Based on the maximum entropy principle a new probability density function (PDF) f(x) for the surface elevation of nonlinear sea waves, X, is derived through performing a coordinate transform of X and solving a var...Based on the maximum entropy principle a new probability density function (PDF) f(x) for the surface elevation of nonlinear sea waves, X, is derived through performing a coordinate transform of X and solving a variation problem subject to three constraint conditions of f( x ). Compared with the maximum entropy PDFs presented previously, the new PDF has the following merits: (1) it has four parameters to be determined and hence can give more refined fit to observed data and has wider suitability for nonlinear waves in different conditions; (2) these parameters are expressed in terms of distribution moments of X in a relatively simple form and hence are easy to be determined from observed data; (3) the PDF is free of the restriction of weak nonlinearity and possible to be used for sea waves in complicated conditions, such as those in shallow waters with complicated topography; and (4) the PDF is simple in form and hence convenient for theoretical and practical uses. l.aboratory wind-wave experiments have been conducted to test the competence of the new PDF for the surface elevation of nonlinear waves. The experimental results manifest that the new PDF gives somewhat better fit to the laboratory wind-wave data than the well-known Gram-Charlier PDF and beta PDF.展开更多
The process of entrainment-mixing between cumulus clouds and the ambient air is important for the development of cumulus clouds.Accurately obtaining the entrainment rate(λ)is particularly important for its parameteri...The process of entrainment-mixing between cumulus clouds and the ambient air is important for the development of cumulus clouds.Accurately obtaining the entrainment rate(λ)is particularly important for its parameterization within the overall cumulus parameterization scheme.In this study,an improved bulk-plume method is proposed by solving the equations of two conserved variables simultaneously to calculateλof cumulus clouds in a large-eddy simulation.The results demonstrate that the improved bulk-plume method is more reliable than the traditional bulk-plume method,becauseλ,as calculated from the improved method,falls within the range ofλvalues obtained from the traditional method using different conserved variables.The probability density functions ofλfor all data,different times,and different heights can be well-fitted by a log-normal distribution,which supports the assumed stochastic entrainment process in previous studies.Further analysis demonstrate that the relationship betweenλand the vertical velocity is better than other thermodynamic/dynamical properties;thus,the vertical velocity is recommended as the primary influencing factor for the parameterization ofλin the future.The results of this study enhance the theoretical understanding ofλand its influencing factors and shed new light on the development ofλparameterization.展开更多
This paper defines a new kind of rule, probability functional dependency rule. The functional dependency degree can be depicted by this kind of rule. Five algorithms, from the simple to the complex, are presefited to ...This paper defines a new kind of rule, probability functional dependency rule. The functional dependency degree can be depicted by this kind of rule. Five algorithms, from the simple to the complex, are presefited to mine this kind of rule in different condition. The related theorems are proved to ensure the high efficiency and the correctness of the above algorithms.展开更多
A novel strategy of probability density function (PDF) shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimiza...A novel strategy of probability density function (PDF) shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimization algorithm. The parameters of the control er are viewed as the space position of a particle in particle swarm optimization algorithm and updated continual y until the control er makes the PDF of the state variable as close as possible to the expected PDF. The proposed PDF shape control technique is compared with the equivalent linearization technique through simulation experiments. The results show the superiority and the effectiveness of the proposed method. The control er is excellent in making the state PDF fol ow the expected PDF and has the very smal error between the state PDF and the expected PDF, solving the control problem of the PDF shape in stochastic systems effectively.展开更多
Probability density function (PDF) method is proposed for analysing the structure of the reconstructed attractor in computing the correlation dimensions of RR intervals of ten normal old men. PDF contains important in...Probability density function (PDF) method is proposed for analysing the structure of the reconstructed attractor in computing the correlation dimensions of RR intervals of ten normal old men. PDF contains important information about the spatial distribution of the phase points in the reconstructed attractor. To the best of our knowledge, it is the first time that the PDF method is put forward for the analysis of the reconstructed attractor structure. Numerical simulations demonstrate that the cardiac systems of healthy old men are about 6-6.5 dimensional complex dynamical systems. It is found that PDF is not symmetrically distributed when time delay is small, while PDF satisfies Gaussian distribution when time delay is big enough. A cluster effect mechanism is presented to explain this phenomenon. By studying the shape of PDFs, that the roles played by time delay are more important than embedding dimension in the reconstruction is clearly indicated. Results have demonstrated that the PDF method represents a promising numerical approach for the observation of the reconstructed attractor structure and may provide more information and new diagnostic potential of the analyzed cardiac system.展开更多
