Monte Carlo Sequence Research Articles

Complexity analysis of quasi continuous level Monte Carlo

Continuous level Monte Carlo is an unbiased, continuous version of the celebrated multilevel Monte Carlo method. The approximation level is assumed to be continuous resulting in a stochastic process describing the quantity of interest. Continuous level Monte Carlo methods allow naturally for samplewise adaptive mesh refinements, which are indicated by (goal-oriented) error estimators. The samplewise refinement levels are drawn in the estimator from an exponentially-distributed random variable. Unfortunately in practical examples this results in higher costs due to high variance in the samples. In this paper we propose a variant of continuous level Monte Carlo, where a quasi Monte Carlo sequence is utilized to ''sample'' the exponential random variable. We provide a complexity theorem for this novel estimator and show that this theoretically and practically results in a variance reduction of the whole estimator.

Fracture Network Characterization Using Stress‐Based Tomography

AbstractInformation on structural features of a fracture network at early stages of Enhanced Geothermal System development is mostly restricted to borehole images and, if available, outcrop data. However, using this information to image discontinuities in deep reservoirs is difficult. Wellbore failure data provides only some information on components of the in situ stress state and its heterogeneity. Our working hypothesis is that slip on natural fractures primarily controls these stress heterogeneities. Based on this, we introduce stress‐based tomography in a Bayesian framework to characterize the fracture network and its heterogeneity in potential Enhanced Geothermal System reservoirs. In this procedure, first a random initial discrete fracture network (DFN) realization is generated based on prior information about the network. The observations needed to calibrate the DFN are based on local variations of the orientation and magnitude of at least one principal stress component along boreholes. A Markov Chain Monte Carlo sequence is employed to update the DFN iteratively by a fracture translation within the domain. The Markov sequence compares the simulated stress profile with the observed stress profiles in the borehole, evaluates each iteration with Metropolis‐Hastings acceptance criteria, and stores acceptable DFN realizations in an ensemble. Finally, this obtained ensemble is used to visualize the potential occurrence of fractures in a probability map, indicating possible fracture locations and lengths. We test this methodology to reconstruct simple synthetic and more complex outcrop‐based fracture networks and successfully image the significant fractures in the domain.

CALCULATION UNCERTAINTIES IN SPENT FUEL INVENTORY DETERMINATION

In the depletion calculation of the nuclear fuel, the uncertainty is of utmost importance, as it affects the uncertainty of the subsequent calculation, when the calculated composition is used. The calculations are even more important when they are safety related, e. g., when determining the reactivity or emissions of radioactivity to the environment. This work covers the depletion model of Gd-2M+ fuel which was developed in ORIGEN-ARP/TRITON calculation sequences, both being parts of a SCALE 6.2.1 package. The uncertainties of the respective calculation model were determined by comparison with experimental data for both calculation sequences. The effects of operational and manufacturing uncertainties on concentration of the most important nuclides using TRITON depletion model of Gd-2M+ fuel were determined. The effect of respective uncertainties manifested in changes of composition on the multiplication factor using Monte-Carlo sequence KENO-VI were also specified.

Generation and application of multivariate polynomial quadrature rules

The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second contribution is the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios.

Application of Monte Carlo simulations to improve basketball shooting strategy

The underlying physics of basketball shooting seems to be a straightforward example of the Newtonian mechanics that can easily be traced by numerical methods. However, a human basketball player does not make use of all the possible basketball trajectories. Instead, a basketball player will build up a database of successful shots and select the trajectory that has the greatest tolerance to small variations of the real world. We simulate the basketball player's shooting training as a Monte Carlo sequence to build optimal shooting strategies, such as the launch speed and angle of the basketball, and whether to take a direct shot or a bank shot, as a function of the player's court positions and height. The phase space volume that belongs to the successful launch velocities generated by Monte Carlo simulations are then used as the criterion to optimize a shooting strategy that incorporates not only mechanical, but human factors as well.

