Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial pertur...Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial perturbation method tends only to capture synoptic scale initial uncertainty rather than mesoscale uncertainty in global ensemble prediction. To address this issue, a multiscale SV initial perturbation method based on the China Meteorological Administration Global Ensemble Prediction System(CMA-GEPS) is proposed to quantify multiscale initial uncertainty. The multiscale SV initial perturbation approach entails calculating multiscale SVs at different resolutions with multiple linearized physical processes to capture fast-growing perturbations from mesoscale to synoptic scale in target areas and combining these SVs by using a Gaussian sampling method with amplitude coefficients to generate initial perturbations. Following that, the energy norm,energy spectrum, and structure of multiscale SVs and their impact on GEPS are analyzed based on a batch experiment in different seasons. The results show that the multiscale SV initial perturbations can possess more energy and capture more mesoscale uncertainties than the traditional single-SV method. Meanwhile, multiscale SV initial perturbations can reflect the strongest dynamical instability in target areas. Their performances in global ensemble prediction when compared to single-scale SVs are shown to(i) improve the relationship between the ensemble spread and the root-mean-square error and(ii) provide a better probability forecast skill for atmospheric circulation during the late forecast period and for short-to medium-range precipitation. This study provides scientific evidence and application foundations for the design and development of a multiscale SV initial perturbation method for the GEPS.展开更多
The singular vector(SV)initial perturbation method can capture the fastest-growing initial perturbation in a tangent linear model(TLM).Based on the global tangent linear and adjoint model of GRAPES-GEPS(Global/Regiona...The singular vector(SV)initial perturbation method can capture the fastest-growing initial perturbation in a tangent linear model(TLM).Based on the global tangent linear and adjoint model of GRAPES-GEPS(Global/Regional Assimilation and Prediction System-Global Ensemble Prediction System),some experiments were carried out to analyze the structure of the moist SVs from the perspectives of the energy norm,energy spectrum,and vertical structure.The conclusions are as follows:The evolution of the SVs is synchronous with that of the atmospheric circulation,which is flowdependent.The moist and dry SVs are located in unstable regions at mid-to-high latitudes,but the moist SVs are wider,can contain more small-and medium-scale information,and have more energy than the dry SVs.From the energy spectrum analysis,the energy growth caused by the moist SVs is reflected in the relatively small-scale weather system.In addition,moist SVs can generate perturbations associated with large-scale condensation and precipitation,which is not true for dry SVs.For the ensemble forecasts,the average anomaly correlation coefficient of large-scale circulation is better for the forecast based on moist SVs in the Northern Hemisphere,and the low-level variables forecasted by the moist SVs are also improved,especially in the first 72 h.In addition,the moist SVs respond better to short-term precipitation according to statistical precipitation scores based on 10 cases.The inclusion of the large-scale condensation process in the calculation of SVs can improve the short-term weather prediction effectively.展开更多
In this study, singular vectors related to a heavy rainfall case over the Korean Peninsula were calculated using the fifth-generation Pennsylvania State University-National Center for Atmospheric Research Mesoscale Mo...In this study, singular vectors related to a heavy rainfall case over the Korean Peninsula were calculated using the fifth-generation Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model （MM5） adjoint modeling system. Tangent linear and adjoint models include moist physical processes, and a moist basic state and a moist total energy norm were used for the singular-vector calculations. The characteristics and nonlinear growth of the first singular vector were analyzed, focusing on the relationship between the basic state and the singular vector. The horizontal distribution of the initial singular vector was closely related to the baroclinicity index and the moisture availability of the basic state. The temperature-component energy at a lower level was dominant at the initial time, and the kinetic energy at upper levels became dominant at the final time in the energy profile of the singular vector. The nonlinear growth of the singular vector appropriately reflects the temporal variations in the basic state. The moisture-component energy at lower levels was dominant at earlier times, indicating continuous moisture transport in the basic state. There were a large amount of precipitation and corresponding latent heat release after that period because the continuous moisture transport created favorable conditions for both convective and nonconvective precipitation. The vertical propagation of the singular-vector energy was caused by precipitation and the corresponding latent heating in the basic state.展开更多
Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight λ, over sl（n,C）. He gave a differential-operator representation of the symmetric group ...Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight λ, over sl（n,C）. He gave a differential-operator representation of the symmetric group Sn on the corresponding space of truncated power series and proved that the solution space of the system is spanned by {σ（1）｜ σ ∈ Sn}. It is known that Sn is also the Weyl group of sl（n, C） and generated by all reflections sα with positive roots α. We present an explicit formula of the solution sα（1） for every positive root α and show directly that sα（1） is a polynomial if and only if （λ＋p, α） is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al..展开更多