By large eddy simulation (LES), turbulent databases of channel flows at different Reynolds numbers were established. Then, the probability distribution functions of the streamwise and wall-normal velocity fluctuatio...By large eddy simulation (LES), turbulent databases of channel flows at different Reynolds numbers were established. Then, the probability distribution functions of the streamwise and wall-normal velocity fluctuations were obtained and compared with the corresponding normal distributions. By hypothesis test, the deviation from the normal distribution was analyzed quantitatively. The skewness and flatness factors were also calculated. And the variations of these two factors in the viscous sublayer, buffer layer and log-law layer were discussed. Still illustrated were the relations between the probability distribution functions and the burst events-sweep of high-speed fluids and ejection of low-speed fluidsIin the viscous sub-layer, buffer layer and loglaw layer. Finally the variations of the probability distribution functions with Reynolds number were examined.展开更多
The joint probability density fimction (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution ...The joint probability density fimction (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a flame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.展开更多
This paper presents a linearized approach for the controller design of the shape of output probability density functions for general stochastic systems. A square root approximation to an output probability density fun...This paper presents a linearized approach for the controller design of the shape of output probability density functions for general stochastic systems. A square root approximation to an output probability density function is realized by a set of B-spline functions. This generally produces a nonlinear state space model for the weights of the B-spline approximation. A linearized model is therefore obtained and embedded into a performance function that measures the tracking error of the output probability density function with respect to a given distribution. By using this performance function as a Lyapunov function for the closed loop system, a feedback control input has been obtained which guarantees closed loop stability and realizes perfect tracking. The algorithm described in this paper has been tested on a simulated example and desired results have been achieved.展开更多
Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equ...Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equations numerically for homoclinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations.The transient joint probability density functions(PDFs)and marginal PDFs of the rolling responses are also obtained.The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint.The homoclinic ship rolling has two rolling states,the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large,and the PDF is remarkably distributed in phase space.These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling,and the ship may lose stability due to unforeseeable facts or conditions.Meanwhile,only one rolling state is observed when the ship is in heteroclinic rolling.As the periodic excitation amplitude grows,the PDF concentration increases and drifts away from the beginning location,suggesting that the ship rolling substantially changes in a cycle and its stability is low.The PDF becomes increasingly uniform and covers a large region as the noise intensity increases,reducing the certainty of ship rolling and navigation safety.The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms.展开更多
The probability distribution function (PDF) of a passive tracer, forced by a "mean gradient", is studied. First, we take two theoretical approaches, the Lagrangian and the conditional closure formalisms, to study ...The probability distribution function (PDF) of a passive tracer, forced by a "mean gradient", is studied. First, we take two theoretical approaches, the Lagrangian and the conditional closure formalisms, to study the PDFs of such an externally forced passive tracer. Then, we carry out numerical simulations for an idealized random flow on a sphere and for European Center for Medium-Range Weather Forecasts (ECMWF) stratospheric winds to test whether the mean-gradient model can be applied to studying stratospheric tracer mixing in midlatitude surf zones, in which a weak and poleward zonal-mean gradient is maintained by tracer leakage through polar and tropical mixing barriers, and whether the PDFs of tracer fluctuations in midlatitudes are consistent with the theoretical predictions. The numerical simulations show that when diffusive dissipation is balanced by the mean-gradient forcing, the PDF in the random flow and the Southern-Hemisphere PDFs in ECMWF winds show time-invariant exponential tails, consistent with theoretical predictions. In the Northern Hemisphere, the PDFs exhibit non-Gaussian tails. However, the PDF tails are not consistent with theoretical expectations. The long-term behavior of the PDF tails of the forced tracer is compared to that of a decaying tracer. It is found that the PDF tails of the decaying tracer are time-dependent, and evolve toward flatter than exponential.展开更多
