This paper is concerned with the problem of designing robust H∞and H2static output feedback controllers for a class of discrete-time piecewise-affine singular systems with norm-bounded time-varying parameters uncerta...This paper is concerned with the problem of designing robust H∞and H2static output feedback controllers for a class of discrete-time piecewise-affine singular systems with norm-bounded time-varying parameters uncertainties. Based on a piecewise singular Lyapunov function combined with S-procedure,Projection lemma and some matrix inequality convexifying techniques,sufficient conditions in terms of linear matrix inequalities are given for the existence of an output-feedback controller for the discrete-time piecewiseaffine singular systems with a prescribed H∞disturbance attenuation level,and the H2norm is smaller than a given positive number. It is shown that the controller gains can be obtained by solving a family of LMIs parameterized by one or two scalar variables. The numerical examples are given to illustrate the effectiveness of the proposed design methods.展开更多
There are several examples of spaces of univariate functions for which we have a characterization of all sets of knots which are poised for the interpolation problem. For the standard spaces of univariate polynomials,...There are several examples of spaces of univariate functions for which we have a characterization of all sets of knots which are poised for the interpolation problem. For the standard spaces of univariate polynomials, or spline functions the mentioned results are well-known. In contrast with this, there are no such results in the bivariate case. As an exception, one may consider only the Pascal classic theorem, in the interpolation theory interpretation. In this paper, we consider a space of bivariate piecewise linear functions, for which we can readily find out whether the given node set is poised or not. The main tool we use for this purpose is the reduction by a basic subproblem, introduced in this paper.展开更多
The quantum hydrodynamic model for ion-acoustic waves in plasmas is studied.First,we design a new disturbance expansion to describe the ion fluid velocity and electric field potential.It should be emphasized that the ...The quantum hydrodynamic model for ion-acoustic waves in plasmas is studied.First,we design a new disturbance expansion to describe the ion fluid velocity and electric field potential.It should be emphasized that the piecewise function perturbation form is new with great difference from the previous perturbation.Then,based on the piecewise function perturbation,a(3+1)-dimensional generalized modified Korteweg–de Vries Zakharov–Kuznetsov(mKdV-ZK)equation is derived for the first time,which is an extended form of the classical mKdV equation and the ZK equation.The(3+1)-dimensional generalized time-space fractional mKdV-ZK equation is constructed using the semi-inverse method and the fractional variational principle.Obviously,it is more accurate to depict some complex plasma processes and phenomena.Further,the conservation laws of the generalized time-space fractional mKdV-ZK equation are discussed.Finally,using the multi-exponential function method,the non-resonant multiwave solutions are constructed,and the characteristics of ion-acoustic waves are well described.展开更多
In this paper,we study smoothing approximations for some piecewise smooth functions.We first present two approaches for one-dimensional case:a global approach is to construct smoothing approximations over the whole do...In this paper,we study smoothing approximations for some piecewise smooth functions.We first present two approaches for one-dimensional case:a global approach is to construct smoothing approximations over the whole domain and a local approach is to construct smoothing approximations within appropriate neighborhoods of the nonsmooth points.We obtain some error estimate results for both approaches and discuss whether the smoothing approximations can inherit the convexity of the original functions.Furthermore,we extend the global approach to some multiple dimensional cases.展开更多
Here,we developed novel extended piecewise bilinear power law(C-m)models to describe flow stresses under broad ranges of strain,strain rate,and temperature for mechanical and metallurgical calculations during metal fo...Here,we developed novel extended piecewise bilinear power law(C-m)models to describe flow stresses under broad ranges of strain,strain rate,and temperature for mechanical and metallurgical calculations during metal forming at elevated temperatures.The traditional C-m model is improved upon by formulating the material parameters C and m,defined at sample strains and temperatures as functions of the strain rate.The coefficients are described as a linear combination of the basis functions defined in piecewise patches of the sample strain and temperature domain.A comparison with traditional closed-form function flow models revealed that our approach using the extended piecewise bilinear C-m model is superior in terms of accuracy,ease of use,and adaptability;additionally,the extended C-m model was applicable to numerical analysis of mechanical,metallurgical,and microstructural problems.Moreover,metallurgy-related values can be calculated directly from the flow stress information.Although the proposed model was developed for materials at elevated temperatures,it can be applied over a broad temperature range.展开更多
