Multimodal Distribution Function Research Articles

Predicting the stochastic behavior of uncertainty sources in planning a stand-alone renewable energy-based microgrid using Metropolis–coupled Markov chain Monte Carlo simulation

Due to the lack of available flexibility sources to cope with different uncertainties in the real-time operation of stand-alone renewable energy-based microgrids, the stochastic behavior of uncertainty sources needs to be included in the planning stage. Since there is a high association between some of the uncertainty sources, defining a proper time series to represent the behavior of each source of uncertainty is a challenging issue. Consequently, uncertainty sources should be modeled in such a way that the designed microgrid be able to cope with all scenarios from probability and impact viewpoints. This paper proposes a modified Metropolis–coupled Markov chain Monte Carlo (MC)3 simulation to predict the stochastic behavior of different uncertainty sources in the planning of a stand-alone renewable energy-based microgrid. Solar radiation, wind speed, the water flow of a river, load consumption, and electricity price have been considered as primary sources of uncertainty. A novel data classification method is introduced within the (MC)3 simulation to model the time-dependency and the association between different uncertainty sources. Moreover, a novel curve-fitting approach is proposed to improve the accuracy of representing the multimodal distribution functions, modeling the Markov chain states, and the long-term probability of uncertainty sources. The predicted representative time series with the proposed modified (MC)3 model is benchmarked against the retrospective model, the long-term historical data, and the simple Monte Carlo simulation model to capture the stochastic behavior of uncertainty sources. The results show that the proposed model represents the probability distribution function of each source of uncertainty, the continuity of samples, time dependency, the association between different uncertainty sources, short-term and long-term trends, and the seasonality of uncertainty sources. Finally, results confirm that the proposed modified (MC)3 can appropriately predict all scenarios with high probability and impact.

A meshless Radial Basis Method (RBM) for solving the detailed population balance equation

The Radial Basis Method (RBM) is introduced for solving the Population Balance Equation (PBE). In this method, the continuous number density function is approximated by a series of symmetric multiquadric basis functions at selected global collocation points along the internal coordinates. Therefore, the RBM is considered as a meshless method. The RBM is convenient and suitable for complex problems involving nucleation, growth, breakage and aggregation phenomena. The method provides naturally a continuously differentiable, or “smooth”, solution and avoids solving ill-conditioned problems. The numerical solutions are validated analytically and numerically using different case-studies for uniform stirred-tank. For these examples, 20 collocation points were found to be enough to achieve accurate predictions. The RBM ability to predict multimodal distribution functions and moving steep fronts is moreover highlighted. The proposed method proves to be very efficient for solving the detailed PBE in terms of accuracy, stability and CPU time requirements.

Large-scale round-trip delay time analysis of IPv4 hosts around the globe

Design and optimization of many network applications, services, protocols, and routing protocols can be improved with delay-related measurement for a better operation over the Internet. Many experimental delay measurements have been performed on predetermined end-to-end connections with a less number of hosts compared to our study. This study aims to investigate up-to-date round-trip delay time measurement results over the Internet through pinging random IPv4 addresses from three vantage points located in the United States, Turkey, and Japan. Considering different time periods in a day and in consecutive 5 years, we performed a large-scale round-trip delay time analysis study by sending more than 300 million ICMP requests to randomly chosen IPv4 addresses. Approximately, 55 million unique Internet hosts replied to ICMP requests and were evaluated for the analysis. The results show that 90\% of IP hosts accomplish their ICMP communication in less than 0.4 s. Mostly the propagation time on backbone links constitutes the larger part of total round-trip delay time. Distribution fitting test results demonstrate that RTTs of distributed hosts around the world could be modeled with multimodal distribution functions. Wakeby distribution function gives best results for modeling RTTs with two different modes according to the Kolmogorov--Simirnov test statistics. Our study also gives perspective about how packet delay values would be, when a message is broadcasted all over the world. Another significant finding is that it gives a point of view where to locate servers to provide a fast Internet service all over the world via Internet.

Integrating MT-DREAMzs and nested sampling algorithms to estimate marginal likelihood and comparison with several other methods

Marginal likelihood (or Bayesian evidence) estimation is the basis for model evaluation and model selection, which is the key research content of the uncertainty quantification of groundwater model structure. Recently, the nested sampling estimator (NSE) has been widely used as an efficient method to estimate the marginal likelihood. The Metropolis-Hasting (M-H) algorithm is usually used to explore the model likelihood space for NSE. However, M-H is inefficient in searching complex likelihood functions, especially for the multimodal distribution functions, because of the sampling structure of M-H algorithm. The multi-try differential evolution adaptive Metropolis (MT-DREAMzs) algorithm is effective and robust in searching complex probability space, and it is incorporated into NSE to improve the performance of marginal likelihood estimation in this study. In addition, the developed MT-DREAMzs-based NSE (NSE_MT) is evaluated by comparing with several other marginal likelihood estimators including DREAMzs-based and M-H-based NSEs (NSE_DR, NSE_MH), Gaussian mixture importance (GAME) sampling and a new nested sampling algorithm (PolyChord). Based on three analytical functions (up to 100 dimensions) and one groundwater solute transport case studies, the results showed that NSE_MT outperforms NSE_MH in the convergence, efficiency and accuracy of marginal estimation, and NSE_MT is slightly more efficient and accurate than NSE_DR. The accuracy of NSE_MH, NSE_DR and NSE_MT deteriorates with increasing dimensionality of target distributions/model parameters. In addition, GAME sampling and PolyChord sampling outperform other three NSEs in the accuracy and stability of estimation for high dimensional (up to 100-variate) distributions, and PolyChord is slightly less accurate and stable than GAME. Compared with GAME and PolyChord, NSE_MT and NSE_DR are competent for less than 10-dimensional distributions with higher computational efficiency.

