Scalar problem

CASE I - An Inherently Scalar Problem of Bulk Amorphous Polymer Systems... 

The extension of the SR model to differential diffusion is outlined in Section 4.7. In an analogous fashion, the LSR model can be used to model scalars with different molecular diffusivities (Fox 1999). The principal changes are the introduction of the conditional scalar covariances in each wavenumber band 4> a4> p) and the conditional joint scalar dissipation rate matrix (e). For example, for a two-scalar problem, the LSR model involves three covariance components (/2, W V/A) and and three joint dissipation... 

Fig. 15.5 Illustration of the full Newton and modified Newton algorithm on a scalar problem, F(y) = 0. The curve represents a nonlinear function F(y), and the solution is the value of y at which the function is zero.

Similar to scalar problems, the first step of the BEM is to discretize the boundary into a series of elements over which the velocity and traction are assumed to vary according to some interpolation functions. 

In the scalar case (i.e., N = 1), wave solutions are easily constructed with the equilibrium diagram y(x) or Y(X). According to the above considerations, typical scalar problems are a binary nonreactive distillation process, a ternary reactive distillation process with a single chemical reaction, a reactive distillation process with Nc components and Nc - 2 chemical reactions, or a chromatographic reactor with Ns solutes... 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

The scalarized problem to be solved in the weighting method (Gass and Saaty, 1955 Zadeh, 1963) is... 

The strength of evolutionary approaches is their wide applicability to, e.g., nondifferentiable and nonconvex problems. We wish to emphasize that this positive feature can be combined with scalarization based approaches by using evolutionary algorithms (i.e., not EMO but single objective optimizers) for solving the scalarized problem. 

A classification is feasible if at least one of the objective functions is allowed to get worse. Then the original multi-objective optimization problem is converted into a scalarized problem using the classification information specified. The solution of the scalarized problem reflects how well the hopes expressed in the classification could be achieved. 

There exist several variants of NIMBUS (Miettinen, 1999 Miettinen and Makela, 1995, 1999, 2000, 2006). Here we concentrate on the synchronous version (Miettinen and Makela, 2006), where several scalarizing functions can be used based on a classification once expressed. Because they take the preference information into account in slightly different ways (Miettinen and Makela, 2002), the DM can learn more about different solutions satisfying his/her hopes and choose the one that best obeys his/her preferences. An example of the scalarized problems used is... 

Once the DM has classified the objective functions, (s)he can decide how many Pareto optimal solutions (between one and four) based on this information (s)he wants to see and compare. Then, as many scalarized problems are formed and solved and the new solutions are shown to the DM together with the current solution. If the DM has found the most preferred solution, the solution process stops. Otherwise, the DM can select a solution as a starting point of a new classification or ask for a desired number of intermediate (Pareto optimal) solutions between any two solutions generated so far. The DM can also save any interesting solutions to a database and return to them later. All the solutions considered are Pareto optimal. For details of the algorithm, see Miettinen and Makela (2006). 

There are different single-objective optimizers available for solving the scalarized problems formed and the user can decide after each classification which optimizer to use or use the default one. The proximal bundle method (Makela and Neittaanmaki, 1992) is a local optimizer and needs initial values for variables as well as (sub)gradients for functions. (The system can generate the latter automatically.) Alternatively, it is possible to use two variants of (global) real-coded genetic algorithms that differ from each other... 

E. Balkovsky and A. Fouxon. Universal long-time properties of La-grangian statistics in the Batchelor regime and their application to the passive scalar problem. Phys. Rev. E, 60 4164-4174, 1999. 

Scalar problems are often easier to solve than the associated vector problem, and the scalar analog of Eq. 2-5 is known as the macroscopic mechanical energy balance. One begins the derivation of this useful result by forming the scalar product of Eq. 1-59 with the velocity vector to obtain... 

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