Conditional Monte Carlo sampling

Prediction of Branched Architectures by Conditional Monte Carlo Sampling... 

Prediction of Branched krchitecturei by Conditional Monte Carlo Sampling 513 9.9.4.3 Characterization of Architectures by Seniorities and Priorities... 

In order to discuss schemes that allow us to include information about rejected states in our Monte Carlo sampling, it is useful to review the basic equations that underlie Metropolis importance sampling. Our aim is to sample a distribution p. The weight of state i is denoted by p i). The probability that the system that is initially in state m will undergo a transition to state n is denoted by TTmn- This probability is normalized as the system must end in some state (possibly the original state o). The normalization condition... 

The calculation implied by Eq. (9) for N(E) (or Eq. (1) for k(T)) is therefore to integrate over phase space (p, q)—in practice, usually with Monte Carlo sampling methods—where each phase point (p, q) serves as the initial conditions for a trajectory that must be run (i.e., numerically integrated) to determine whether Xr is 1 or 0, i.e., whether or not this phase point contributes to the integral. Because the flux, Eq. (6), contains the factor 8[/(q)], all trajectories begin on the dividing surface/(q) = 0. 

Figure 6 Diagram illustrating kth step of the construction of a square king lattice of L X L spins with the stochastic models (SM) method solid circles denote lattice sites already filled with spins ( 1) in preceding steps of the process open circles denote the still empty lattice sites. The linear nature of the buildup construction is achieved by using spiral boundary conditions (i.e., the first spin in a row interacts with the last spin of the preceding row). Whereas all the L uncovered spins (at sites k - L,k — L + 1,..., k - ) determine the transition probability for selecting spin k, the spins in close proximity to k k - 1, k - L, etc.) have the largest effect. The local states method is based on the SM construction. Thus, the transition probabilities for spin k are obtained from a Metropolis Monte Carlo sample by calculating the number of occurrences of the various local states, (a, a) = n k-v k-2 k-L k-L v k-L 2 k-L 3 l- transition probability is jS(cT d ) = (cr, ff)/[ (a = 1,ct) -I- n(a = These transition...

Comparison of these two sets of boundary conditions is not easy, since the two calculations also differed in other ways. In that of Patey and Valleau (PV) the solvent molecules were hard spheres with embedded point dipoles, such that the solvent s dielectric constant was about 8, while in that of Rasaiah and McDonald (RM) they were Stockmayer particles, with dipole moment corresponding to a somewhat larger (but then necessarily less certain) dielectric constant. In both cases the ions were equidistant from the center of the central box, but in RM the line joining them was an axis of synunetry of the system, while in PV the results were averaged over all orientations of that line. (Some more discussion of the Monte Carlo sampling is given below.)... 

Other devices for improving the efficiency of the sampling techniques in Monte Carlo calculations are Russian roulette and splitting described by Kahn [4], the conditional Monte Carlo of Trotter and Tukey [6], and the antithetic variates of Hammersley and Morton [7 8]. It is likely that further powerful techniques of this sort await discovery, whereby the usefulness of Monte Carlo for practical problems may be greatly extended. 

H. F. Trotter, and J. W. Tukey, Conditional Monte Carlo for normal samples, Florida Symposium, John Wiley Sons, 1954, p. 64. 

Market simulations are run many times for a range of initial aquifer levels, with input obtained via Monte Carlo sampling from a joint, multivariate distribution created from inflow and withdrawal data. Simulated supply and demand conditions are translated into market prices for each transfer type, and the expected cost and reliability of various combinations, or portfolios, of transfer types can be computed. The transfer types are specified for each scenario, and a sequential search method is then used to identify minimum cost portfolios that meet designated supply-reliability constraints (Figure 3). Differences in the cost of the respective portfolios indicate the value of including each transaction type in the market, as well as how the cost of market-based approaches compares to the development of the least expensive new water source (Carrizo Aquifer). 