This paper presents an application of the sparse Bayesian learning(SBL)algorithm to linear inverse problems with a high order total variation(HOTV)sparsity prior.For the problem of sparse signal recovery,SBL often pro...This paper presents an application of the sparse Bayesian learning(SBL)algorithm to linear inverse problems with a high order total variation(HOTV)sparsity prior.For the problem of sparse signal recovery,SBL often produces more accurate estimates than maximum a posteriori estimates,including those that useℓ1 regularization.Moreover,rather than a single signal estimate,SBL yields a full posterior density estimate which can be used for uncertainty quantification.However,SBL is only immediately applicable to problems having a direct sparsity prior,or to those that can be formed via synthesis.This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis,and then utilizes SBL.This expands the class of problems available to Bayesian learning to include,e.g.,inverse problems dealing with the recovery of piecewise smooth functions or signals from data.Numerical examples are provided to demonstrate how this new technique is effectively employed.展开更多
Detection of edges in piecewise smooth functions is important in many applications.Higher order reconstruction algorithms in image processing and post processing of numerical solutions to partial differential equation...Detection of edges in piecewise smooth functions is important in many applications.Higher order reconstruction algorithms in image processing and post processing of numerical solutions to partial differential equations require the identification of smooth domains,creating the need for algorithms that will accurately identify discontinuities in a given function as well as those in its gradient.This work expands the use of the polynomial annihilation edge detector,(Archibald,Gelb and Yoon,2005),to locate discontinuities in the gradient given irregularly sampled point values of a continuous function.The idea is to preprocess the given data by calculating the derivative,and then to use the polynomial annihilation edge detector to locate the jumps in the derivative.We compare our results to other recently developed methods.展开更多
The intuitionistic fuzzy set(I-fuzzy set)plays an effective role in game theory when players face‘neither this nor that’situation to set their goals.This study presents a maxmin–minmax solution to multi-objective t...The intuitionistic fuzzy set(I-fuzzy set)plays an effective role in game theory when players face‘neither this nor that’situation to set their goals.This study presents a maxmin–minmax solution to multi-objective two person zero-sum matrix games with I-fuzzy goals.In this article,a class of piecewise linear membership and non-membership functions for I-fuzzy goals is constructed.These functions are more effective in real games because marginal rate of increase(decrease)of such membership functions(non-membership functions)is different in different intervals of tolerance errors.Finally,one numerical example is given to examine the effectiveness and advantages of the proposed results.展开更多
Tropospheric delay is an important factor affecting high precision Global Navigation Satellite System(GNSS)positioning and also the basic data for GNSS atmospheric research.However,the existing tropospheric delay mode...Tropospheric delay is an important factor affecting high precision Global Navigation Satellite System(GNSS)positioning and also the basic data for GNSS atmospheric research.However,the existing tropospheric delay models have some problems,such as only a single function used for the entire atmosphere.In this paper,an ERA5-based(the fifth generation of European Centre for Medium-Range Weather Forecasts Reanalysis)global model for vertical adjustment of Zenith Tropospheric Delay(ZTD)using a piecewise function is developed.The ZTD data at 611 radiosonde stations and the MERRA-2(second Modern-Era Retrospective analysis for Research and Applications)atmospheric reanalysis data were used to validate the model reliability.The Global Zenith Tropospheric Delay Piecewise(GZTD-P)model has excellent performance compared with the Global Pressure and Temperature(GPT3)model.Validated at radiosonde stations,the performance of the GZTD-P model was improved by 0.96 cm(23%)relative to the GPT3 model.Validated with MERRA-2 data,the quality of the GZTD-P model is improved by 1.8 cm(50%)compared to the GPT3 model,showing better accuracy and stability.The ZTD vertical adjustment model with different resolutions was established to enrich the model's applicability and speed up the process of tropospheric delay calculation.By providing model parameters with different resolutions,users can choose the appropriate model according to their applications.展开更多
