Multiple scales

R. C. Y. Chin, G. W. Hedstrom, and F. A. Howes. Considerations on Solving Problems with Multiple Scales. Academic Press, Orlando, Florida, 1985. 

HOTMAC/RAPTAD requires very extensive meteorological and terrain data input. The program user s guide and diagnostics are inadequate. HOTMAC does not model multiple scale eddy turbulence and does not provide for dispersion of gases that are denser-than air. It must be tailored to reflect the climatic characteristics of specific sites. 

III. Wavelet Decomposition Extraction of Trends at Multiple Scales... 

For solving the pattern recognition problem encountered in the operation of chemical processes, the analysis of measured process data and extraction of process trends at multiple scales constitutes the feature extraction, whereas induction via decision trees is used for inductive... 

Consideration of Multiple Scales—local, landscape, national, regional, and global. 

There is a need for large-scale differential-algebraic methods for simulating systems at multiple scales (e.g., fluid mechanics and molecular dynamics), a capability that is still at a very early stage. 

Steel EA, Lange IA (2007) Using wavelet analysis to detect changes in water temperature regimes at multiple scales effects of multi-purpose dams in the Willamette River basin. River Res Appl 23 351-359... 

Multiple-scale perturbation analysis and numerical simulation of the unsteady-state IEM model. Chemical Engineering Science 45, 2857-2876. 

When going beyond the lowest-order term in derivatives, we need a counting scheme. For theories with only one relevant scale (such as QCD at zero chemical potential), each derivative is suppressed by a factor of This is not the case for theories with multiple scales. In the CFL phase, we have both and the gap, A, and the general form of the chiral expansion is [31] ... 

In this chapter, we review the elements of G3 theory and related techniques of computational thermochemistry. This review is restricted almost exclusively to the techniques that we have developed and the reader is referred to the remaining chapters in this volume for other complementary approaches. An important part of the development of such quantum-chemical methods is their critical assessment on test sets of accurate experimental data. Section 3.2 provides a brief description of the comprehensive G3/99 test set [26] of experimental data that we have collected. Section 3.3 discusses the components of G3 theory as well as the approximate versions such as G3(MP3) [22] and G3(MP2) [23], and their performance for the G3/99 test set. The G3S method [29] that includes multiplicative scale factors is presented in section 3.4 along with other related variants. Section 3.5 discusses the recently developed G3X method [30] that corrects for most of the deficiencies of G3 theory for larger molecules. The performance of these methods is compared to... 

A new family of methods, referred to as G3S (G3 Scaled), has been developed recently [29], where the additive higher-level correction is replaced by a multiplicative scaling of the correlation and Hartree-Fock components of the G3 energy. The scale factors have been obtained by fitting to the G2/97 test set of energies. This test set is substantially larger than that used in previous fits and can provide a reliable assessment of the use of such a scaling approach to computational thermochemistry. 

In a similar manner, the approximate G3(MP3) method can be modified to use multiplicative scale factors. The resulting G3S(MP3) energy expression is... 

G3 theory based on multiplicative scaling of the energy terms (G3S) instead of the additive higher-level correction has a mean absolute deviation of 1.08 for the G3/99 test set, an increase from 0.99 for the G2/97 test set. As in the case of G3 theory, the increase is largely due to the new non-hydrogen species in the test set. However, systems such as the highly strained P4 molecule perform poorly with the scaled methods. 

In this section, we describe the role of fhe specific membrane environment on proton transport. As we have already seen in previous sections, it is insufficient to consider the membrane as an inert container for water pathways. The membrane conductivity depends on the distribution of water and the coupled dynamics of wafer molecules and protons af multiple scales. In order to rationalize structural effects on proton conductivity, one needs to take into account explicit polymer-water interactions at molecular scale and phenomena at polymer-water interfaces and in wafer-filled pores at mesoscopic scale, as well as the statistical geometry and percolation effects of the phase-segregated random domains of polymer and wafer at the macroscopic scale. 

Geochemistry of Archean sulfidic black shale horizons combining data at multiple scales for improved targeting in VMS exploration... 

Fig. 2. DPF simulation has to face multiple scales. Figure from Konstandopoulos and Kostoglou (2004).

Rahman et al. (1999) extended the algorithm of Moore et al. They use a Gaussian to compute the blurred image and perform color correction on multiple scales. This method is not only used for color constancy but also for dynamic range compression. Output color o, is computed as... 

Color correction on multiple scales, no gain factor. 

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.

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