Deterministic simulations

It might seem that transition rules that are predominantly random would not give rise to interesting behavior, but this is not entirely true. Semirandom rules have a role in adding noise to deterministic simulations and, thus, leading to a simulation that is closer to reality, but even without this role such rules can be of interest. 

XPPAUT (http //www.math.pitt.edu/ bard/xpp/xpp.html) offers deterministic simulations with a set of very good stiff solvers. It also offers fitting, stability analysis, nonlinear systems analysis, and time-series analysis, like histograms. The GUI is simple. It is mainly available under Linux, but also mns on Windows. 

Fig. 7.3 Deterministic (a) and noisy (b) computer simulations of the time course of affective disorders showing the intervals between successive disease episodes (interval duration) as a function of a disease variable S and examples of episode generation from different disease states (figure modified after [2]). In deterministic simulations (a), there is a progression from steady state (S = 18) to subthreshold oscillations (S = 22) with immediate onset of periodic event generation at a certain value of S (slightly below S = 60). With further increase of S, the intervals between successive episodes are continuously...

In deterministic simulations, these subthreshold oscillations may cover only a narrow regime. However, with the addition of noise a broad regime of functionally most interesting patterns can develop exactly in and around the area of subthreshold oscillations. Noise can introduce a random mixture of subthreshold oscillations... 

In this chapter, we explain the technique of sequential bifurcation and add some new results for random (as opposed to deterministic) simulations. In a detailed case study, we apply the resulting method to a simulation model developed for Ericsson in Sweden. In Sections 1.1 to 1.3, we give our definition of screening, discuss our view of simulation versus real-world experiments, and give a brief indication of various screening procedures. 

The technique of sequential bifurcation is an important and useful method for identifying important factors in experiments with simulation models that involve a large number of factors. We have demonstrated the steps of this technique through a case study on three supply chain configurations in the Swedish mobile communications industry. We have formalized the assumptions of the technique and found that, in practice, these assumptions may not be too restrictive, as our case study illustrates. We have extended the technique of sequential bifurcation to random (as opposed to deterministic) simulations. 

Whatever the geologic causes, there are several purely statistical inferences to be drawn from Figure 16 which bear directly on the issue of reservoir simulation. The size of grid four may be a natural choice for the grid block size in a deterministic simulation model. Such a selection would minimize the variation between blocks and may, in fact, make stochastic assignments of secondary importance (thus, reducing the differences between realizations). The variation of the fifth scale would be incorporated as pseudo functions or megascopic dispersivity into individual blocks. 

The kinetic Monte Carlo (KMC) simulation method focuses on the state-to-state dynamic transitions and neglects the short-time system fluctuations. This approximation allows much longer timescales to be reached, without chemically relevant compromise in the resolution of the simulation, especially for solid-state systems. This is particularly important, since the diffusion of an oxygen ion on the surface of a YSZ electrolyte (among defect sites) requires approximately 1 ps, and the adsorption of one molecular oxygen onto the YSZ at 0.01 atm pressure requires approximately 0.5 ps [32]. Thus, deterministic simulation methods, like MD, are not easily able to capture this behavior, so other methods must be employed. 

Prior to model estimation the question that it will be used to answer and the specific manner in which it will be used should be explicitly stated. Using a model to answer a question is the act of simulation. There are two types of simulation deterministic and stochastic. In a deterministic simulation, the statistical model is ignored and no error is introduced into the model—the results are error-free. For example, given data from single-dose administration of a drug it may be of interest to predict the typical concentration-time profile at steady-state under a repeated dose administration regimen. A deterministic simulation would be useful in this case. 

Once an input-output model is coupled to a trial execution model, the trial may be simulated either once or many times. Each time the simulation is executed is referred to as an iteration, replicate, or run. For troubleshooting purposes, one should always confirm that the simulation works using a few replicate runs, two to five should be sufficient, before progressing to executing the simulation many times. Many stochastic simulations may take days to run and if the simulation does not work as performed using a single run, then days may be wasted before finding this out. In a deterministic simulation, there is no need to replicate the simulation many times because the outcome will always be the same. However, with a stochastic simulation, the outcome could be different every time the simulation is replicated. 

