Discrete variable representation analysis

Bacic Z, Kress J D, Parker G A and Pack R T 1990 Quantum reactive scattering in 3 dimensions using hyperspherical (APH) coordinates. 4. discrete variable representation (DVR) basis functions and the analysis of accurate results for F + Hg d. Chem. Phys. 92 2344... 

The potential energy surface (PES) of ammonia has been studied repeatedly by many authors ([1-11], and references therein), and continues to be an object of active theoretical interest. Most authors start their analysis with an abinitio (or semi-empirical) calculation of the PES and then perform an additional refinement to achieve an agreement between the calculated and experimental vibrational frequencies. Lately, the discrete variable representation has received particular attention and is currently one of the preferred methods [3,7,8,10-12],... 

Needless to mention, the exact capturing of time presents further challenges in the analysis. Fundamentally, a decision has to be made on how the time horizon has to be represented. Early methods relied on even discretization of the time horizon (Kondili et al., 1993), although there are still methods published to date that still employ this concept. The first drawback of even time discretization is that it inherently results in a very large number of binary variables, particularly when the granularity of the problem is too small compared to the time horizon of interest. The second drawback is that accurate representation of time might necessitate even smaller time intervals with more binary variables. Even discretization of time is depicted in Fig. 1.8a. 

Aside from the continuity assumption and the discrete reality discussed above, deterministic models have been used to describe only those processes whose operation is fully understood. This implies a perfect understanding of all direct variables in the process and also, since every process is part of a larger universe, a complete comprehension of how all the other variables of the universe interact with the operation of the particular subprocess under study. Even if one were to find a real-world deterministic process, the number of interrelated variables and the number of unknown parameters are likely to be so large that the complete mathematical analysis would probably be so intractable that one might prefer to use a simpler stochastic representation. A small, simple stochastic model can often be substituted for a large, complex deterministic model since the need for the detailed causal mechanism of the latter is supplanted by the probabilistic variation of the former. In other words, one may deliberately introduce simplifications or errors in the equations to yield an analytically tractable stochastic model from which valid statistical inferences can be made, in principle, on the operation of the complex deterministic process. 

Many analytical measures cannot be represented as a time-series in the form of a spectrum, but are comprised of discrete measurements, e.g. compositional or trace analysis. Data reduction can still play an important role in such cases. The interpretation of many multivariate problems can be simplified by considering not only the original variables but also linear combinations of them. That is, a new set of variables can be constructed each of which contains a sum of the original variables each suitably weighted. These linear combinations can be derived on an ad hoc basis or more formally using established mathematical techniques. Whatever the method used, however, the aim is to reduce the number of variables considered in subsequent analysis and obtain an improved representation of the original data. The number of variables measured is not reduced. 

Wavelet analysis was also proposed for variable reduction problems and, in particular, the wavelet coefficients obtained from discrete wavelet transforms (DWT) were proposed as a molecular representation in PEST descriptor methodology and their sums as molecular descriptors [Breneman, Sundhng et al., 2003 Lavine, Davidson et al, 2003]. 

Reliability analysis aims at capturing the probabihs-tic nature of the failures to which a system is subject. One powerful theoretical framework frequently used in reliability analysis is the Markov chain. Markov chains are based on a state representation of a system in which the next future state only depends on the current state and not on the previous history of the system (this assmnption is referred as the Markov property). Mathematically, a discrete-time Markov chain X n = 0,1,... is defined as a discrete-time, discrete-value random sequence such that given Wq,. .., X , the next random variable X +i depends only onX through the transition probabihty expressed in Equation 1. 

The representation of a hybrid system model by means of a bond graph with system mode independent causalities has the advantage that a unique set of equations can be derived from the bond graph that holds for all system modes. Discrete switch state variables in these equations account for the system modes. In this chapter, this bond graph representation is used to derive analytical redundancy relations (ARRs) from the bond graph. The result of their numerical evaluation called residuals can serve as fault indicator. Analysis of the structure of ARRs reveals which system components, sensors, actuators or controllers contribute to a residual if faults in these devices happen. This information is usually expressed in a so-called structural fault signature matrix (FSM). As ARRs derived from the bond graph of a hybrid system model contain discrete switch state variables, the entries in a FSM are mode dependent. Moreover, the FSM is used to decide if a fault has occurred and whether it can unequivocally be attributed to a component. Finally, the chapter discusses the numerical computation of ARRs. 

Seismic Analysis of Steel and Composite Bridges Numerical Modeling, Fig. 23 Representation of piers through discrete beam mass models - pier with (a) constant and (b) variable dimensions... 

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