Cascade algorithm

In a few (important) special cases the basis function has a closed form, but more generally adequate approximations can be evaluated numerically by the cascade algorithm. This amounts to just applying the subdivision process repeatedly to initial cardinal data. There is an efficient way of doing this. 

As described above, the cascade algorithm is regarded as producing the basis function as the pointwise limit of a sequence of polygons. There is, however, another way of looking at it. 

The basis function can be expressed as the sum of scaled copies of itself, and the scaling factors are just the entries in the mask. The cascade algorithm can then be viewed as making the basis function from tinier and tinier copies of itself as iteration proceeds. 

An even more important theoretical result is that if we take some other function (with some small print) bo(t) and apply the cascade algorithm by... 

This definition is also justified by the view taken of the cascade algorithm at the end of section 12.1. 

D.Chen and M.Han Convergence of cascade algorithm for individual initial function and arbitrary refinement masks. Science in China A 48(3) pp350-359, 2005... 

The same example was solved using MINOPT (Rojnuckarin and Floudas, 1994) by treating the PFR model as a differential model. The required input files are shown in the MINOPT manual. Kokossis and Floudas (1990) applied the presented approach for large-scale systems in which the reactor network superstructure consisted of four CSTRs and four PFR units interconnected in all possible ways. Each PFR unit was approximated by a cascade of equal volume CSTRs (up to 200-300 CSTRs in testing the approximation). Complex reactions taking place in continuous and semibatch reactors were studied. It is important to emphasize that despite the complexity of the postulated superstructure, relatively simple structure solutions were obtained with the proposed algorithmic strategy. 

The analysis of the incident showed that this reactor was equipped with an indirect heating-cooling system with oil circulation and computerized temperature control. In this plant, to obtain a nervous temperature control, the control algorithm of the cascade controller (see Section 9.2.4.3) was adjusted to have an on-off behaviour, by calculating the set temperature of the jacket proportional to the squared difference between the actual and set values of the reaction medium temperature as... 

A major problem in predicting protein structure is the computational intractability. A short, 100-residue protein will contain at least 100 side-chain-to-side-chain or side-chain-to-solvent interactions. The orientation of each of these interactions will lead to cascading effects throughout the protein. Comparative modeling, threading algorithms, and de novo predictions seek to predict protein structure in reasonable execution times. 

Providing external reset for the cascade master from the slave measurement is always recommended. This guarantees bumpless transfer when the operator switches the loop from slave control to cascade control (Figure 2.45). The internal logic of the master controller algorithm is such that as long as its output signal (m) does not equal its external reset (ER), the value of m is set to be the sum of the ER and the proportional correction (Kc(e)) only. 

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