We investigate the possibility for two-mode probability density function (PDF) to have a non-zero flux steady state solution. We take the large volume limit so that the space of modes becomes continuous. It is shown...We investigate the possibility for two-mode probability density function (PDF) to have a non-zero flux steady state solution. We take the large volume limit so that the space of modes becomes continuous. It is shown that in this limit all the steady-state twoor higher-mode PDFs are the product of one-mode PDFs. The flux of this steady-state solution turns out to be zero for any finite mode PDF.展开更多
The principal resonance of Duffing random external excitation was investigated. oscillator to combined deterministic and The random excitation was taken to be white noise or harmonic with separable random amplitude an...The principal resonance of Duffing random external excitation was investigated. oscillator to combined deterministic and The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.展开更多
Turbulent gas-particle flows are studied by a kinetic description using a prob- ability density function (PDF). Unlike other investigators deriving the particle Reynolds stress equations using the PDF equations, the...Turbulent gas-particle flows are studied by a kinetic description using a prob- ability density function (PDF). Unlike other investigators deriving the particle Reynolds stress equations using the PDF equations, the particle PDF transport equations are di- rectly solved either using a finite-difference method for two-dimensional (2D) problems or using a Monte-Carlo (MC) method for three-dimensional (3D) problems. The proposed differential stress model together with the PDF (DSM-PDF) is used to simulate turbulent swirling gas-particle flows. The simulation results are compared with the experimental results and the second-order moment (SOM) two-phase modeling results. All of these simulation results are in agreement with the experimental results, implying that the PDF approach validates the SOM two-phase turbulence modeling. The PDF model with the SOM-MC method is used to simulate evaporating gas-droplet flows, and the simulation results are in good agreement with the experimental results.展开更多
We propose a new flame index for the transported probability density function(PDF) method. The flame index uses mixing flux projections of Lagrangian particles on mixture fraction and progress variable directions as t...We propose a new flame index for the transported probability density function(PDF) method. The flame index uses mixing flux projections of Lagrangian particles on mixture fraction and progress variable directions as the metrics to identify the combustion mode, with the Burke-Schumann solution as a reference. A priori validation of the flame index is conducted with a series of constructed turbulent partially premixed reactors. It indicates that the proposed flame index is able to identify the combustion mode based on the subgrid mixing information. The flame index is then applied the large eddy simulation/PDF datasets of turbulent partially premixed jet flames. Results show that the flame index separate different combustion modes and extinction correctly. The proposed flame index provides a promising tool to analyze and model the partially premixed flames adaptively.展开更多
A new identification method of neuro-uzzy Hammerstein model based on probability density function(PDF) is presented,which is different from the idea that mean squared error(MSE) is employed as the index function in tr...A new identification method of neuro-uzzy Hammerstein model based on probability density function(PDF) is presented,which is different from the idea that mean squared error(MSE) is employed as the index function in traditional identification methods.Firstly,a neuro-fuzzy based Hammerstein model is constructed to describe the nonlinearity of Hammerstein process without any prior process knowledge.Secondly,a kind of special test signal is used to separate the link parts of the Hammerstein model.More specifically,the conception of PDF is introduced to solve the identification problem of the neuro-fuzzy Hammerstein model.The antecedent parameters are estimated by a clustering algorithm,while the consequent parameters of the model are identified by designing a virtual PDF control system in which the PDF of the modeling error is estimated and controlled to converge to the target.The proposed method not only guarantees the accuracy of the model but also dominates the spatial distribution of PDF of the model error to improve the generalization ability of the model.Simulated results show the effectiveness of the proposed method.展开更多
The probability distributions of small sample data are difficult to determine,while a large proportion of samples occur in the early failure period,so it is particularly important to make full use of these data in the...The probability distributions of small sample data are difficult to determine,while a large proportion of samples occur in the early failure period,so it is particularly important to make full use of these data in the statistical analysis.Based on gamma distribution,four methods of probability density function(PDF)reconstruction with early failure data are proposed,and then the mean time between failures(MTBF)evaluation expressions are concluded from the reconstructed PDFs.Both theory analysis and an example show that method 2 is the best evaluation method in dealing with early-failure-small-sample data.The reconstruction methods of PDF also have certain guiding significance for other distribution types.展开更多
