In the standard approach, particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction on the discretization parameter as well as on the diierential cross section in the case of the general Boltzmann equation. Recently, it was shown, how to construct an implicit ...

We propose a new algorithm, called CILA, for discretization of continuous attribute. The CILA algorithm can be used with any class labeled data. The tests performed using the CILA algorithm show that it generates discretization schemes with almost always the highest dependence between the class labels and the discrete intervals, and always with significantly lower number of intervals, when comp...

Although versions of Poisson’s Summation Formula (PSF) have already been studied extensively, there seems to be no theorem that relates discretization to periodization and periodization to discretization in a simple manner. In this study, we show that two complementary formulas, both closely related to the classical Poisson Summation Formula, are needed to form a reciprocal Discretization-Perio...

In supervised machine learning, some algorithms are restricted to discrete data and thus need to discretize continuous attributes. In this paper, we present a new discretization method called MODL, based on a Bayesian approach. The MODL method relies on a model space of discretizations and on a prior distribution defined on this model space. This allows the setting up of an evaluation criterion...

We study uniform bounds associated with the Allen–Cahn equation and its numerical discretization schemes. These uniform bounds are different from, and weaker than, the conventional energy dissipation and the maximum principle, but they can be helpful in the analysis of numerical methods. In particular, we show that finite difference spatial discretization, like the original continuum model, sha...

This paper argues that two commonly-used discretization approaches, fixed k-interval discretization and entropy-based discretization have sub-optimal characteristics for naive-Bayes classification. This analysis leads to a new discretization method, Proportional k-Interval Discretization (PKID), which adjusts the number and size of discretized intervals to the number of training instances, thus...

We compare four discretization methods, all based on entropy: the original C4.5 approach to discretization, two globalized methods, known as equal interval width and equal frequency per interval, and a relatively new method for discretization called multiple scanning using the C4.5 decision tree generation system. The main objective of our research is to compare the quality of these four method...

In this paper we consider a level set equation, the solution of which (called level set function) is used to capture a moving interface denoted by Γ. We assume that this level set function is close to a signed distance function. For discretization of the linear hyperbolic level set equation we use standard polynomial finite element spaces with SUPG stabilization combined with a CrankNicolson ti...

Abstract. We consider a parabolic interface problem which models the transport of a dissolved species in two-phase incompressible flow problems. Due to the so-called Henry interface condition the solution is discontinuous across the interface. We use an extended finite element space combined with a method due to Nitsche for the spatial discretization of this problem and derive optimal discretiz...