Traditional Monte Carlo Sampling Research Articles

Analytical marginalization over photometric redshift uncertainties in cosmic shear analyses

ABSTRACT As the statistical power of imaging surveys grows, it is crucial to account for all systematic uncertainties. This is normally done by constructing a model of these uncertainties and then marginalizing over the additional model parameters. The resulting high dimensionality of the total parameter spaces makes inferring the cosmological parameters significantly more costly using traditional Monte Carlo sampling methods. A particularly relevant example is the redshift distribution, p($z$ ), of the source samples, which may require tens of parameters to describe fully. However, relatively tight priors can be usually placed on these parameters through calibration of the associated systematics. In this paper, we show, quantitatively, that a linearization of the theoretical prediction with respect to these calibrated systematic parameters allows us to analytically marginalize over these extra parameters, leading to a factor of ∼30 reduction in the time needed for parameter inference, while accurately recovering the same posterior distributions for the cosmological parameters that would be obtained through a full numerical marginalization over 160 p($z$ ) parameters. We demonstrate that this is feasible not only with current data and current achievable calibration priors but also for future Stage-IV data sets.

Estimating space–time wave statistics using a sequential sampling method and Gaussian process regression

The traditional Monte Carlo sampling of waves requires generating tens of thousands of random waves to achieve stable statistics in the tail of the distribution, which is computationally costly for numerical simulations and time-consuming and often impractical for experiments. To improve the sampling efficiency, we present a sequential sampling method, which predicts the wave statistics based on the nonlinear response of deterministic wave groups. This data-driven method starts with parameterising the linear random wave field with a series of Gaussian wave groups. The nonlinear system response from the water wave system is then approximated by observations through the nonlinear evolution of wave groups with carefully designed initial conditions. A sequential sampling method is introduced during this sampling procedure, which determines the next best sampling point based on the previous observations and the distance metric. We examine the performance of the proposed method with Monte Carlo simulation of random wave fields as well as a grid search sampling strategy. The results show the proposed method can achieve computational cost savings over several orders of magnitude for numerical simulations. The results also suggest such a sequential sampling method can be used to determine the test matrix of the wave experiments with better coverage over the parameter space.

A Day-Ahead Chance Constrained Volt/Var Control Scheme With Renewable Energy Sources by Novel Scenario Generation Method in Active Distribution Networks

With the integration of huge renewable energy sources (RESs) into active distribution networks, how to address the uncertainty outputs of RESs for the day-ahead volt/var control (VVC) is a significant challenge. This paper presents a chance constrained mixed integer second order cone model to handle the nodal power uncertainties and nonlinear branch flow equations. A direct and fast scenario generation method is proposed by employing the group division method and the seven-step probability distribution model of RESs outputs. An efficient and accurate solution method which only uses few larger probability level scenarios instead of all reserved scenarios is also proposed. Numerical simulations on the IEEE 69 standard system show the superiority of the proposed algorithm over traditional Monte Carlo sampling (MCS)-based methods.

Enhancing risk control ability of distribution network for improved renewable energy integration through flexible DC interconnection

The increased penetration of distributed renewable energy into modern distribution networks has resulted in a significant rise in the system operational risk. However, the risk control ability of traditional open-loop AC distribution networks is greatly limited due to the inherent restriction of the network structure, which hinders the integration of larger-scale renewable energy into the power grid. To address this issue, this paper developed an effective risk assessment model and risk averse method for fully utilizing flexible DC interconnections to enhance the risk control ability and to improve the penetration of renewable energy in distribution networks. Moreover, aiming at the complex coupling risk characteristics between the AC and DC system that are imposed by the DC interconnection and considering the inefficiency of traditional Monte Carlo sampling risk assessment methods, an effective point-estimation-based risk assessment method is proposed that conducts a limited number of sequential AC/DC power flow calculations with estimation points constructed from random input variables, thereby the operational risk of the system is effectively realized. On this basis, a risk-averse multi-objective optimization model is further established for minimizing both the operational risk and the expected network power loss, and an improved kriging-model-assisted multi-object gray wolf algorithm is proposed for quickly supporting the model solution. Finally, a fuzzy clustering method is presented for effectively identifying the optimal Pareto solution. A case study is conducted on an AC/DC hybrid system with three AC networks and one DC network. The results demonstrate that the proposed method can substantially improve the risk assessment efficiency compared with the traditional Monte Carlo method. In addition, the risk-averse optimization method can effectively reduce the system risk and improve the operational reliability as well as the penetration of renewable energy.