Based on probability density functions,we present a theoretical model to explain filtered ghost imaging(FGI)we first proposed and experimentally demonstrated in 2017[Opt.Lett.425290(2017)].An analytic expression for t...Based on probability density functions,we present a theoretical model to explain filtered ghost imaging(FGI)we first proposed and experimentally demonstrated in 2017[Opt.Lett.425290(2017)].An analytic expression for the joint intensity probability density functions of filtered random speckle fields is derived according to their probability distributions.Moreover,the normalized second-order intensity correlation functions are calculated for the three cases of low-pass,bandpass and high-pass filterings to study the resolution and visibility in the FGI system.Numerical simulations show that the resolution and visibility predicted by our model agree well with the experimental results,which also explains why FGI can achieve a super-resolution image and better visibility than traditional ghost imaging.展开更多
The self-excited second harmonic in radio-frequency capacitively coupled plasma was significantly enhanced by adjusting the external variable capacitor.At a lower pressure of 3 Pa,the excitation of the second harmonic...The self-excited second harmonic in radio-frequency capacitively coupled plasma was significantly enhanced by adjusting the external variable capacitor.At a lower pressure of 3 Pa,the excitation of the second harmonic caused an abnormal transition of the electron energy probability function,resulting in abrupt changes in the electron density and temperature.Such changes in the electron energy probability function as well as the electron density and temperature were not observed at the higher pressure of 16 Pa under similar harmonic changes.The phenomena are related to the influence of the second harmonic on stochastic heating,which is determined by both amplitude and the relative phase of the harmonics.The results suggest that the self-excited high-order harmonics must be considered in practical applications of lowpressure radio-frequency capacitively coupled plasmas.展开更多
The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macrosc...The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation(MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional(2D), six-dimensional(6D), and eight-dimensional(8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints.展开更多
The joint probability distribution of wind speed and significant wave height in the Bohai Bay was investigated by comparing the Gurnbel logistic model, the Gumbel-Hougaard (GH) copula function, and the Clayton copul...The joint probability distribution of wind speed and significant wave height in the Bohai Bay was investigated by comparing the Gurnbel logistic model, the Gumbel-Hougaard (GH) copula function, and the Clayton copula function. Twenty years of wind data from 1989 to 2008 were collected from the European Centre for Medium-Range Weather Forecasts (ECMWF) database and the blended wind data of the Quick Scatterometer (QSCAT) satellite data set and re-analysis data from the United States National Centers for Environmental Prediction (NCEP). Several typhoons were taken into account and merged with the background wind fields from the ECMWF or QSCAT/NCEP database. The 20-year data of significant wave height were calculated with the unstructured-grid version of the third-generation wind wave model Simulating WAves Nearshore (SWAN) under extreme wind process conditions. The Gumbel distribution was used for univariate and marginal distributions. The distribution parameters were estimated with the method of L-moments. Based on the marginal distributions, the joint probability distributions, the associated return periods, and the conditional probability distributions were obtained. The GH copula function was found to be optimal according to the ordinary least squares (OLS) test. The results show that wind waves are the prevailing type of wave in the Bohai Bay.展开更多
基金This workis financially supported by the National Natural Science Foundation of China (Grant No.40490263 andNo.40276006)
文摘Based on the maximum entropy principle a new probability density function (PDF) f(x) for the surface elevation of nonlinear sea waves, X, is derived through performing a coordinate transform of X and solving a variation problem subject to three constraint conditions of f( x ). Compared with the maximum entropy PDFs presented previously, the new PDF has the following merits: (1) it has four parameters to be determined and hence can give more refined fit to observed data and has wider suitability for nonlinear waves in different conditions; (2) these parameters are expressed in terms of distribution moments of X in a relatively simple form and hence are easy to be determined from observed data; (3) the PDF is free of the restriction of weak nonlinearity and possible to be used for sea waves in complicated conditions, such as those in shallow waters with complicated topography; and (4) the PDF is simple in form and hence convenient for theoretical and practical uses. l.aboratory wind-wave experiments have been conducted to test the competence of the new PDF for the surface elevation of nonlinear waves. The experimental results manifest that the new PDF gives somewhat better fit to the laboratory wind-wave data than the well-known Gram-Charlier PDF and beta PDF.