Deterministic simulation will be used to illustrate the effect of fitting a less complex to model to data arising from a more complex model. Concentrationtime data were simulated from a 3-compartment model with the following parameters CL = 1.5 Q2 = 0.15 Q3 = 0.015 VI = 1, V2 = 0.5 V3 = 0.25. A bolus unit dose of 100 was injected into the central compartment (VI). A total of 50 samples were collected equally spaced for 48-time units. The data are shown in Fig. 3.15 and clearly demonstrate triphasic elimination kinetics. Now suppose the LLOQ of the assay was 0.01. Data above this line are consistent with biphasic elimination kinetics. What hap-... 

Example 4.9 Stochastic versus deterministic simulation of a virus... 

Deterministic simulation of reaction A h- B — C compared to stochastic simulation. 185... 

Next we explore the effect of increasing the initial number of A molecules on a single simulation. The results for 1000 and 4000 initial A molecules are shown in Figures 4.30 and 4.31, respectively. We see the random fluctuations become less pronounced. Notice that even With only 4000 starting molecules, Figure 4.31 compares very favorably with the deterministic simulation shown in Figure 4.11 of Section 4.2. 

See Law (2007) or Carson (2004) for a more detailed discussion about these prerequisites. The sub-class of deterministic simulation is typically referred to as computer experiments, see Santner et al. (2003) for more details. In this work the focus is on stochastic simulation models. 

The regional earthquake hazard scenario can be probabilistic or deterministic. The variation of earthquake ground shaking parameters (peak ground accelerations, PGA and spectral accelerations, SA at T = 0.2 s and 1 s at the engineering bedrock outcrop) are determined independently within the investigated area for a specified level of exceedance probability or based on deterministic simulations. 

Execution of the method requires the physical domain to be divided into a distribution of conqiutational cells. The cells provide geometric boundaries and volumes, which are used to sample macroscopic properties. Also, only molecules located within the same cell, at a given time, are allowed to collide. The DSMC simulation proceeds from a set of prescribed initial condition. The molecules randomly populate the computational domain. These simulated molecules are assigned random velocities, usually based on the equilibrium distribution. The simulated representative particles move for a certain time step. This molecule motion is modeled deterministically. This process enforces the boundary conditions. With the simulated particles being appropriately indexed, the molecular collision process can be performed. The collision process is modeled statistically, which is different from deterministic simulation methods such as the molecular dynamics methods. In general, only particles within the same computational cell are considered to be possible collision partners. Mthin each cell, collision pairs are selected randomly and a representative set of collisions is performed. The post-collision velocities are determined. There are several... 

Poole, D. and Raftery, A.E. 2000. Inference for deterministic simulation models the Bayesian melding approach. Journal of the American Statistical Association, 95, 1244-1255. 

In MD, time is a clearly singled out variable in a deterministic simulation based on a postulated force field and on the classical equations of motion. For the simulation of an evolving crystal aggregate, MD has the obvious advantage that the kinetics of the process is transparent, as accretion rates can be immediately described as a function of computational time, although the rate of any molecular process is obviously dependent on the postulated force model. In contrast, there is no apparent time variable in an MC simulation, because evolution steps are random and may randomly affect molecular evolutions which in reality happen on different timescales. If, as is often the case, time in MC is taken as proportional to the number of moves, one is implicitly assuming that all molecular moves occur on the same timescale, perhaps not a very severe approximation in studies of molecular aggregates bound by nearly isotropic van der Waals forces. In a variant of the MC formulation, called kinetic Monte Carlo (KMC)... 

Fig. 22. Probability distribution computed by time averaging over the corresponding phase space trajectories under the conditions of Figure 19 (a) deterministic simulation (b) master equation simulation.

Shared variables must be used with caution to ensure that multiple assignments to the same variable in different processes are correctly synchronized. There must be no possibility that two processes could be updating the same variable concurrently. This can lead to a design with unpredictable (non-deterministic) simulation and synthesis results. 

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