Reliability and sensitivity analysis of more‐electric aircraft based on cross‐entropy

The traditional Monte Carlo sampling method has several deficiencies in evaluating the reliability of more-electric aircraft (MEA), such as more sampling times and longer simulation time, due to the complex topology and low probability of failure of the aircraft. In this Letter, the optimal probability distribution of multi-component systems is introduced by cross-entropy, which theoretically makes the variance of reliability evaluation zero. Based on this, an evaluation method for the power-supply reliability of MEA is presented. For further analysis of the system vulnerabilities, the sensitivity of MEA is modelled based on reliability calculation. Finally, the hybrid MEA and high-voltage direct current MEA power systems are used as application scenarios to calculate the reliability and sensitivity. The results show that the proposed method has advantages in evaluating small probability events, and it is useful for the optimisation of MEA's power-supply system structure.

Efficient Information Flow Maximization in Probabilistic Graphs

Reliable propagation of information through large networks, e.g., communication networks, social networks or sensor networks is very important in many applications concerning marketing, social networks, and wireless sensor networks. However, social ties of friendship may be obsolete, and communication links may fail, inducing the notion of uncertainty in such networks. In this paper, we address the problem of optimizing information propagation in uncertain networks given a constrained budget of edges. We show that this problem requires to solve two NP-hard subproblems: the computation of expected information flow, and the optimal choice of edges. To compute the expected information flow to a source vertex, we propose the F-tree as a specialized data structure, that identifies independent components of the graph for which the information flow can either be computed analytically and efficiently, or for which traditional Monte-Carlo sampling can be applied independently of the remaining network. For the problem of finding the optimal edges, we propose a series of heuristics that exploit properties of this data structure. Our evaluation shows that these heuristics lead to high quality solutions, thus yielding high information flow, while maintaining low running time.

Response surface method for assessing energy production from geopressured geothermal reservoirs

Developing low-enthalpy geothermal resources along the US Gulf Coast is attractive for reducing global warming and providing clean energy. In this work, synthetic yet representative models for typical geopressured geothermal reservoirs located along the US Gulf Coast are considered. A Box–Behnken experimental design is used to select a small set of these models to perform detailed reservoir simulation runs. Full quadratic linear models are fit to the simulation results, and their sufficiency is confirmed by comparing them to kriging response surfaces. To achieve a higher degree of efficiency in using the response surfaces, Hammersley sequence sampling (HSS) method is used instead of traditional Monte Carlo sampling. HSS ensures that the factor space is sampled more uniformly and the response distribution is converged in less time. By evaluating these proxy models in the sampled factor space, the sensitivity and uncertainty of the response to the factors can be assessed. In this work, the sensitivity and uncertainty of engineered convection is assessed. For quantifying engineered convection, five uncertain reservoir attributes were selected. The response was defined as the net extracted enthalpy. In particular, two different designs for harvesting energy from geothermal reservoirs were compared using the response surfaces. In the modeled systems, results show that the regular design is more effective than the reverse design for extracting energy from geopressured geothermal reservoirs.

Analysis and computation of the elastic wave equation with random coefficients

We consider the stochastic initial–boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo sampling method for approximating quantities with high stochastic regularity.

Gradient-domain path tracing

We introduce gradient-domain rendering for Monte Carlo image synthesis. While previous gradient-domain Metropolis Light Transport sought to distribute more samples in areas of high gradients, we show, in contrast, that estimating image gradients is also possible using standard (non-Metropolis) Monte Carlo algorithms, and furthermore, that even without changing the sample distribution, this often leads to significant error reduction. This broadens the applicability of gradient rendering considerably. To gain insight into the conditions under which gradient-domain sampling is beneficial, we present a frequency analysis that compares Monte Carlo sampling of gradients followed by Poisson reconstruction to traditional Monte Carlo sampling. Finally, we describe Gradient-Domain Path Tracing (G-PT), a relatively simple modification of the standard path tracing algorithm that can yield far superior results.