On the use of randomization tests following adaptive designs

ABSTRACTRandomization tests (sometimes referred to as “re-randomization” tests) are used in clinical trials, either as an assumption-free confirmation of parametric analyses, or as an independent analysis based on the principle of randomization-based inference. In the context of adaptive randomization, either restricted or response-adaptive procedures, it is unclear how accurate such Monte Carlo approximations are, or how many Monte Carlo sequences to generate. In this paper, we describe several randomization procedures for which there is a known exact or asymptotic distribution of the randomization test. For a special class of procedures, called , and binary responses, the exact test statistic has a simple closed form. For the limited subset of existing procedures with known exact and asymptotic distributions, we can use these as a benchmark for the accuracy of Monte Carlo randomization techniques. We conclude that Monte Carlo tests are very accurate, and require minimal computation time. For simple tests with binary response in the class of procedures, the exact distribution provides the best test, but Monte Carlo approximations can be used when the exact distribution is difficult to compute.

High-performance financial simulation using randomized quasi-Monte Carlo methods

Graphics Processing Unit (GPU) computing has become popular in computational finance, and many financial institutions are moving their CPU-based applications to the GPU platform. Since most Monte Carlo algorithms are embarrassingly parallel, they benefit greatly from parallel implementations, and consequently Monte Carlo has become a focal point in GPU computing. GPU speed-up examples reported in the literature often involve Monte Carlo algorithms, and there are software tools commercially available that help migrate Monte Carlo financial pricing models to GPU. We present a survey of Monte Carlo and randomized quasi-Monte Carlo methods, and discuss existing (quasi) Monte Carlo sequences in GPU libraries. We discuss specific features of GPU architecture relevant for developing efficient (quasi) Monte Carlo methods. We introduce a recent randomized quasi-Monte Carlo method, and compare it with some of the existing implementations on GPU, when they are used in pricing caplets in the LIBOR market model and mortgage-backed securities.

Statistical Impact Prediction of Decaying Objects

This paper describes a method for statistically predicting the impact time and location of an orbital object under 200 km altitude. It also investigates the influence of the chief parameter uncertainties on the statistical dispersion. Unlike the main methods used nowadays, which derive drag parameters from historical state data, a shape model is used to acquire aerodynamic coefficients in the various flow regimes. As a test object, the Delta-K rocket stage is chosen, due to the high data availability for validating the tool. Using a Monte Carlo sequence, multiple six-degrees-of-freedom simulations are executed, in which the initial state and the density output of the atmosphere model are varied within their uncertainty windows. By kernel density estimation, the resulting data are used to derive a probability density function of the impact time and infer a ground track expressing impact probability. In a comparative study with the Tracking and Impact Predictions published by the United States Space Surveillance Network, the method’s performance is tested. In this comparison, a decrease in impact window size is observed, while maintaining the reliability. Moreover, by a variance-based sensitivity analysis, the uncertainty in the density model is identified as the prime contributor to footprint dispersion and it is shown that knowledge of the rotational state can be critical in decreasing impact windows.

Diagnosing the Relative Impact of “Sneaks,” “Phantoms,” and Volatility in Sequences of Lagged Ensemble Probability Forecasts with a Simple Dynamic Decision Model

AbstractMonte Carlo simulation of sequences of lagged ensemble probability forecasts is undertaken using Markov transition law estimated from a reforecast ensemble. A simple three-state, three-action dynamic decision model is then applied to the Monte Carlo sequence realizations using a basket of cost functions, and the resulting expense incurred by the decision model is conditioned upon the structure of the sequence realizations. Findings show that the greatest average expense is incurred by “sneak” and “volatile” sequence structures, which are structures characterized by large and rapid increases in event probability at short lag times. These findings are simple quantitative illustration of the adage that large run-to-run variability of forecasts can be troublesome to a decision maker. The experiments also demonstrate how even small improvements in the amount of advance warning of an event can translate into a substantial reduction in decision expense. In general, the conditioned decision expense is found to be sensitive to sequence structure for a given cost function, to the parameters of a given cost function, and to the choice of cost function itself.

A structure and evolution-guided Monte Carlo sequence selection strategy for multiple alignment-based analysis of proteins

Various multiple sequence alignment-based methods have been proposed to detect functional surfaces in proteins, such as active sites or protein interfaces. The effect that the choice of sequences has on the conclusions of such analysis has seldom been discussed. In particular, no method has been discussed in terms of its ability to optimize the sequence selection for the reliable detection of functional surfaces. Here we propose, for the case of proteins with known structure, a heuristic Metropolis Monte Carlo strategy to select sequences from a large set of homologues, in order to improve detection of functional surfaces. The quantity guiding the optimization is the clustering of residues which are under increased evolutionary pressure, according to the sample of sequences under consideration. We show that we can either improve the overlap of our prediction with known functional surfaces in comparison with the sequence similarity criteria of selection or match the quality of prediction obtained through more elaborate non-structure based-methods of sequence selection. For the purpose of demonstration we use a set of 50 homodimerizing enzymes which were co-crystallized with their substrates and cofactors.