基金supported by the National Natural Science Foundation of China(Grant Nos.42175099,42027804,42075073)the Innovative Project of Postgraduates in Jiangsu Province in 2023(Grant No.KYCX23_1319)+3 种基金supported by the National Natural Science Foundation of China(Grant No.42205080)the Natural Science Foundation of Sichuan(Grant No.2023YFS0442)the Research Fund of Civil Aviation Flight University of China(Grant No.J2022-037)supported by the National Key Scientific and Technological Infrastructure project“Earth System Science Numerical Simulator Facility”(Earth Lab)。
文摘The process of entrainment-mixing between cumulus clouds and the ambient air is important for the development of cumulus clouds.Accurately obtaining the entrainment rate(λ)is particularly important for its parameterization within the overall cumulus parameterization scheme.In this study,an improved bulk-plume method is proposed by solving the equations of two conserved variables simultaneously to calculateλof cumulus clouds in a large-eddy simulation.The results demonstrate that the improved bulk-plume method is more reliable than the traditional bulk-plume method,becauseλ,as calculated from the improved method,falls within the range ofλvalues obtained from the traditional method using different conserved variables.The probability density functions ofλfor all data,different times,and different heights can be well-fitted by a log-normal distribution,which supports the assumed stochastic entrainment process in previous studies.Further analysis demonstrate that the relationship betweenλand the vertical velocity is better than other thermodynamic/dynamical properties;thus,the vertical velocity is recommended as the primary influencing factor for the parameterization ofλin the future.The results of this study enhance the theoretical understanding ofλand its influencing factors and shed new light on the development ofλparameterization.
文摘This paper defines a new kind of rule, probability functional dependency rule. The functional dependency degree can be depicted by this kind of rule. Five algorithms, from the simple to the complex, are presefited to mine this kind of rule in different condition. The related theorems are proved to ensure the high efficiency and the correctness of the above algorithms.
基金supported by the National Natural Science Fundation of China(61273127)the Specialized Research Fund of the Doctoral Program in Higher Education(20106118110009+2 种基金20116118110008)the Scientific Research Plan Projects of Shaanxi Education Department(12JK0524)the Young Teachers Scientific Research Fund of Xi’an University of Posts and Telecommunications(1100434)
文摘A novel strategy of probability density function (PDF) shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimization algorithm. The parameters of the control er are viewed as the space position of a particle in particle swarm optimization algorithm and updated continual y until the control er makes the PDF of the state variable as close as possible to the expected PDF. The proposed PDF shape control technique is compared with the equivalent linearization technique through simulation experiments. The results show the superiority and the effectiveness of the proposed method. The control er is excellent in making the state PDF fol ow the expected PDF and has the very smal error between the state PDF and the expected PDF, solving the control problem of the PDF shape in stochastic systems effectively.
