Simultaneous, computations

Many institutions have hundreds, or even thousands, of powerful work stations that are idle for much of the day. There is often vastiy more power available in these machines than in any supercomputer center, the only problem being how to harness the power already available. There are network load-distribution tools that allocate individual jobs to unused computers on a network, but this is different from having many computers simultaneously cooperating on the solution of a single problem. 

It is noted that the partial derivatives in the above variance expressions depend on time t and therefore the variances should be computed simultaneously with the state variables and sensitivity coefficients. Finally, the confidence intervals of the cumulative production of each well and of the total reservoir are calculated by integration with respect to time (Kalogerakis and Tomos, 1995). 

What determines the number of rows and columns The number of rows is determined by the number of coefficients that are to be calculated. In this example, therefore, we will compute a set (sets, actually, as we will see) of seven coefficients. The number of columns is determined by the degree of the polynomial that will be used as the fitting function. The number of columns also determines the maximum order of derivative that can be computed. In our example we will use a third-power fitting function and we can produce up to a third derivative. As we shall see, coefficients for lower-order derivatives are also computed simultaneously. 

While the number of independent variables is arbitrary in our definitions, it makes a tremendous difference in computations. Simultaneous solution of n equations and minimization in n dimensions are much more difficult than in one dimension. The main difference between one and several dimensions is that in one dimension it is possible to "bracket" a root or a local minimum point between some bracketing values, and then to tighten the interval of uncertainty. This gives rise to special algorithms, and hence the solution of a single equation and minimization in one variable will be discussed separately from the multidimensional methods. 

Citral has three sites of hydrogenation and as shown in (Fig. 1) the mechanism for its reduction is complex. However we have shown previously (ref. 3) that, after the elimination of time as an independent variable, the LANGMUIR-HINSELWOOD rate equation for each component in the liquid phase can be integrated. The integrated equations B = f( A ), C = g( 8 ), i01 = h ( B ), E = i ( B ) and F = j( 8 ) depend on six ratios k g/k, k /k, kgb g/k b. ... k bp/k b which have been computed simultaneously. The agrement between the calculated product compositions as a function of hydrogen consumed and the experimental data is excellent as illustrated in (Fig. 2). Therefore the computed ratio k o /kjb for each step has been used as selectivity criteria. 

Simultaneous. In order to circumvent the in-efficiencies associated with loop within loop struct-tures forced by the module design and sequential modular approach, there has been considerable academic effort to investigate how to perform all computations simultaneously. The potential advantages of this global (or "equation oriented") approach are generally recognized but acceptance of the approach has been slow due to a number of reasons ... 

The experiments were carried out in the presence of 50 jAf Neutral Red, 40 mg Brij 58/ml, corresponding with 500(jlM of micellar concentration), 1 mAf of 2-naphthol-3,6-disulfonate, and KC1 as indicated. The values appearing in the table were computed simultaneously with k. ... 

As alternative expression to be used for the terms R, A, D and X, (Guo, 1996) provides a detailed description of different source and sink sub-models in the User s Guide of his Z-30 lAQ simulator. This very useful compilation of existing widely used source and sink sub-models includes 25 source and 6 sink sub-models. The 25 source sub-models include 3 for constant source, 7 for a first-order (exponential) decay source, 3 for a higher order decaying source, 1 for an instant source, 1 for a time-varying source, 2 for ambient air as an indoor source and 8 mass transfer sub-models. The 6 sink sub-models include one irreversible and five reversible sub-models. Among the five reversible sink sub-mod-els two are equilibrium models and the remaining three compute simultaneously both adsorption and desorption rates. 

III. For this purpose one sets p = 0 in Eq. (2.2) and makes the substitution < = (uT and Gp Up) = m(p)>. Expressions for several values of p can be computed simultaneously and virtually for the price of one. We explicitly mention the important special case obtained by choosing for the general operator X the evolution operator G itself. This yields the following estimator for the dominant eigenvalue A0 of G ... 

The details of the memory redefinition process depend on the hardware design of the computer. Simultaneous redefinition of all addressable memory (bank switching) is found on some micro-computers. This is not well-suited to the implementation of large data structures because the program must be re-mapped along with the data. 

This periodic boundary model of a pseudoinfinite system does not provide the freedom that would be enjoyed by particles in a genuinely infinite space, but it renders the problem computationally tractable. Using the new generation of vector processor computers, which permit several independent interactions to be computed simultaneously, can lead to a more efficient solution of the equations of motion for a given N if the shape of the simulation box and its images are not a cube but a higher order polyhedron. ... 

A key difference between the complement method and the LP method is that the latter requires the solution of a linear program, whereas the former is a direct application of the CSTR attainability condition. In the LP approach, all points on the AR boundary are computed simultaneously—via the solution of a large linear program—in a single calculation step. In order for this result to be achieved, the candidate region boundary points must be expressed in terms of all other boundary points in space using a superstructure formulation, which is termed the total connectivity model. 

The mnnber of different summation techniques developed over the years is nearly coimtless. Many of these are based on variations of the parallel prefix principle. Ladner and Fischer [10] showed that the outputs of each finite, determined automaton can be computed simultaneously with techniques based on the solution ofthe parallel prefix sum problem. In their work, Brent and Kimg [2] presented the first representation of a parallel prefix adder working in 0(21og2n-1) time that could be visualized by using black and gray processors. The carry-skip principle [3] [5] [6] can also by applied... 

Another solution to improve the execution time is to compute several indices (z,j. A ) in parallel with i and k fixed. Let c be the number of j-indices such that nodes (i,i, fc) are computed simultaneously for i and k fixed. This leads to an array with the same number of cells as in well-known solutions (namely n ), but with a lower execution time An -I- 7), which comes at the price of an increase in the period ( n < 3n instead of n). 

Note that if the concentrations and reaction rates have already been computed and stored, then this is a set of linear differential equations (with non-constant coefficients) for the sensitivity coefficients. Note also that each sensitivity coefficient depends only on the sensitivity coefficients of the other species to the same parameter, but not on sensitivity coefficients with respect to other parameters. Thus, the sensitivity coefficients with respect to a given parameter (K of them) are coupled and must be computed simultaneously, but it is not necessary to solve for all the sensitivity coefficients (2 x / x of them) simultaneously. 

This equation can either be integrated forward in time with the state equations or backward in time w ith the adjoint equations. The choice of the backward in time integration is more efficient since we do not have to store the values of p2- Instead y can be computed simultaneously while computing the adjoint variables. 

The lag of the master-slave processing cycle is part of the protections of the common mode against an external disturbance (EMC) or electromagnetic pulse (EMI) affecting both computers simultaneously. 

Subsurface stresses are computed simultaneously for points lying at the same depth directly below the grid nodes. Therefore, the fatigue integral F is also computed layer by layer, as follows ... 

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