Simulation Estimation of Mixed Discrete Choice Models with the Use of Randomized Quasi–Monte Carlo Sequences

The overall performance of the quasi–Monte Carlo (QMC) sequences proposed by Halton and Faure, as well as their scrambled versions, are numerically compared against each other and against the Latin hypercube sampling sequence in the context of the simulated likelihood estimation of a mixed multinomial logit model of choice. In addition, the efficiency of the QMC sequences generated with and without scrambling is compared across observations, and the performance of the Box–Muller and inverse normal transform procedures is tested. Numerical experiments were performed in five dimensions with 25, 125, and 625 draws and in 10 dimensions with 100 draws. Results indicate that the Faure sequence consistently outperforms the Halton sequence and that the scrambled versions of the Faure sequence perform best overall.

Bayesian Markov chain Monte Carlo sequence analysis reveals varying neutral substitution patterns in mammalian evolution.

We describe a model of neutral DNA evolution that allows substitution rates at a site to depend on the two flanking nucleotides ("context"), the branch of the phylogenetic tree, and position within the sequence and implement it by using a flexible and computationally efficient Bayesian Markov chain Monte Carlo approach. We then apply this approach to characterize phylogenetic variation in context-dependent substitution patterns in a 1.7-megabase genomic region in 19 mammalian species. In contrast to other substitution types, CpG transition substitutions have accumulated in a relatively clock-like fashion. More broadly, our results support the notion that context-dependent DNA replication errors, cytosine deamination, and biased gene conversion are major sources of naturally occurring mutations whose relative contributions have varied in mammalian evolution as a result of changes in generation times, effective population sizes, and recombination rates.

Computing Discrepancies of Smolyak Quadrature Rules

In recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as possible competition to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadrature formulas consists in computing theirL2-discrepancy. Especially for larger dimensions, such computations are a highly complex task. In this paper we develop a fast recursive algorithm for computing theL2-discrepancy (and related quality measures) of general Smolyak quadratures. We carry out numerical comparisons between the discrepancies of certain Smolyak rules and Hammersley and Monte Carlo sequences.

Diffusion correlation in random alloys

Abstract The solid state diffusion process in random binary alloys has been simulated on a computer array of A and B atoms, coded in the relevant cubic symmetry. The process was a Monte Carlo sequence on which were superimposed the appropriate conditions of concentration and defect-atom exchange frequencies. Correlation factors for atoms and vacancies have been obtained for b.c.c. and f.c.c. random alloys. Their validity was verified for monatomic solids of the same symmetry, for which correlation factors are known to a high degree of accuracy from random walk analysis. The reproducibility of the simulation method varies between 4% and 1% contingent upon the degree of correlation. Semi-empirical polynomials of degree z−1 (z is the coordination number for the particular lattice symmetry) for − t versus concentration describe the results obtained accurately (t is defined by ƒ = (1+t)/(1−t)) . Comparison with the calculations by Manning shows discrepancies which are only significant when there is considerable correlation. There are, however, substantial differences between the present results and those of Kikuchi and Sato over the entire concentration and frequency ranges.

Computer simulation of the gas/liquid surface

The gas/liquid surface of a system of 255 Lennard-Jones (12,6) molecules has been simulated by Monte Carlo sequences at three reduced temperatures which span most of the liquid range. The results at the two higher temperatures show a monotonic profile of density as a function of height, but those at a temperature close to the triple point suggest that the liquid just below the surface can form layers; that is, that local density is an oscillatory function of height. This phenomenon is compared with similar previous results and it is concluded that it may well be the consequence of the constraints applied to the simulated system rather than an inherent property of a free liquid surface.

Statistical studies of protein sequences: Pair and triplet analysis in helical conformations

Nearest neighbor pair and triplet associations in helical conformations of protein chains were compared with the same associations in Monte Carlo sequences generated from equivalent populations of suborns. The analysis was performed with different groupings of amino acids: on the basis of chemical similarities, side-chain volume and side-chain orientation in crystals. The results of the tests with chemical similarity groupings of residues give evidence of specific associations of amino acids in the proteins analyzed, but the basis of this association is unknown.