Computational Issues

There are two major approaches to elaboration of computationally efficient algorithms. These are based on Lagrangian relaxation and Bender s decomposition. A short overview of these methods is provided here. Readers are referred to Avriel and Golany (1996) for a detailed coverage of mathematical programming. 

Pirkul and Jayaraman (1998) successfully applied the Lagrangian relaxation problem for the supply chain configuration problem. Similar results have been obtained by Jang et al. (2002) and Amiri (2006). The supply chain configuration model by Pirkul and Jayaraman (1998) locates a specified number of manufacturing facilities and warehouses to minimize fixed and transformation costs subject to customer demand satisfaction and capacity constraints. 

The mathematical representation of their model is as follows. Parameters of the model are  

Ciji—the variable cost to distribute a unit of product I from warehouse j to customer zone  

Tjki a unit cost to transport product I from plant k to warehouse f 

For calculating [St] from Eq. 10.4.30 we may make use of Sylvester s formula as follows  

To calculate [A ] we would normally not bother to compute the matrix [St], only its eigenvalues. Equations 8.4.24 or 8.4.31 may be used directly as written to compute the multicomponent mass transfer coefficients with 

We may identify [A] as the matrix j[[Sc] - [/]]. If the matrix j[[Sc] - [/]] will have eigenvalues lying between —1 and +1 then the term ln[[/] + j[[Sc] — [/]]] may be evaluated from the convergent series (Eq. 10.4.37). The matrix [St] may then be computed from Eq. 10.4.30 using only elementary matrix computations. For gas mixtures the eigenvalues of [Sc] are of order 1 and this procedure can, therefore, be used. 

For computation of the fluxes themselves. Algorithms 8.1, 8.4, and 8.5 may be used more or less as written simply replace all quantities in molar units by the corresponding quantities in the turbulent eddy diffusivity model. 

---

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

Shah, N., Pantelides, C.C., and Sargent, R.W.H., A general algorithm for short-term scheduling of batch operations. II Computational issues. Comput. Chem. Eng. 17(2), 229-244 (1993). 

In principle, the relationships described by equations 66-9 (a-c) could be used directly to construct a function that relates test results to sample concentrations. In practice, there are some important considerations that must be taken into account. The major consideration is the possibility of correlation between the various powers of X. We find, for example, that the correlation coefficient of the integers from 1 to 10 with their squares is 0.974 - a rather high value. Arden describes this mathematically and shows how the determinant of the matrix formed by equations 66-9 (a-c) becomes smaller and smaller as the number of terms included in equation 66-4 increases, due to correlation between the various powers of X. Arden is concerned with computational issues, and his concern is that the determinant will become so small that operations such as matrix inversion will be come impossible to perform because of truncation error in the computer used. Our concerns are not so severe as we shall see, we are not likely to run into such drastic problems. 

The high dimensional nature of LIBS signals can lead to several computational issues when used in conjunction with many machine learning techniques. Dimensionality reduction is the process by which the high dimensional signals are mapped into a lower dimensional space. The resulting lower dimensional space can enable more robust performance when used in conjunction with pattern recognition techniques. 

Shah, N. C. C. Pantelides and R. W. H. Sargent. A General Algorithm for Short-Term Scheduling of Batch Operations. II. Computational Issues. Comput Chem Eng 17(2) 229-244 (1993). 

Manning, R.A., Manhardt, P.D., Orzechowski, J.A., Cathles, L.M., Baker, A.J. 1993. A parallel 3-D geologic basin modeling code. Technical Report for the SIAM Conference on Mathematical and Computational Issues in the Geosciences, Knoxville. 

In addition to phase change and pyrolysis, mixing between fuel and oxidizer by turbulent motion and molecular diffusion is required to sustain continuous combustion. Turbulence and chemistry interaction is a key issue in virtually all practical combustion processes. The modeling and computational issues involved in these aspects have been covered well in the literature [15, 20-22]. An important factor in the selection of sub-models is computational tractability, which means that the differential or other equations needed to describe a submodel should not be so computationally intensive as to preclude their practical application in three-dimensional Navier-Stokes calculations. In virtually all practical flow field calculations, engineering approximations are required to make the computation tractable. 

Bernardo, F.P., Pistikopoulos, E.N., and Saraiva, P.M. (1999) Integration and computational issues in stochastic design and planning optimization problems. Industrial cj Engineering Chemistry Research, 38, 3056. 

Consider the following first-order linear model problem [215], which has been designed to illustrate some of the computational issues associated with stiffness ... 

An exposition of theoretical, algorithmic, and computational issues of the above classes of algorithms is provided in Nemhauser and Wolsey (1988) and Parker and Rardin (1988). 

Because this contribution is dedicated to the physics of solvation and not to computational issues, we do not add other comments on these methods, except to remark that a full understanding of the basic justifications of such methods is necessary to avoid misunderstandings and erroneous conclusions in their use. 

Sauro, H. M. Ingalls, B. Conservation analysis in biochemical networks computational issues for software writers. Biophys Chem 2004,109 1-15. 

To simplify our discussion, we will ignore the presence of the high-frequency intramolecular degrees of freedom present in most molecular solvents, though they do pose interesting conceptual and computational issues. 

A mathematics layer to deal with computational issues independent from the physics. 

We have discussed some of the broader issues facing computational supramolecular chemistry, and we will now discuss some of the specific computational issues that we have experienced in the areas explored in this review. 

Electronic Records and Electronic Signatures Number of Paragraphs in Warning Letter Dealing with Different Computer Issues... 

Computer-issued prescriptions must conform to recommendations of professional bodies. If altered by hand (undesirable), the alteration must be signed. 

Hard memory errors are related to a hardware failure and are reproducible. When a hard memory error occurs, the computer might issue either a Parity Error or a 201 BIOS Error or issue a series of beeps (upon startup). For example, if you are using AMI BIOS, and the computer issues one long and three short beeps on startup (other BIOSs will use different beep codes), you have a hardware-related memory error. A table of BIOS error codes is included in Chapter 10, Hardware Troubleshooting. ... 

---