Methods of Discretization

1 Methods of Discretization. - We consider particular discretizations of (213) applied to the more general form 

1 Methods which Lead to a Tridiagonal Form of the Matrix A. - Fack and Vanden Berghe100 have used the following method of discretization 

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Partial order ranking (POR) is based on elementary methods of discrete mathematics (e.g., Hasse diagrams) — if A < B and B < C, then A < C in the ranking procedures. POR does not assume linearity or any assumptions about distribution properties such as normality. The disadvantage is that often a preprocessing of data is needed to avoid the effects of stochastic noise. Combining POR with PCA may improve its usefulness. POR can only be applied for interpolation. 

The smaller the squares of the grid, the better the resolution of the representation of D by the animals. By approximately filling up the interior D of J by animals at various levels of resolution, a shape characterization of the continuous Jordan curve J can be obtained by the shape characterization of animals. The animals contain a finite number of square cells, consequently, their shape characterization can be accomplished using the methods of discrete mathematics. As a result, one obtains an approximate, discrete characterization of the shape of the Jordan curve (i.e., the shape of a continuum). The level of resolution can be represented indirectly, by the number of cells of the animals. In particular, one can show [240,243] that the number of cells required to distinguish between two Jordan curves provides a numerical measure of their similarity. 

Tabular method of discrete optimization. The basis of dynamic programming is the principle of prefix optimality. This principle states that the optimal solution to the optimization problem can be composed of optimal solutions of a limited number of smaller instances of the same type of problem. The tabular method incrementally solves subinstances of increas-... 

This example illustrates that the step size, h, plays a crucial role in the convergence of the numerical solution of a differential equation. Depending on the equation to be solved and the method of discretization, h can be adjusted to obtain convergence of a numerical solution. 

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses). 

In this paper we demonstrate that all known feedback-mediated control methods based on the phenomenon of resonant drift can be considered in the framework of a unified theoretical approach. This approach allows to analyze existing methods of discrete and continuous control, and helps to elaborate novel control algorithms. The theoretical predictions are confirmed by numerical computations and experimental studies. 

B. G. Carlson and K. D. Lathrop, Transport Theory—The Method of Discrete Ordinates, in H. Greenspan, G N. Kelber, and D. Okrent (eds.), Computing Methods in Reactor Physics, Gordon and Breach, New York, 1968. 

Recently, much progress has been achieved in various methods of discretization [156,157] of protein conformational space [42] and in the analysis of the fidelity of such discretizations [14,156,158-160]. 

Numerov Discretization. - Flack and Vanden Berghe101 have used the well known Numerov method of discretization which leads to a tridiagonal (not necessarily symmetric) form for A and B. The method is of algebraic order 4, phase-lag order 4. The interval of periodicity is (0,6). 

The methods of discrete mathematics, introduced in this chapter, were sufficient for describing chemical compounds as discrete structures. However, once the relationships between properties and structures have to be modelled, non-discrete methods are required. Methods from supervised statistical learning theory and machine learning are particularly useful and thus some of these will be introduced in the next chapter. 

Structural properties or loads varying randomly in space are modeled here as multi-dimensional random processes, or random fields. For finite element implementation, it is necessary to discretize such fields into random vector representations. Two methods of discretization are investigated here. In one method, the field value over an element is represented by its value at the midpoint of the element. In the second method, the value for an element is represented by the spatial average of the process over the element, as originally suggested by Vanmarcke et al. [27]. For a two-dimensional process X(jc,y), the above representations for an element / are... 

If the random field is Gaussian, the joint distribution of Xi for various elements remains Gaussian for both the midpoint and spatial averaging methods. However, for non-Gaussian fields the distribution of X, defined by Eq. 21 is difhcult or impossible to obtain. An approximate description of the distribution for that case has been suggested elsewhere [8]. Here, attention is restricted to Gaussian random fields so that a fair comparison between the two methods of discretization can be made. 

Doicu, A., Wriedt, T., Formulation of the Extended Boundary Condition Method for Three-dimensional Scattering Using the Method of Discrete Sources,/. Mod Opt, 1998, 45, 199-213. 

An analytic solution to Eq. (9.1.24) has been obtained by the method of discrete ordinates (Samuelson, 1983), which is a generalization of the two-stream approximation considered in this chapter. The solution is then used to obtain expressions for the flux from Eq. (9.1.18) and the Planck intensity from Eq. (9.1.17). The upper boundary condition... 

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