In this paper,we investigate the relationship between deep neural net works(DNN)with rectified linear unit(ReLU)function as the activation function and continuous piecewise linear(CPWL)functions,especially CPWL functi...In this paper,we investigate the relationship between deep neural net works(DNN)with rectified linear unit(ReLU)function as the activation function and continuous piecewise linear(CPWL)functions,especially CPWL functions from the simplicial linear finite element method(FEM).We first consider the special case of FEM.By exploring the DNN representation of its nodal basis functions,we present a ReLU DNN representation of CPWL in FEM.We theoretically establish that at least 2 hidden layers are needed in a ReLU DNN to represent any linear finite element functions inΩ■R^2 when d≥2.Consequently,for d=2,3 which are often encountered in scientific and engineering computing,the minimal number of two hidden layers are necessary and sufficient for any CPWL function to be represented by a ReLU DNN.Then we include a detailed account on how a general CPWL in R^d can be represented by a ReLU DNN with at most[log2(d+1)]|hidden layers and we also give an estimation of the number of neurons in DNN that are needed in such a represe ntation.Furthermore,using the relationship bet ween DNN and FEM,we theoretically argue that a special class of DNN models with low bit-width are still expected to have an adequate representation power in applications.Finally,as a proof of concept,we present some numerical results for using ReLU DNNs to solve a two point boundary problem to demonstrate the potential of applying DNN for numerical solution of partial differential equations.展开更多
The aim of this paper is to obtain the numerical solution of singular ordinary differential equations using the Haar-wavelet approach.The proposed method is mathematically simple and provides highly accurate solutions...The aim of this paper is to obtain the numerical solution of singular ordinary differential equations using the Haar-wavelet approach.The proposed method is mathematically simple and provides highly accurate solutions.In this method,we derive the Haar operational matrix using Haar function.Haar operational matrix is a basic tool and applied in system analysis to evaluate the numerical solution of differential equations.The convergence of the proposed method is discussed through its error analysis.To illustrate the efficiency of this method,solutions of four singular differential equations are obtained.展开更多
基金Sponsored by the National Natural Science Foundation of China(Grant No.61004038)
文摘This paper is concerned with the problem of designing robust H∞and H2static output feedback controllers for a class of discrete-time piecewise-affine singular systems with norm-bounded time-varying parameters uncertainties. Based on a piecewise singular Lyapunov function combined with S-procedure,Projection lemma and some matrix inequality convexifying techniques,sufficient conditions in terms of linear matrix inequalities are given for the existence of an output-feedback controller for the discrete-time piecewiseaffine singular systems with a prescribed H∞disturbance attenuation level,and the H2norm is smaller than a given positive number. It is shown that the controller gains can be obtained by solving a family of LMIs parameterized by one or two scalar variables. The numerical examples are given to illustrate the effectiveness of the proposed design methods.
文摘There are several examples of spaces of univariate functions for which we have a characterization of all sets of knots which are poised for the interpolation problem. For the standard spaces of univariate polynomials, or spline functions the mentioned results are well-known. In contrast with this, there are no such results in the bivariate case. As an exception, one may consider only the Pascal classic theorem, in the interpolation theory interpretation. In this paper, we consider a space of bivariate piecewise linear functions, for which we can readily find out whether the given node set is poised or not. The main tool we use for this purpose is the reduction by a basic subproblem, introduced in this paper.
基金Project supported by the National Natural Science Foundation of China(Grant No.11975143)the Natural Science Foundation of Shandong Province of China(Grant No.ZR2018MA017)+1 种基金the Taishan Scholars Program of Shandong Province,China(Grant No.ts20190936)the Shandong University of Science and Technology Research Fund(Grant No.2015TDJH102).