In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors of tensors is studied, which have found many applications in diverse areas. The main results are:(ⅰ)each(Z-)eigenvector/singul...In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors of tensors is studied, which have found many applications in diverse areas. The main results are:(ⅰ)each(Z-)eigenvector/singular vector tuple of a generic tensor is nondegenerate, and(ⅱ) each nonzero Zeigenvector/singular vector tuple of an orthogonally decomposable tensor is nondegenerate.展开更多
In ensemble forecast,by summing up ensemble members,filtering the uncertainty,and retaining the common component,the ensemble mean with a better result can be achieved.However,the filtering works only when the initial...In ensemble forecast,by summing up ensemble members,filtering the uncertainty,and retaining the common component,the ensemble mean with a better result can be achieved.However,the filtering works only when the initial perturbation develops nonlinearly.If the initial perturbation propagates in a linear space,the positive and negative members will counteract,leading to little difference between ensemble mean and control forecast and finally insignificant ensemble result.In 1-2-day ensemble forecast,based on singular vector（SV） calculations,to avoid this insignificance,the counteracting members originated from the same SV are advised not to put into the ensemble system together;the only candidate should be the one with the better forecast.Based on the ingredient analysis of initial perturbation development,a method to select ensemble members is presented in this paper,which can fulfill the above requirement.The regional model MM5V1 of NCAR/PSU（National Center for Atmosphere Research/Pennsylvania State University） and its corresponding tangent adjoint model are used.The ensemble spread and forecast errors are calculated with dry energy norm.Two mesoscale lows on the Meiyu front along the Yangtze River are examined.According to the analysis of the perturbation ingredient,among couples of counteracting members from different SVs, those members performing better always have smaller or greater spread compared with other members. Following this thinking,an optimized ensemble and an inferior ensemble are identified.The ensemble mean of the optimized ensemble is more accurate than that of the inferior ensemble,and the former also performs better than the traditional ensemble with positive and negative members simultaneously.As for growth of the initial perturbation,those initial perturbations originated from the summed SVs grow more quickly than those from the single SV,and they enlarge the range of spread of the ensemble effectively,thus leading to better performance of ensemble members.展开更多
Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra ac...Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.展开更多
A novel concept of nonlinear singular vector and nonlinear singular value is introduced, which is a natural generalization of the classical linear singular vector and linear singular value to the nonlinear category. T...A novel concept of nonlinear singular vector and nonlinear singular value is introduced, which is a natural generalization of the classical linear singular vector and linear singular value to the nonlinear category. The optimization problem related to the determination of nonlinear singular vectors and singular values is formulated. The general idea of this approach is demonstrated by a simple two-dimensional quasigeostrophic model in the atmospheric and oceanic sciences. The advantage and its applications of the new method to the predictability, ensemble forecast and finite-time nonlinear instability are discussed. This paper makes a necessary preparation for further theoretical and numerical investigations.展开更多
In this paper the method and technique of the diagonalization are employed to transform a vector second-order nonlinear system into two first-order approximate diagonalized systems. The existence and the asymptotic be...In this paper the method and technique of the diagonalization are employed to transform a vector second-order nonlinear system into two first-order approximate diagonalized systems. The existence and the asymptotic behavior of the solutions are obtained for a vector second-order nonlinear Robin problem of singular perturbation type.展开更多
In this article, we study a kind of vector singularly perturbed delay-amerenum equation. Using boundary layer function method and geometric analysis skill, the asymptotic expression of the system is constructed and th...In this article, we study a kind of vector singularly perturbed delay-amerenum equation. Using boundary layer function method and geometric analysis skill, the asymptotic expression of the system is constructed and the uniform validity of asymptotic solution is also proved.展开更多
An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution o...An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences converge more quickly is to apply to them vector extrapolation methods. Two types of methods exist in the literature: polynomial type methods and epsilon algorithms. In most applications, the polynomial type methods have proved to be superior convergence accelerators. Three polynomial type methods are known, and these are the minimal polynomial extrapolation (MPE), the reduced rank extrapolation (RRE), and the modified minimal polynomial extrapolation (MMPE). In this work, we develop yet another polynomial type method, which is based on the singular value decomposition, as well as the ideas that lead to MPE. We denote this new method by SVD-MPE. We also design a numerically stable algorithm for its implementation, whose computational cost and storage requirements are minimal. Finally, we illustrate the use of SVD-MPE with numerical examples.展开更多