文摘Probability density function (PDF) method is proposed for analysing the structure of the reconstructed attractor in computing the correlation dimensions of RR intervals of ten normal old men. PDF contains important information about the spatial distribution of the phase points in the reconstructed attractor. To the best of our knowledge, it is the first time that the PDF method is put forward for the analysis of the reconstructed attractor structure. Numerical simulations demonstrate that the cardiac systems of healthy old men are about 6-6.5 dimensional complex dynamical systems. It is found that PDF is not symmetrically distributed when time delay is small, while PDF satisfies Gaussian distribution when time delay is big enough. A cluster effect mechanism is presented to explain this phenomenon. By studying the shape of PDFs, that the roles played by time delay are more important than embedding dimension in the reconstruction is clearly indicated. Results have demonstrated that the PDF method represents a promising numerical approach for the observation of the reconstructed attractor structure and may provide more information and new diagnostic potential of the analyzed cardiac system.
文摘By large eddy simulation (LES), turbulent databases of channel flows at different Reynolds numbers were established. Then, the probability distribution functions of the streamwise and wall-normal velocity fluctuations were obtained and compared with the corresponding normal distributions. By hypothesis test, the deviation from the normal distribution was analyzed quantitatively. The skewness and flatness factors were also calculated. And the variations of these two factors in the viscous sublayer, buffer layer and log-law layer were discussed. Still illustrated were the relations between the probability distribution functions and the burst events-sweep of high-speed fluids and ejection of low-speed fluidsIin the viscous sub-layer, buffer layer and loglaw layer. Finally the variations of the probability distribution functions with Reynolds number were examined.
基金the National Natural Science Foundation of Chinafor Innovative Research Groups Under Grant No.50621062the National Natural Science Foundation of China forYoung Scholars Under Grant No.10402030
文摘The joint probability density fimction (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a flame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.
文摘This paper presents a linearized approach for the controller design of the shape of output probability density functions for general stochastic systems. A square root approximation to an output probability density function is realized by a set of B-spline functions. This generally produces a nonlinear state space model for the weights of the B-spline approximation. A linearized model is therefore obtained and embedded into a performance function that measures the tracking error of the output probability density function with respect to a given distribution. By using this performance function as a Lyapunov function for the closed loop system, a feedback control input has been obtained which guarantees closed loop stability and realizes perfect tracking. The algorithm described in this paper has been tested on a simulated example and desired results have been achieved.
基金the National Natural Science Foundation of China(Nos.52088102,51875540)。
文摘Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equations numerically for homoclinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations.The transient joint probability density functions(PDFs)and marginal PDFs of the rolling responses are also obtained.The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint.The homoclinic ship rolling has two rolling states,the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large,and the PDF is remarkably distributed in phase space.These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling,and the ship may lose stability due to unforeseeable facts or conditions.Meanwhile,only one rolling state is observed when the ship is in heteroclinic rolling.As the periodic excitation amplitude grows,the PDF concentration increases and drifts away from the beginning location,suggesting that the ship rolling substantially changes in a cycle and its stability is low.The PDF becomes increasingly uniform and covers a large region as the noise intensity increases,reducing the certainty of ship rolling and navigation safety.The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms.
基金This work is supported by the National Natural Science Foundation of China (NSFC) under Grants Nos. 40575031 and 40533016by the Ministry of Education of China under Grant No. 106002.
