Simultaneous Localization and Mapping(SLAM)is the foundation of autonomous navigation for unmanned systems.The existing SLAM solutions are mainly divided into the visual SLAM(vSLAM)equipped with camera and the lidar S...Simultaneous Localization and Mapping(SLAM)is the foundation of autonomous navigation for unmanned systems.The existing SLAM solutions are mainly divided into the visual SLAM(vSLAM)equipped with camera and the lidar SLAM equipped with lidar.However,pure visual SLAM have shortcomings such as low positioning accuracy,the paper proposes a visual-inertial information fusion SLAM based on Runge-Kutta improved pre-integration.First,the Inertial Measurement Unit(IMU)information between two adjacent keyframes is pre-integrated at the front-end to provide IMU constraints for visual-inertial information fusion.In particular,to improve the accuracy in pre-integration,the paper uses the RungeKutta algorithm instead of Euler integral to calculate the pre-integration value at the next moment.Then,the IMU pre-integration value is used as the initial value of the system state at the current frame time.We combine the visual reprojection error and imu pre-integration error to optimize the state variables such as speed and pose,and restore map points’three-dimensional coordinates.Finally,we set a sliding window to optimize map points’coordinates and state variables.The experimental part is divided into dataset experiment and complex indoor-environment experiment.The results show that compared with pure visual SLAM and the existing visual-inertial fusion SLAM,our method has higher positioning accuracy.展开更多
We propose a symplectic partitioned Runge-Kutta (SPRK) method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this te...We propose a symplectic partitioned Runge-Kutta (SPRK) method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete (NAD) operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal derivatives.The stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction (LWC) method and the eighth-order staggered-grid (SG)method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical dispersion.These numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic exploration and seismology research.展开更多
A newly proposed competent population-based optimization algorithm called RUN,which uses the principle of slope variations calculated by applying the Runge Kutta method as the key search mechanism,has gained wider int...A newly proposed competent population-based optimization algorithm called RUN,which uses the principle of slope variations calculated by applying the Runge Kutta method as the key search mechanism,has gained wider interest in solving optimization problems.However,in high-dimensional problems,the search capabilities,convergence speed,and runtime of RUN deteriorate.This work aims at filling this gap by proposing an improved variant of the RUN algorithm called the Adaptive-RUN.Population size plays a vital role in both runtime efficiency and optimization effectiveness of metaheuristic algorithms.Unlike the original RUN where population size is fixed throughout the search process,Adaptive-RUN automatically adjusts population size according to two population size adaptation techniques,which are linear staircase reduction and iterative halving,during the search process to achieve a good balance between exploration and exploitation characteristics.In addition,the proposed methodology employs an adaptive search step size technique to determine a better solution in the early stages of evolution to improve the solution quality,fitness,and convergence speed of the original RUN.Adaptive-RUN performance is analyzed over 23 IEEE CEC-2017 benchmark functions for two cases,where the first one applies linear staircase reduction with adaptive search step size(LSRUN),and the second one applies iterative halving with adaptive search step size(HRUN),with the original RUN.To promote green computing,the carbon footprint metric is included in the performance evaluation in addition to runtime and fitness.Simulation results based on the Friedman andWilcoxon tests revealed that Adaptive-RUN can produce high-quality solutions with lower runtime and carbon footprint values as compared to the original RUN and three recent metaheuristics.Therefore,with its higher computation efficiency,Adaptive-RUN is a much more favorable choice as compared to RUN in time stringent applications.展开更多
This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. ...This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. A new convergence concept, called DA-convergence, is introduced. The DA-convergence result for the methods is derived. At the end, a numerical example is given to verify the computational effectiveness and the theoretical result.展开更多
The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta (RK) method is an important part of the former, and the ...The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta (RK) method is an important part of the former, and the continuous finite element method (CFEM) belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms --the regular method. Finally, numerical experiments are given to verify the theoretical results.展开更多
Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic ...Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations (PDEs) derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.展开更多
Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are establi...Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.展开更多
The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)...The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)+by(t-τ)+cy’(t-τ), t>0, y(t)=g(t), -τ≤t≤0, with a,b,c∈[FK(W+3mm\.3mm][TPP129A,+3mm?3mm,BP], τ>0 and g(t) is a continuous real value function. In this paper we are concerned with the dependence of stability region on a fixed but arbitrary delay τ. In fact, it is one of the N.Guglielmi open problems to investigate the delay dependent stability analysis for NDDEs. The results that the 2,3 stages non natural R-K methods are unstable as Radau IA and Lobatto IIIC are proved. And the s stages Radau IIA methods are unstable, however all Gauss methods are compatible.展开更多
An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditio...An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.展开更多