基金supported by the Joint Funds of the Chinese National Natural Science Foundation (NSFC)(Grant No.U2242213)the National Key Research and Development (R&D)Program of the Ministry of Science and Technology of China(Grant No. 2021YFC3000902)the National Science Foundation for Young Scholars (Grant No. 42205166)。
文摘Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial perturbation method tends only to capture synoptic scale initial uncertainty rather than mesoscale uncertainty in global ensemble prediction. To address this issue, a multiscale SV initial perturbation method based on the China Meteorological Administration Global Ensemble Prediction System(CMA-GEPS) is proposed to quantify multiscale initial uncertainty. The multiscale SV initial perturbation approach entails calculating multiscale SVs at different resolutions with multiple linearized physical processes to capture fast-growing perturbations from mesoscale to synoptic scale in target areas and combining these SVs by using a Gaussian sampling method with amplitude coefficients to generate initial perturbations. Following that, the energy norm,energy spectrum, and structure of multiscale SVs and their impact on GEPS are analyzed based on a batch experiment in different seasons. The results show that the multiscale SV initial perturbations can possess more energy and capture more mesoscale uncertainties than the traditional single-SV method. Meanwhile, multiscale SV initial perturbations can reflect the strongest dynamical instability in target areas. Their performances in global ensemble prediction when compared to single-scale SVs are shown to(i) improve the relationship between the ensemble spread and the root-mean-square error and(ii) provide a better probability forecast skill for atmospheric circulation during the late forecast period and for short-to medium-range precipitation. This study provides scientific evidence and application foundations for the design and development of a multiscale SV initial perturbation method for the GEPS.
基金the National Key R&D Program of China(Grant Nos.2017YFC1502102 and 2017YFC1501803).
文摘The singular vector(SV)initial perturbation method can capture the fastest-growing initial perturbation in a tangent linear model(TLM).Based on the global tangent linear and adjoint model of GRAPES-GEPS(Global/Regional Assimilation and Prediction System-Global Ensemble Prediction System),some experiments were carried out to analyze the structure of the moist SVs from the perspectives of the energy norm,energy spectrum,and vertical structure.The conclusions are as follows:The evolution of the SVs is synchronous with that of the atmospheric circulation,which is flowdependent.The moist and dry SVs are located in unstable regions at mid-to-high latitudes,but the moist SVs are wider,can contain more small-and medium-scale information,and have more energy than the dry SVs.From the energy spectrum analysis,the energy growth caused by the moist SVs is reflected in the relatively small-scale weather system.In addition,moist SVs can generate perturbations associated with large-scale condensation and precipitation,which is not true for dry SVs.For the ensemble forecasts,the average anomaly correlation coefficient of large-scale circulation is better for the forecast based on moist SVs in the Northern Hemisphere,and the low-level variables forecasted by the moist SVs are also improved,especially in the first 72 h.In addition,the moist SVs respond better to short-term precipitation according to statistical precipitation scores based on 10 cases.The inclusion of the large-scale condensation process in the calculation of SVs can improve the short-term weather prediction effectively.
基金funded by the Korea Meteorological Administration Research and Development Program (Grant No.RACS 2010-2016)supported by Leading Foreign Research Institute Recruitment Program through the National Research Foundation of Korea (NRF)funded by the Ministry of Education,Science and Technology (MEST) (2010-00715)the Brain Korea 21Project
文摘In this study, singular vectors related to a heavy rainfall case over the Korean Peninsula were calculated using the fifth-generation Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model （MM5） adjoint modeling system. Tangent linear and adjoint models include moist physical processes, and a moist basic state and a moist total energy norm were used for the singular-vector calculations. The characteristics and nonlinear growth of the first singular vector were analyzed, focusing on the relationship between the basic state and the singular vector. The horizontal distribution of the initial singular vector was closely related to the baroclinicity index and the moisture availability of the basic state. The temperature-component energy at a lower level was dominant at the initial time, and the kinetic energy at upper levels became dominant at the final time in the energy profile of the singular vector. The nonlinear growth of the singular vector appropriately reflects the temporal variations in the basic state. The moisture-component energy at lower levels was dominant at earlier times, indicating continuous moisture transport in the basic state. There were a large amount of precipitation and corresponding latent heat release after that period because the continuous moisture transport created favorable conditions for both convective and nonconvective precipitation. The vertical propagation of the singular-vector energy was caused by precipitation and the corresponding latent heating in the basic state.