The Monte Carlo sampling of initial conditions in the method of classical trajectories is illustrated here for the particular problem of computing the reaction cross-section ctr given in terms of the opacity function P b) by, Eq. (3.14),... 

J. Monte Carlo sampling of initial conditions, (a) Discuss why the reaction cross-section computed using Eq. (A.5.7) need not change in value if the value, B, of the maximal impact parameter for reaction is taken larger than it has to be. (b) The reaction rate constant for thermal reactants is obtained. Section 3.1, as an average over the reaction cross-section. Determine a sampling of the relative... 

K. Limitations on the Monte Carlo sampling of initial conditions. Say you want to compute the reaction rate constant when there is a rather small steric factor and/or a rather high activation energy, (a) Will you use a Monte Carlo sampling of initial conditions Discuss in detail why we think that getting an accurate result will require a considerable consumption of computer time. 

The above presentation describes how the collision impact parameter is sampled to calculate reaction cros.s-sections. In the following, Monte Carlo sampling of the reactant s Cartesian coordinates and momenta is de.scribed for atom -I- polyatomic collisions. Initial energies are chosen for the reactants, which correspond to quantum mechanical vibrational-rotational energy levels. This is the quasiclassical model. " Extensive reviews have been given of the procedure for selecting initial conditions for atom + diatom collisions. 

The approach is therefore analogous to the classical approaches described above, but instead of directly calculating the conditional variances using, e.g., FAST or Monte Carlo samples, now a metamodel is developed first and the sensitivity indices are calculated using the metamodel. 

The efficiency and accuracy of Subset Simulation also depends on the set of univariate proposal PDFs qiJ, k = I,..., d, that are used within the Modified Metropolis algorithm for sampling fi om the conditional distributions n -IFi). To see this, note that in contract to the Monte Carlo samples xq n -) which are i.i.d., the... 

Fig. 6 Samples generated by Subset Simulation red samples are Monte Carlo samples generated at the 0th unconditional level, purple samples are MCMC sample generated at the 1st conditional level, etc. The dashed lines represent the boundaries of intermediate failure domains f , / =...

Since the early 1960s classical trajectory simulations, with Monte Carlo sampling of initial conditions, have been widely used to study the uni-molecular and intramolecular dynamics of molecules and clusters reactive and nonreactive collisions between atoms, molecules, and clusters and the collisions of these species with surfaces. " For a classical trajectory study of a system, the motions of the atoms for the system under study are determined by numerically integrating the system s classical equations of motion. These equations are usually expressed in either Hamilton s form ... 

An ensemble of trajectories is calculated in a trajectory simulation, with each trajectory in the ensemble specified by the system s initial set of momenta p and coordinates q. The initial ensemble of p and q is chosen to represent the experiment under investigation or chosen so that a particular dynamical attribute of the system may be studied. Distribution functions are usually sampled randomly in choosing the ensemble of initial conditions and the methodology of sampling is often called Monte Carlo sampling. Procedures for choosing trajectory initial conditions to represent unimolecular and bimolecular reactions, and gas-surface collisions are described in the section on trajectory initial conditions. 

Procedures for selecting initial values of coordinates and momenta for an ensemble of trajectories has been described in detail in recent chapters entitled Monte Carlo Sampling for Classical Trajectory Simulations and Classical Trajectory Simulations Initial Conditions. In this section a brief review is given of methods for selecting initial conditions for trajectory simulations of unimolecular and bimolecular reactions and gas-surface collisions. 

In the Monte Carlo sampling Step 1, the calculations have to be carried out with many groups of input data samples to obtain the statistical distribution of MCST. In order to reduce the required computing time and to ensure accuracy, a subchannel procedure, which is applied in a conservative manner, is adopted for this study. The subchannel analysis for the hot assembly in the core is performed for each group of samples. The hot channel and the hot spot are assumed to be in the hot assembly. Therefore, this closed hot assembly not only has the maximum assembly power, but also has the worst thermal condition in the core. Some results of the three-dimensional core design are used to determine the nominal power distributiOTi and other parameters in the hot assembly. 

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