Sign problem and Monte Carlo calculations beyond Lefschetz thimbles

We point out that Monte Carlo simulations of theories with severe sign problems can be profitably performed over manifolds in complex space different from the one with fixed imaginary part of the action (“Lefschetz thimble”). We describe a family of such manifolds that interpolate between the tangent space at one critical point (where the sign problem is milder compared to the real plane but in some cases still severe) and the union of relevant thimbles (where the sign problem is mild but a multimodal distribution function complicates the Monte Carlo sampling). We exemplify this approach using a simple 0+1 dimensional fermion model previously used on sign problem studies and show that it can solve the model for some parameter values where a solution using Lefschetz thimbles was elusive.

Sign problem and Monte Carlo calculations beyond Lefschetz thimbles

We point out that Monte Carlo simulations of theories with severe sign problems can be profitably performed over manifolds in complex space different from the one with fixed imaginary part of the action. We describe a family of such manifolds that interpolate between the tangent space at one critical point, where the sign problem is milder compared to the real plane but in some cases still severe, and the union of relevant thimbles, where the sign problem is mild but a multimodal distribution function complicates the Monte Carlo sampling. We exemplify this approach using a simple 0 + 1 dimensional fermion model previously used on sign problem studies and show that it can solve the model for some parameter values where a solution using Lefshetz thimbles was elusive.

A Fast Field Cycling Nuclear Magnetic Resonance Relaxometry Study of Natural Soils

This study used nuclear magnetic resonance (NMR) relaxometry at different Larmor frequencies to investigate water dynamics in the pore space of natural porous media. Spin‐lattice NMR relaxation times (T1) were determined in purified fine sand and two natural soils, Kaldenkirchen sandy loam and Merzenhausen silt loam, by means of fast field cycling relaxometry. This technique investigates relaxation processes as a function of the Larmor frequency ν in the range between 0.005 and 20 MHz, yielding so‐called relaxation dispersion curves (1/T1vs. log ν). The data were further analyzed by means of inverse Laplace transformation to calculate theT1relaxation time distribution functions. Only the fine sand was characterized by monomodal distribution withT1of about 1 s at 20 MHz, whereas the natural soil samples showed multimodal distribution functions in the range between 2 and 70 ms. With decreasing Larmor frequency, all distribution functions kept their shapes but were shifted to faster relaxation times. The corresponding relaxation dispersion curves indicate predominance of two‐dimensional diffusion of water in the soils, whereas in the macroporous sand, diffusion behaved like unrestricted three‐dimensional diffusion. In terms of the Brownstein–Tarr model, the decrease in theT1relaxation times with increasing silt and clay content can be explained by an increase of the surface/volume ratios (S/V) of these porous media, i.e., by a decrease in the pore sizes. Finally, distribution functions of the pore size parameterV/Swere obtained from the spin‐lattice relaxation time distributions by normalizing on the specific surface area. They ranged from submicrometers in the Merzenhausen soil to micrometers and submillimeters in the Kaldenkirchen soil and fine sand, respectively.

On the multimodality of random probability measures

Nonparametric methods for density estimation are examined here. Within a Bayesian setting the construction of an absolutely continuous random probability measure is often required for nonparametric statistical analysis. To achieve this we propose a partial convexification procedure of a process, such as the Dirichlet, resulting in a multimodal distribution function with a finite expected number of modes. In agreement with convexity theory results, it is shown that the derived random probability measure admits a density with respect to Lebesgue measure.

Role of the surface heterogeneity in adsorption of hydrogen ions on metal oxides: Theory and simulations

In this article we study the effect of energetic heterogeneity of a crystalline surface on the adsorption of hydrogen ions (protons) from the liquid phase. In particular, we examine the influence of the shape of the adsorption energy distribution on the equilibrium isotherms of hydrogen ions. To that purpose, a few popular distribution functions, including rectangular, exponential, and asymmetric Gaussian are considered. Additionally, multimodal distribution functions, which may correspond to the adsorption on different crystal planes of the oxide, are also used. Lateral interactions between adsorbed charges are modeled using the potential function proposed by Borkovec et al., which accounts also for polarization of the liquid medium. The results presented here are obtained using both Monte Carlo (MC) simulations and theoretical calculations involving Mean Field Approximation (MFA). They indicate that increased energetic heterogeneity of the adsorbing surface may, in general, considerably change the behavior of the adsorption isotherms, regardless of the assumed distribution function. It is also shown that the predictions of the proposed theory are consistent with the data obtained from the MC simulations.

Impact crushing of concrete for liberation and recycling

A test rig (large-scale pneumatic cannon) has been designed, constructed and tested. This apparatus allows the adjustment of intensive stressing conditions, e.g., impact and double impact, single and multiple stressing, velocity and energy. The accelerating tube diameter limit was given by the condition of a representative feed sample (model concrete spheres 150 mm sample diameter limited by an upper particle size of aggregate d o=16 mm). The crushed fragments are described as subcollectives (truncated logarithmic normal distributions) of a multimodal distribution function. For impact stressing, liberation grades between 0.3 and 0.6 (0% adherent hardened cement paste) as well as more than 0.9 (0–50% adherent hardened cement paste) were obtained.

The role of the fracture diagonostic data on the estimation of parameters of weibull's multimodal distribution function.

The role of the fracture diagonostic data on the estimation of parameters of weibull's multimodal distribution function.

On multimodal distributions

As a sequel to khinchine's definition of unimodality a multimodal distribution function is defined. A characterization for such a distribution is given. The convolution of two such distributions is studied.