文摘The probability distribution function (PDF) of a passive tracer, forced by a "mean gradient", is studied. First, we take two theoretical approaches, the Lagrangian and the conditional closure formalisms, to study the PDFs of such an externally forced passive tracer. Then, we carry out numerical simulations for an idealized random flow on a sphere and for European Center for Medium-Range Weather Forecasts (ECMWF) stratospheric winds to test whether the mean-gradient model can be applied to studying stratospheric tracer mixing in midlatitude surf zones, in which a weak and poleward zonal-mean gradient is maintained by tracer leakage through polar and tropical mixing barriers, and whether the PDFs of tracer fluctuations in midlatitudes are consistent with the theoretical predictions. The numerical simulations show that when diffusive dissipation is balanced by the mean-gradient forcing, the PDF in the random flow and the Southern-Hemisphere PDFs in ECMWF winds show time-invariant exponential tails, consistent with theoretical predictions. In the Northern Hemisphere, the PDFs exhibit non-Gaussian tails. However, the PDF tails are not consistent with theoretical expectations. The long-term behavior of the PDF tails of the forced tracer is compared to that of a decaying tracer. It is found that the PDF tails of the decaying tracer are time-dependent, and evolve toward flatter than exponential.
基金Project supported by the Korean Research Foundation of the Korea Government (MEST) (Grant No. 2009-0073081)
文摘We investigate the possibility for two-mode probability density function (PDF) to have a non-zero flux steady state solution. We take the large volume limit so that the space of modes becomes continuous. It is shown that in this limit all the steady-state twoor higher-mode PDFs are the product of one-mode PDFs. The flux of this steady-state solution turns out to be zero for any finite mode PDF.
基金Project supported by the National Natural Science Foundation of China (Key Program) (No.10332030)the Natural Science Foundation of Guangdong Province of China (No.04011640)
文摘The principal resonance of Duffing random external excitation was investigated. oscillator to combined deterministic and The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.
基金supported by the National Natural Science Foundation of China(No.51390493)
文摘Turbulent gas-particle flows are studied by a kinetic description using a prob- ability density function (PDF). Unlike other investigators deriving the particle Reynolds stress equations using the PDF equations, the particle PDF transport equations are di- rectly solved either using a finite-difference method for two-dimensional (2D) problems or using a Monte-Carlo (MC) method for three-dimensional (3D) problems. The proposed differential stress model together with the PDF (DSM-PDF) is used to simulate turbulent swirling gas-particle flows. The simulation results are compared with the experimental results and the second-order moment (SOM) two-phase modeling results. All of these simulation results are in agreement with the experimental results, implying that the PDF approach validates the SOM two-phase turbulence modeling. The PDF model with the SOM-MC method is used to simulate evaporating gas-droplet flows, and the simulation results are in good agreement with the experimental results.
基金sponsored by King Abdullah University of Science and Technology(KAUST)the National Natural Science Foundation of China(Grant No.91841302)。
文摘We propose a new flame index for the transported probability density function(PDF) method. The flame index uses mixing flux projections of Lagrangian particles on mixture fraction and progress variable directions as the metrics to identify the combustion mode, with the Burke-Schumann solution as a reference. A priori validation of the flame index is conducted with a series of constructed turbulent partially premixed reactors. It indicates that the proposed flame index is able to identify the combustion mode based on the subgrid mixing information. The flame index is then applied the large eddy simulation/PDF datasets of turbulent partially premixed jet flames. Results show that the flame index separate different combustion modes and extinction correctly. The proposed flame index provides a promising tool to analyze and model the partially premixed flames adaptively.
基金National Natural Science Foundation of China(No.61374044)Shanghai Municipal Science and Technology Commission,China(No.15510722100)+2 种基金Shanghai Municipal Education Commission,China(No.14ZZ088)Shanghai Talent Development Plan,ChinaShanghai Baoshan Science and Technology Commission,China(No.bkw2013120)
文摘A new identification method of neuro-uzzy Hammerstein model based on probability density function(PDF) is presented,which is different from the idea that mean squared error(MSE) is employed as the index function in traditional identification methods.Firstly,a neuro-fuzzy based Hammerstein model is constructed to describe the nonlinearity of Hammerstein process without any prior process knowledge.Secondly,a kind of special test signal is used to separate the link parts of the Hammerstein model.More specifically,the conception of PDF is introduced to solve the identification problem of the neuro-fuzzy Hammerstein model.The antecedent parameters are estimated by a clustering algorithm,while the consequent parameters of the model are identified by designing a virtual PDF control system in which the PDF of the modeling error is estimated and controlled to converge to the target.The proposed method not only guarantees the accuracy of the model but also dominates the spatial distribution of PDF of the model error to improve the generalization ability of the model.Simulated results show the effectiveness of the proposed method.