文摘The quantum hydrodynamic model for ion-acoustic waves in plasmas is studied.First,we design a new disturbance expansion to describe the ion fluid velocity and electric field potential.It should be emphasized that the piecewise function perturbation form is new with great difference from the previous perturbation.Then,based on the piecewise function perturbation,a(3+1)-dimensional generalized modified Korteweg–de Vries Zakharov–Kuznetsov(mKdV-ZK)equation is derived for the first time,which is an extended form of the classical mKdV equation and the ZK equation.The(3+1)-dimensional generalized time-space fractional mKdV-ZK equation is constructed using the semi-inverse method and the fractional variational principle.Obviously,it is more accurate to depict some complex plasma processes and phenomena.Further,the conservation laws of the generalized time-space fractional mKdV-ZK equation are discussed.Finally,using the multi-exponential function method,the non-resonant multiwave solutions are constructed,and the characteristics of ion-acoustic waves are well described.
基金This work was supported in part by the National Natural Science Foundation of China(No.11431004)the Innovation Program of Shanghai Municipal Education Commission.
文摘In this paper,we study smoothing approximations for some piecewise smooth functions.We first present two approaches for one-dimensional case:a global approach is to construct smoothing approximations over the whole domain and a local approach is to construct smoothing approximations within appropriate neighborhoods of the nonsmooth points.We obtain some error estimate results for both approaches and discuss whether the smoothing approximations can inherit the convexity of the original functions.Furthermore,we extend the global approach to some multiple dimensional cases.
基金financially supported by the Ministry of Trade,Industry and Energy(MOTIE),Korea Institute for Advancement of Technology(KIAT)through the International Cooperative R&D program(Project No.P0011877)MOTIE as a part of the joint R&D project(Project No.10081334)。
文摘Here,we developed novel extended piecewise bilinear power law(C-m)models to describe flow stresses under broad ranges of strain,strain rate,and temperature for mechanical and metallurgical calculations during metal forming at elevated temperatures.The traditional C-m model is improved upon by formulating the material parameters C and m,defined at sample strains and temperatures as functions of the strain rate.The coefficients are described as a linear combination of the basis functions defined in piecewise patches of the sample strain and temperature domain.A comparison with traditional closed-form function flow models revealed that our approach using the extended piecewise bilinear C-m model is superior in terms of accuracy,ease of use,and adaptability;additionally,the extended C-m model was applicable to numerical analysis of mechanical,metallurgical,and microstructural problems.Moreover,metallurgy-related values can be calculated directly from the flow stress information.Although the proposed model was developed for materials at elevated temperatures,it can be applied over a broad temperature range.
基金supported in part by NSF-DMS 1502640,NSF-DMS 1912685,AFOSR FA9550-18-1-0316Office of Naval Research MURI grant N00014-20-1-2595.
文摘This paper presents an application of the sparse Bayesian learning(SBL)algorithm to linear inverse problems with a high order total variation(HOTV)sparsity prior.For the problem of sparse signal recovery,SBL often produces more accurate estimates than maximum a posteriori estimates,including those that useℓ1 regularization.Moreover,rather than a single signal estimate,SBL yields a full posterior density estimate which can be used for uncertainty quantification.However,SBL is only immediately applicable to problems having a direct sparsity prior,or to those that can be formed via synthesis.This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis,and then utilizes SBL.This expands the class of problems available to Bayesian learning to include,e.g.,inverse problems dealing with the recovery of piecewise smooth functions or signals from data.Numerical examples are provided to demonstrate how this new technique is effectively employed.
基金This work was partially supported by NSF grants CNS 0324957DMS 0617867+2 种基金DMS 0608844(AG)DMS 0510813(AG and HM)DMS 0421846(AG and HM).
文摘Detection of edges in piecewise smooth functions is important in many applications.Higher order reconstruction algorithms in image processing and post processing of numerical solutions to partial differential equations require the identification of smooth domains,creating the need for algorithms that will accurately identify discontinuities in a given function as well as those in its gradient.This work expands the use of the polynomial annihilation edge detector,(Archibald,Gelb and Yoon,2005),to locate discontinuities in the gradient given irregularly sampled point values of a continuous function.The idea is to preprocess the given data by calculating the derivative,and then to use the polynomial annihilation edge detector to locate the jumps in the derivative.We compare our results to other recently developed methods.
