustering, the so-called generalized improved partition FCM algorithm, and several versions of the suppressed FCM (s-FCM) in its conventional and generalized form, are included in this study. Accuracy is measured as the average color difference between pixels of the input and output image, while efficiency is mostly characterized by the total runtime of the performed color reduction. Nu- merical evaluation found all enhanced FCM algorithms more accurate, and four out of seven enhanced algorithms faster than FCM. All tested algorithms can create reduced color images of acceptable quality

Well-Posed Initial-Boundary Evolution in General Relativity

Maximally dissipative boundary conditions are applied to the initial-boundary value problem for Einstein's equations in harmonic coordinates to show that it is well-posed for homogeneous boundary data and for boundary data that is small in a linearized sense. The method is implemented as a nonlinear evolution code which satisfies convergence tests in the nonlinear regime and is robustly stable in the weak field regime. A linearized version has been stably matched to a characteristic code to compute the gravitational waveform radiated to infinity.Comment: 5 pages, 6 figures; added another convergence plot to Fig. 2 + minor change

Unambiguous determination of gravitational waveforms from binary black hole mergers

Gravitational radiation is properly defined only at future null infinity (\scri), but in practice it is estimated from data calculated at a finite radius. We have used characteristic extraction to calculate gravitational radiation at \scri for the inspiral and merger of two equal mass non-spinning black holes. Thus we have determined the first unambiguous merger waveforms for this problem. The implementation is general purpose, and can be applied to calculate the gravitational radiation, at \scri, given data at a finite radius calculated in another computation.Comment: 4 pages, 3 figures, published versio

Modelling and Simulation of Particle Size Distribution of Precipitates in Continuous Tubular Crystallizers

This paper presents a population balance model for describing the temporal evolution of the particle size distribution of a precipitate produced in a laboratory scale tubular crystallizer, including nucleation, growth and agglomeration of crystals. The quadrature method of moments is used to calculate the moments of the crystal size distribution. The gamma probability density function with the sixth order Laguerre polynomials are used to reproduce the particle size distribution from moments. A sensitivity analyses is carried out by simulation that can be helpful when designing a crystallizer. Influence of the design and process parameters on the particle size distribution of product is analysed

Numerical stability for finite difference approximations of Einstein's equations

We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in Numerical Relativity. By analyzing the symbol of the second order system, we obtain necessary and sufficient conditions for stability in a discrete norm containing one-sided difference operators. We prove stability for certain toy models and the linearized Nagy-Ortiz-Reula formulation of Einstein's equations. We also find that, unlike in the fully first order case, standard discretizations of some well-posed problems lead to unstable schemes and that the Courant limits are not always simply related to the characteristic speeds of the continuum problem. Finally, we propose methods for testing stability for second order in space hyperbolic systems.Comment: 18 pages, 9 figure

Testing numerical relativity with the shifted gauge wave

Computational methods are essential to provide waveforms from coalescing black holes, which are expected to produce strong signals for the gravitational wave observatories being developed. Although partial simulations of the coalescence have been reported, scientifically useful waveforms have so far not been delivered. The goal of the AppleswithApples (AwA) Alliance is to design, coordinate and document standardized code tests for comparing numerical relativity codes. The first round of AwA tests have now being completed and the results are being analyzed. These initial tests are based upon periodic boundary conditions designed to isolate performance of the main evolution code. Here we describe and carry out an additional test with periodic boundary conditions which deals with an essential feature of the black hole excision problem, namely a non-vanishing shift. The test is a shifted version of the existing AwA gauge wave test. We show how a shift introduces an exponentially growing instability which violates the constraints of a standard harmonic formulation of Einstein's equations. We analyze the Cauchy problem in a harmonic gauge and discuss particular options for suppressing instabilities in the gauge wave tests. We implement these techniques in a finite difference evolution algorithm and present test results. Although our application here is limited to a model problem, the techniques should benefit the simulation of black holes using harmonic evolution codes.Comment: Submitted to special numerical relativity issue of Classical and Quantum Gravit

Algebraic stability analysis of constraint propagation

The divergence of the constraint quantities is a major problem in computational gravity today. Apparently, there are two sources for constraint violations. The use of boundary conditions which are not compatible with the constraint equations inadvertently leads to 'constraint violating modes' propagating into the computational domain from the boundary. The other source for constraint violation is intrinsic. It is already present in the initial value problem, i.e. even when no boundary conditions have to be specified. Its origin is due to the instability of the constraint surface in the phase space of initial conditions for the time evolution equations. In this paper, we present a technique to study in detail how this instability depends on gauge parameters. We demonstrate this for the influence of the choice of the time foliation in context of the Weyl system. This system is the essential hyperbolic part in various formulations of the Einstein equations.Comment: 25 pages, 5 figures; v2: small additions, new reference, publication number, classification and keywords added, address fixed; v3: update to match journal versio