Numerical solution of the Duffing equation with random coefficients

The main aim of this work is numerical solution of the nonlinear vibrations of micro-resonators exhibiting bounded and Gaussian uncertainty in their parameters. The mechanical response in deterministic situation is described by the Duffing equation, whose numerical solution is obtained with the Runge–Kutta–Fehlenberg algorithm, while probabilistic analysis is carried out using the generalized stochastic perturbation technique enriched with automatic optimization of the approximating polynomial. Basic solution to this nonlinear vibration in the deterministic context is obtained with the use of the computer algebra system MAPLE, where all additional probabilistic procedures are also implemented. We compare each time expectations, coefficients of variation, skewness and kurtosis for the structural response to show probabilistic sensitivity of the MEMS accelerometer with respect to its design parameter expectation and coefficient of variation. An additional comparison of the proposed technique with the traditional Monte-Carlo sampling for the first four probabilistic moments is also provided.

A comparison of approximation techniques for variance-based sensitivity analysis of biochemical reaction systems

BackgroundSensitivity analysis is an indispensable tool for the analysis of complex systems. In a recent paper, we have introduced a thermodynamically consistent variance-based sensitivity analysis approach for studying the robustness and fragility properties of biochemical reaction systems under uncertainty in the standard chemical potentials of the activated complexes of the reactions and the standard chemical potentials of the molecular species. In that approach, key sensitivity indices were estimated by Monte Carlo sampling, which is computationally very demanding and impractical for large biochemical reaction systems. Computationally efficient algorithms are needed to make variance-based sensitivity analysis applicable to realistic cellular networks, modeled by biochemical reaction systems that consist of a large number of reactions and molecular species.ResultsWe present four techniques, derivative approximation (DA), polynomial approximation (PA), Gauss-Hermite integration (GHI), and orthonormal Hermite approximation (OHA), for analytically approximating the variance-based sensitivity indices associated with a biochemical reaction system. By using a well-known model of the mitogen-activated protein kinase signaling cascade as a case study, we numerically compare the approximation quality of these techniques against traditional Monte Carlo sampling. Our results indicate that, although DA is computationally the most attractive technique, special care should be exercised when using it for sensitivity analysis, since it may only be accurate at low levels of uncertainty. On the other hand, PA, GHI, and OHA are computationally more demanding than DA but can work well at high levels of uncertainty. GHI results in a slightly better accuracy than PA, but it is more difficult to implement. OHA produces the most accurate approximation results and can be implemented in a straightforward manner. It turns out that the computational cost of the four approximation techniques considered in this paper is orders of magnitude smaller than traditional Monte Carlo estimation. Software, coded in MATLAB®, which implements all sensitivity analysis techniques discussed in this paper, is available free of charge.ConclusionsEstimating variance-based sensitivity indices of a large biochemical reaction system is a computationally challenging task that can only be addressed via approximations. Among the methods presented in this paper, a technique based on orthonormal Hermite polynomials seems to be an acceptable candidate for the job, producing very good approximation results for a wide range of uncertainty levels in a fraction of the time required by traditional Monte Carlo sampling.

Techniques for assessing the effects of uncertainties in thermodynamic models and data

Thermodynamic models and experimental data exhibit the usual systematic and random errors. The severity of their errors depends on their use, such as for process calculations in a process simulator. Similarly, the value of better thermodynamic models and/or data should be measured with reference to such use. We have developed techniques for quantification of such thermodynamic-induced process uncertainties via Monte Carlo simulation, regression analyses, and analogies to optimization. The influence of experimental data sources and data types on the uncertainty of thermodynamic models is studied. Details and applications of our new sampling strategy (EPS), which accounts for the high degree of correlation between thermodynamics model parameters, is given. This procedure directly uses the regression results in a way that is much more powerful and mathematically accurate than traditional covariance matrix techniques. Level sets are used for the Monte Carlo samples so that unbiased accurate sampling of the entire feasible region is obtained. Comparison with traditional Monte Carlo sampling, Latin Hypercube sampling (LHS), and Shifted Hammersley sampling (SHS) are shown. The result is an unbiased estimate of uncertainties that reduces the over- and under-estimations common in traditional techniques. The approaches presented can be used for safety-factor/risk analysis, guidelines for simulator use, experimental design, and model comparisons. They allow determinations of the value of obtaining additional phase-equilibrium data and the potential value of improved phase-equilibrium models. Examples and case studies of these applications are provided.