Multiscale representation

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

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... 

To avoid the evaluation of these quantities, the continuous phase equilibrium function is expressed in terms of the single-space basis e.g. K T,p, F ) G Vn- Using the fast Wavelet transform T the single-scale representation of the trial solution is obtained and the residuum of (1.3) can be formulated in terms of the single-scale basis The residuum is thus expressed by means of scaling functions of common level n. A subsequent fast wavelet transform provides the residuum of (1.3) in the multiscale representation, whose inner products with the weight functions can easily be evaluated due to orthogonality. This approach exactly recovers the discrete model without any deviations due to the continuous model formulation if the Haar basis is used. 

As noted, the wavefunction-wavelet reconstruction formula is equivalent to the zero scale limit of the scaling transform lim, n 5 (a. h) = (6). This defines an explicitly scale-translation dependent, multiscale, representation for the Dirac measure, S(x — b) = lima o Similarly, the... 

MRF representation can be derived from a multiscale representation for the Dirac measure which does not explicitly depend on the scale variable. This is discussed in Sec. 1.2.3.1. This alternative derivation overlaps with the Distributed Approximating Functionals approach of Hoffman et al (1991). 

This expansion recovers the Dirac distribution through a multiscale representation involving configurations that vary over decreasing length scales, as the order of the expansion is increased, n -A oo. 

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. 

Based on these HRTEM data, a sketch of the multiscale organisation of the composite is proposed13 in Figure 6. Even if this representation is very schematic, it illustrates the two types of carbon short and very disordered layers mainly in the fiber, and a developed preferential orientation of longer and better stacked layers in the pyrocarbon coating. 

For I and r one can use exact formulas (34) and (36) or zero-one asymptotic representations based on Equations (37) and (35) for multiscale systems. This approximation (44) could be improved by iterative methods, if necessary. 

As it is demonstrated, dynamics of this system approximates relaxation of the whole network in subspace = 0. Eigenvalues for Equation (45) are —k, (i < n), the corresponded eigenvectors are represented by Equations (34), (36) and zero-one multiscale asymptotic representation is based on Equations (37) and (35). 

Fig. 14.11 Schematic representation of fiber spinning process simulation scheme showing the multiple scale simulation analysis down to the molecular level. This is the goal of the Clemson University-MIT NSF Engineering Research Center for Advanced Engineering Fibers and Films (CAEFF) collaboration. CAEFF researchers are addressing fiber and film forming and structuring by creating a multiscale model that can be used to predict optimal combinations of materials and manufacturing conditions, for these and other processes.

X. Yin, S. Lee, W. Chen, W.K. Liu, M.F. Horstemeyer A Multiscale Design Approach with Random Field Representation of Material Uncertainty, ASME DETC08. NY, New York (2008)... 

Fig. 1. Schematic representation depicting scales and various simulators. Most multiscale work has focused on the simplest, one-way information passing, usually from the finest to the coarsest scale model. On the other hand, most processes exhibit strong coupling between scales or lack...

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