基金National Science and Technology Major Project of China(No.2016ZX04003001)。
文摘The probability distributions of small sample data are difficult to determine,while a large proportion of samples occur in the early failure period,so it is particularly important to make full use of these data in the statistical analysis.Based on gamma distribution,four methods of probability density function(PDF)reconstruction with early failure data are proposed,and then the mean time between failures(MTBF)evaluation expressions are concluded from the reconstructed PDFs.Both theory analysis and an example show that method 2 is the best evaluation method in dealing with early-failure-small-sample data.The reconstruction methods of PDF also have certain guiding significance for other distribution types.
基金Project supported by the National Key Research and Development Program of China(Grant No.2018YFB0504302)the Project of Innovation and Entrepreneurship Training Program for college students of Liaoning University(Grant No.S202110140003)。
文摘Based on probability density functions,we present a theoretical model to explain filtered ghost imaging(FGI)we first proposed and experimentally demonstrated in 2017[Opt.Lett.425290(2017)].An analytic expression for the joint intensity probability density functions of filtered random speckle fields is derived according to their probability distributions.Moreover,the normalized second-order intensity correlation functions are calculated for the three cases of low-pass,bandpass and high-pass filterings to study the resolution and visibility in the FGI system.Numerical simulations show that the resolution and visibility predicted by our model agree well with the experimental results,which also explains why FGI can achieve a super-resolution image and better visibility than traditional ghost imaging.
文摘The self-excited second harmonic in radio-frequency capacitively coupled plasma was significantly enhanced by adjusting the external variable capacitor.At a lower pressure of 3 Pa,the excitation of the second harmonic caused an abnormal transition of the electron energy probability function,resulting in abrupt changes in the electron density and temperature.Such changes in the electron energy probability function as well as the electron density and temperature were not observed at the higher pressure of 16 Pa under similar harmonic changes.The phenomena are related to the influence of the second harmonic on stochastic heating,which is determined by both amplitude and the relative phase of the harmonics.The results suggest that the self-excited high-order harmonics must be considered in practical applications of lowpressure radio-frequency capacitively coupled plasmas.
基金Project supported by the National Natural Science Foundation of China (Grant No.12172226)。
文摘The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation(MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional(2D), six-dimensional(6D), and eight-dimensional(8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints.
基金supported by the Science Fund for Creative Research Groups of the National Natural ScienceFoundation of China (Grant No. 51021004)the National High Technology Research and DevelopmentProgram of China (863 Program, Grants No. 2012AA112509 and 2012AA051702)
文摘The joint probability distribution of wind speed and significant wave height in the Bohai Bay was investigated by comparing the Gurnbel logistic model, the Gumbel-Hougaard (GH) copula function, and the Clayton copula function. Twenty years of wind data from 1989 to 2008 were collected from the European Centre for Medium-Range Weather Forecasts (ECMWF) database and the blended wind data of the Quick Scatterometer (QSCAT) satellite data set and re-analysis data from the United States National Centers for Environmental Prediction (NCEP). Several typhoons were taken into account and merged with the background wind fields from the ECMWF or QSCAT/NCEP database. The 20-year data of significant wave height were calculated with the unstructured-grid version of the third-generation wind wave model Simulating WAves Nearshore (SWAN) under extreme wind process conditions. The Gumbel distribution was used for univariate and marginal distributions. The distribution parameters were estimated with the method of L-moments. Based on the marginal distributions, the joint probability distributions, the associated return periods, and the conditional probability distributions were obtained. The GH copula function was found to be optimal according to the ordinary least squares (OLS) test. The results show that wind waves are the prevailing type of wave in the Bohai Bay.