文摘The intuitionistic fuzzy set(I-fuzzy set)plays an effective role in game theory when players face‘neither this nor that’situation to set their goals.This study presents a maxmin–minmax solution to multi-objective two person zero-sum matrix games with I-fuzzy goals.In this article,a class of piecewise linear membership and non-membership functions for I-fuzzy goals is constructed.These functions are more effective in real games because marginal rate of increase(decrease)of such membership functions(non-membership functions)is different in different intervals of tolerance errors.Finally,one numerical example is given to examine the effectiveness and advantages of the proposed results.
基金National Natural Science Foundation of China(41704027)Guangxi Natural Science Foundation of China(2020GXNSFBA297145,2020GXNSFBA159033)“Ba Gui Scholars”program of the provincial government of Guangxi.
文摘Tropospheric delay is an important factor affecting high precision Global Navigation Satellite System(GNSS)positioning and also the basic data for GNSS atmospheric research.However,the existing tropospheric delay models have some problems,such as only a single function used for the entire atmosphere.In this paper,an ERA5-based(the fifth generation of European Centre for Medium-Range Weather Forecasts Reanalysis)global model for vertical adjustment of Zenith Tropospheric Delay(ZTD)using a piecewise function is developed.The ZTD data at 611 radiosonde stations and the MERRA-2(second Modern-Era Retrospective analysis for Research and Applications)atmospheric reanalysis data were used to validate the model reliability.The Global Zenith Tropospheric Delay Piecewise(GZTD-P)model has excellent performance compared with the Global Pressure and Temperature(GPT3)model.Validated at radiosonde stations,the performance of the GZTD-P model was improved by 0.96 cm(23%)relative to the GPT3 model.Validated with MERRA-2 data,the quality of the GZTD-P model is improved by 1.8 cm(50%)compared to the GPT3 model,showing better accuracy and stability.The ZTD vertical adjustment model with different resolutions was established to enrich the model's applicability and speed up the process of tropospheric delay calculation.By providing model parameters with different resolutions,users can choose the appropriate model according to their applications.
基金This work is partially supported by Beijing International Center for Mat hematical Research,the Elite Program of Computational and Applied Mathematics for PhD Candidates of Peking University,NSFC Grant 91430215,NSF Grants DMS-1522615,DMS-1819157.
文摘In this paper,we investigate the relationship between deep neural net works(DNN)with rectified linear unit(ReLU)function as the activation function and continuous piecewise linear(CPWL)functions,especially CPWL functions from the simplicial linear finite element method(FEM).We first consider the special case of FEM.By exploring the DNN representation of its nodal basis functions,we present a ReLU DNN representation of CPWL in FEM.We theoretically establish that at least 2 hidden layers are needed in a ReLU DNN to represent any linear finite element functions inΩ■R^2 when d≥2.Consequently,for d=2,3 which are often encountered in scientific and engineering computing,the minimal number of two hidden layers are necessary and sufficient for any CPWL function to be represented by a ReLU DNN.Then we include a detailed account on how a general CPWL in R^d can be represented by a ReLU DNN with at most[log2(d+1)]|hidden layers and we also give an estimation of the number of neurons in DNN that are needed in such a represe ntation.Furthermore,using the relationship bet ween DNN and FEM,we theoretically argue that a special class of DNN models with low bit-width are still expected to have an adequate representation power in applications.Finally,as a proof of concept,we present some numerical results for using ReLU DNNs to solve a two point boundary problem to demonstrate the potential of applying DNN for numerical solution of partial differential equations.
文摘The aim of this paper is to obtain the numerical solution of singular ordinary differential equations using the Haar-wavelet approach.The proposed method is mathematically simple and provides highly accurate solutions.In this method,we derive the Haar operational matrix using Haar function.Haar operational matrix is a basic tool and applied in system analysis to evaluate the numerical solution of differential equations.The convergence of the proposed method is discussed through its error analysis.To illustrate the efficiency of this method,solutions of four singular differential equations are obtained.