文摘Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight λ, over sl（n,C）. He gave a differential-operator representation of the symmetric group Sn on the corresponding space of truncated power series and proved that the solution space of the system is spanned by {σ（1）｜ σ ∈ Sn}. It is known that Sn is also the Weyl group of sl（n, C） and generated by all reflections sα with positive roots α. We present an explicit formula of the solution sα（1） for every positive root α and show directly that sα（1） is a polynomial if and only if （λ＋p, α） is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al..
基金supported by National Natural Science Foundation of China(Grant No.11771328)Young Elite Scientists Sponsorship Program by Tianjin and the Natural Science Foundation of Zhejiang Province of China(Grant No.LD19A010002)。
文摘In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors of tensors is studied, which have found many applications in diverse areas. The main results are:(ⅰ)each(Z-)eigenvector/singular vector tuple of a generic tensor is nondegenerate, and(ⅱ) each nonzero Zeigenvector/singular vector tuple of an orthogonally decomposable tensor is nondegenerate.
基金Supported by the National Natural Science Foundation of China under Grant No.40405020
文摘In ensemble forecast,by summing up ensemble members,filtering the uncertainty,and retaining the common component,the ensemble mean with a better result can be achieved.However,the filtering works only when the initial perturbation develops nonlinearly.If the initial perturbation propagates in a linear space,the positive and negative members will counteract,leading to little difference between ensemble mean and control forecast and finally insignificant ensemble result.In 1-2-day ensemble forecast,based on singular vector（SV） calculations,to avoid this insignificance,the counteracting members originated from the same SV are advised not to put into the ensemble system together;the only candidate should be the one with the better forecast.Based on the ingredient analysis of initial perturbation development,a method to select ensemble members is presented in this paper,which can fulfill the above requirement.The regional model MM5V1 of NCAR/PSU（National Center for Atmosphere Research/Pennsylvania State University） and its corresponding tangent adjoint model are used.The ensemble spread and forecast errors are calculated with dry energy norm.Two mesoscale lows on the Meiyu front along the Yangtze River are examined.According to the analysis of the perturbation ingredient,among couples of counteracting members from different SVs, those members performing better always have smaller or greater spread compared with other members. Following this thinking,an optimized ensemble and an inferior ensemble are identified.The ensemble mean of the optimized ensemble is more accurate than that of the inferior ensemble,and the former also performs better than the traditional ensemble with positive and negative members simultaneously.As for growth of the initial perturbation,those initial perturbations originated from the summed SVs grow more quickly than those from the single SV,and they enlarge the range of spread of the ensemble effectively,thus leading to better performance of ensemble members.
基金supported by National Natural Science Foundation of China(Grant No.11326059)
文摘Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.
文摘A novel concept of nonlinear singular vector and nonlinear singular value is introduced, which is a natural generalization of the classical linear singular vector and linear singular value to the nonlinear category. The optimization problem related to the determination of nonlinear singular vectors and singular values is formulated. The general idea of this approach is demonstrated by a simple two-dimensional quasigeostrophic model in the atmospheric and oceanic sciences. The advantage and its applications of the new method to the predictability, ensemble forecast and finite-time nonlinear instability are discussed. This paper makes a necessary preparation for further theoretical and numerical investigations.
文摘In this paper the method and technique of the diagonalization are employed to transform a vector second-order nonlinear system into two first-order approximate diagonalized systems. The existence and the asymptotic behavior of the solutions are obtained for a vector second-order nonlinear Robin problem of singular perturbation type.
基金supported by the National Science Foundation of China(11071075)Introducing Talents Program of SIT (YJ2013-33)
文摘In this article, we study a kind of vector singularly perturbed delay-amerenum equation. Using boundary layer function method and geometric analysis skill, the asymptotic expression of the system is constructed and the uniform validity of asymptotic solution is also proved.
文摘An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences converge more quickly is to apply to them vector extrapolation methods. Two types of methods exist in the literature: polynomial type methods and epsilon algorithms. In most applications, the polynomial type methods have proved to be superior convergence accelerators. Three polynomial type methods are known, and these are the minimal polynomial extrapolation (MPE), the reduced rank extrapolation (RRE), and the modified minimal polynomial extrapolation (MMPE). In this work, we develop yet another polynomial type method, which is based on the singular value decomposition, as well as the ideas that lead to MPE. We denote this new method by SVD-MPE. We also design a numerically stable algorithm for its implementation, whose computational cost and storage requirements are minimal. Finally, we illustrate the use of SVD-MPE with numerical examples.
