Multiscale

Cahn, R.W. (2000) Historical overview, in Multiscale Phenomena in Plasticity (NATO ASI) eds. Saada, G. et al. (Kluwer Academic Publishers, Dordrecht) p. 1. 

Multiscale modeling is an approach to minimize system-dependent empirical correlations for drag, particle-particle, and particle-fluid interactions [19]. This approach is visualized in Eigure 15.6. A detailed model is developed on the smallest scale. Direct numerical simulation (DNS) is done on a system containing a few hundred particles. This system is sufficient for developing models for particle-particle and particle-fluid interactions. Here, the grid is much smaller... 

A. The Content of Process Trends Local in Time and Multiscale.488... 

A systematic analysis of a process signal over (1) different segments of its time record and (2) various ranges of frequency (or scale) can provide a local (in time) and multiscale hierarchical description of the signal. Such description is needed if an intelligent computer-aided tool is to be con--structed in order to (1) localize in time the step and spike from the equipment faults (Fig. 1), or the onset of change in sensor noise characteristics, and (2) extract the slow drift and the periodic load disturbance. 

The engineering context of the need for multiscale representation of process trends can be best seen within the framework of the hierarchical... 

The extraction, though, of the so-called pivotal features from operating data, encounters the same impediments that we discussed earlier on the subject of process trends representation (1) localization in time of operating features and (2) the multiscale content of operating trends. It is clear, therefore, that any systematic and sound methodology for the identification of patterns between process data and operating conditions can be built only on formal and sound descriptions of process trends. 

Scale-space filtering provides a multiscale description of a signal s trends in terms of its inflexion points (second-order zero crossings). The only legal sequences of triangles between two adjacent inflexion points are (in terms of triangular episodes) ... 

Let us now see how the theory of the wavelet-based decomposition and reconstruction of discrete-time functions can be converted into an efficient numerical algorithm for the multiscale analysis of signals. From Eq. (6b) it is easy to see that, given a discrete-time signal, FqU) we have... 

Fig, 10. Methodology for multiscale (a) decomposition and (b) reconstruction, using wavelets, with uniform sampling (m, n) e Z. ... 

These disadvantages are overcome by the methodology we will describe in the subsequent paragraph developed by Bakshi and Stephanopoulos. Effects of the curse of dimensionality may be decreased by using the hierarchical representation of process data, described in Section III. Such a multiscale representation of process data permits hierarchical development of the empirical model, by increasing the amount of input information in a stepwise and controlled manner. An explicit model between the features in the process trends, and the process conditions may be learned... 

Consider a measured operating variable, xit), and its M distinct measurement records, [)], / = 1,2,..., A/ over the same range of time. Using the multiscale decomposition of measured variables, discussed in Section III, we can represent each measurement record, [x(t)], / = 1,2,..., M by a finite state of trends, where each trend is a pattern of triangular episodes ... 

Multiscale modeling of process operations. The description of process variables at different scales of abstraction implies that one could create models at several scales of time in such a way that these models communicate with each other and thus are inherently consistent with each other. The development of multiscale models is extremely important and constitutes the pivotal issue that must be resolved before the long-sought integration of operational tasks (e.g., planning, scheduling, control) can be placed on a firm foundation. 

Multiscale process identification and control. Most of the insightful analytical results in systems identification and control have been derived in the frequency domain. The design and implementation, though, of identification and control algorithms occurs in the time domain, where little of the analytical results in truly operational. The time-frequency decomposition of process models would seem to offer a natural bridge, which would allow the use of analytical results in the time-domain deployment of multiscale, model-based estimation and control. 

Mailat, S., and Zhong, S., Characterization of signals from multiscale edges, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-14(7), 710-732 (1992). 

Atkas, O., Aluru, N. R., A combined continuum/DSMC technique for multiscale analysis of microfluidic filters,... 

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