Simulation error

Tlierc are two major sources of error associated with the calculation of free energies fi computer simulations. Errors may arise from inaccuracies in the Hamiltonian, be it potential model chosen or its implementation (the treatment of long-range forces, e j lie second source of error arises from an insufficient sampling of phase space. 

Understanding and Improving Free Energy Calculations in Molecular Simulations Error Analysis and Reduction Methods... 

The behavior of the detection algorithm is illustrated by adding a bias to some of the measurements. Curves A, B, C, and D of Fig. 3 illustrate the absolute values of the innovation sequences, showing the simulated error at different times and for different measurements. These errors can be easily recognized in curve E when the chi-square test is applied to the whole innovation vector (n = 4 and a = 0.01). Finally, curves F,G,H, and I display the ratio between the critical value of the test statistic, r, and the chi-value that arises from the source when the variance of the ith innovation (suspected to be at fault) has been substantially increased. This ratio, which is approximately equal to 1 under no-fault conditions, rises sharply when the discarded innovation is the one at fault. 

When PSpice runs a Transient Analysis, it solves differential equations to find voltages and currents versus time. The time between simulation points is chosen to be as large as possible while keeping the simulation error below a specified maximum. In some cases, where PSpice can take large time steps, you may get a graph that does not look sinusoidal ... 

FIGURE 3.10 Force-distance curves. The curvilinear line is the prediction of the coulombic attraction theory. The straight line has been drawn through simulated error bars in the previous line to illustrate the approximately linear In P vs. x (or In P vs. r) plots expected from the coulombic attraction theory. The upper dotted line is the corresponding Yukawa prediction. If we scale the plot such that the 1996 data of Crocker and Grier [14] lies along the upper dotted line, the data for the same sample in their 1994 paper [13] is as represented by the lower dotted line. 

We present below some easily implementable methods for improving the robustness and efficiency of feasible path dynamic optimization codes which have proved useful in our work. Here, we cover methods for preventing simulation error from disrupting optimization, representation of path constraints, and handling poor local approximations during the optimization. 

Spreadsheet Summary In Chapter 12 of Applications of Microsoft Excel in Analytical Chemistry, we explore errors in spectrophotometric mea.surements by simulating error curves such as those shown in Figure 26-11. 

Fig. 9.2-2 By combining the effect of ten (left) or one thousand (right) iteration steps in each row, the computational error can be reduced tenfold or one-thousand-fold respectively. In the latter case, experimental errors will most likely exceed any simulation errors.

Fig. 9.2-5 Left explicit simulation of the establishment of the monomer-dimer equilib -rium (9.2-25) with o = 0-8. b0 = 1.2, k = 1, and k = 1. Right the simulation error aslmui -...

Column and high performance liquid chromatography (HPLC) methods for measurement of solubility, octanol-water partition coefficient, and vapor pressure which are replacing the older equilibrium methods tend to underestimate aqueous solubility and vapor pressure and tend to overestimate the octanol-water partition coefficient. The standard deviation for both the equilibrium and dynamic systems are similar, but calibration between systems is necessary to insure that they agree. The range of errors for both types of measurement as mentioned in the literature are well within the range predicted by the computer-simulated error distributions generated in this report. The measurement error... 

The effects of systematic errors are best studied by analyses of accurate Burnett data with superimposed simulated errors. For a relative pressure offset of 0.003%, which is comparable with the accuracy of piston gauges, the ethylene second virial coefficient of —167 cm3/mol changes only by 0.02 cm3/mol. Thus, this type of error is largely cancelled in the Burnett method. An offset in N of 11 ppm, which is comparable with the N variation we expect, changes the same second virial coefficient by 0.1 cm3/mol. Errors resulting from truncation of the series... 

McCormick developed two complementary sets of equations involving quadratic integrals for determining the albedo and an arbitrary number of scattering expansion coefficients, provided unpolarized intensity measurements are dependent on both the polar and azimuthal angles [7]. Numerical tests were performed by Oelund and McCormick [11] that demonstrated the sensitivity of the estimated parameters to simulated errors in the measurements, and provided insight into which of the two independent sets of equations was better. 

In the high reflux ratio case, the reflux flow rate is set by the level controller. The reflux flow rate is measured and sent to a multiplier whose other input is the reciprocal of the desired reflux ratio (D/R). The output signal from the multiplier goes to the SP of a distillate flow controller, which is on cascade. A common simulation error is to send the output signal of the multiplier directly to a control valve. This is an obvious and serious error, but is one that is often made. These control configurations will be used in examples in later chapters. 

It has been shown the beginning of a process which will lead to with the development of a simple procedure for the calibration of laser tracker systems. The work done so far include the tests on SMR and planning of them with Active Target. Also the definition of LT models to study and geometric errors is at an advanced stage along with the definition of an automatic generator of synthetic data with simulated errors. 

Wagner, B. Estimation of Simulation Errors and Investigations of Operating Range Extensions for the European Transonic Windtunnel ETW. BMFT-FB, Bericht No. W 82-003 (1982)... 

One critical factor to keep in mind when building and evaluating predictive models is that every experimental data point has an error associated with it. For example, if we measure the Log 5 of a compound as -6 and that data point has an error of 0.3 log units, the acmal value could be anywhere between -6.3 and -5.7. In a 2009 paper, Brown and coworkers [41] examined the relationship between experimental error and model performance. They carried out a series of theoretical experiments where Gaussian distributed random values were added to data to simulate experimental errors. The authors then calculated the correlation between the measured values and the same values with this simulated error. This correlation can be thought of as the maximum correlation possible given the error in the measurement. As we saw earlier. 

Nagy T, Turdnyi T. Reduction of very large reaction mechanism using methods based on simulation error minimization. Combust Hame 2009 156 417-28. 

The distribution of local normal vapour flux, j, over the droplet surface is shown in Fig. 2. It is done using a rescaled, normalised vapour flux j/jn/iiL Tcc), where A/2(L,7 oo) is defined by Eq. (2). The line with triangles (Fig. 2), gives the result, which coincides with the analytical solution (the straight line) j/U/2 L,Too) = 1 within the simulation error bar. 

It was early realised (for example by Crank and Furzeland in 1977 [40]) that the singularity at such edges will degrade overall accuracy in a simulation. In fact, Gavaghan points out [44, 94] that because of this effect, the simulation error in calculated concentrations is of h being the interval size in space, near the... 

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