This paper mainly presents Euler method and fourth-order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and pr...This paper mainly presents Euler method and fourth-order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and practically well suited for solving these problems. In order to verify the ac-curacy, we compare numerical solutions with the exact solutions. The numerical solutions are in good agreement with the exact solutions. Numerical comparisons between Euler method and Runge Kutta method have been presented. Also we compare the performance and the computational effort of such methods. In order to achieve higher accuracy in the solution, the step size needs to be very small. Finally we investigate and compute the errors of the two proposed methods for different step sizes to examine superiority. Several numerical examples are given to demonstrate the reliability and efficiency.展开更多
A class of parallel Runge-Kutta Methods for differential-algebraic equations of index 2are constructed for multiprocessor system. This paper gives the order conditions and investigatesthe convergence theory for such m...A class of parallel Runge-Kutta Methods for differential-algebraic equations of index 2are constructed for multiprocessor system. This paper gives the order conditions and investigatesthe convergence theory for such methods.展开更多
This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods....This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods. It is proved that for the suitable selection of the diagonal matrix D, the method based on Radau IIA is δ-convergent, and the estimates for the non-stiff speed and the stiff speed of convergence are given.展开更多
基金supported by the China Postdoctoral Science Foundation under Grant 2019M653870XBNational Natural Science Foundation of Shanxi Province under Grants No.2020GY-003 and 2021GY-036+1 种基金National Natural Science Foundation of China under Grants 62001340Fundamental Research Funds for the Central Universities,China,XJS211306 and JC2007
文摘Simultaneous Localization and Mapping(SLAM)is the foundation of autonomous navigation for unmanned systems.The existing SLAM solutions are mainly divided into the visual SLAM(vSLAM)equipped with camera and the lidar SLAM equipped with lidar.However,pure visual SLAM have shortcomings such as low positioning accuracy,the paper proposes a visual-inertial information fusion SLAM based on Runge-Kutta improved pre-integration.First,the Inertial Measurement Unit(IMU)information between two adjacent keyframes is pre-integrated at the front-end to provide IMU constraints for visual-inertial information fusion.In particular,to improve the accuracy in pre-integration,the paper uses the RungeKutta algorithm instead of Euler integral to calculate the pre-integration value at the next moment.Then,the IMU pre-integration value is used as the initial value of the system state at the current frame time.We combine the visual reprojection error and imu pre-integration error to optimize the state variables such as speed and pose,and restore map points’three-dimensional coordinates.Finally,we set a sliding window to optimize map points’coordinates and state variables.The experimental part is divided into dataset experiment and complex indoor-environment experiment.The results show that compared with pure visual SLAM and the existing visual-inertial fusion SLAM,our method has higher positioning accuracy.
基金This research was supported by the National Natural Science Foundation of China (Nos. 41230210 and 41204074), the Science Foundation of the Education Department of Yunnan Province (No. 2013Z152), and Statoil Company (Contract No. 4502502663).
文摘We propose a symplectic partitioned Runge-Kutta (SPRK) method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete (NAD) operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal derivatives.The stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction (LWC) method and the eighth-order staggered-grid (SG)method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical dispersion.These numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic exploration and seismology research.
文摘A newly proposed competent population-based optimization algorithm called RUN,which uses the principle of slope variations calculated by applying the Runge Kutta method as the key search mechanism,has gained wider interest in solving optimization problems.However,in high-dimensional problems,the search capabilities,convergence speed,and runtime of RUN deteriorate.This work aims at filling this gap by proposing an improved variant of the RUN algorithm called the Adaptive-RUN.Population size plays a vital role in both runtime efficiency and optimization effectiveness of metaheuristic algorithms.Unlike the original RUN where population size is fixed throughout the search process,Adaptive-RUN automatically adjusts population size according to two population size adaptation techniques,which are linear staircase reduction and iterative halving,during the search process to achieve a good balance between exploration and exploitation characteristics.In addition,the proposed methodology employs an adaptive search step size technique to determine a better solution in the early stages of evolution to improve the solution quality,fitness,and convergence speed of the original RUN.Adaptive-RUN performance is analyzed over 23 IEEE CEC-2017 benchmark functions for two cases,where the first one applies linear staircase reduction with adaptive search step size(LSRUN),and the second one applies iterative halving with adaptive search step size(HRUN),with the original RUN.To promote green computing,the carbon footprint metric is included in the performance evaluation in addition to runtime and fitness.Simulation results based on the Friedman andWilcoxon tests revealed that Adaptive-RUN can produce high-quality solutions with lower runtime and carbon footprint values as compared to the original RUN and three recent metaheuristics.Therefore,with its higher computation efficiency,Adaptive-RUN is a much more favorable choice as compared to RUN in time stringent applications.
文摘This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. A new convergence concept, called DA-convergence, is introduced. The DA-convergence result for the methods is derived. At the end, a numerical example is given to verify the computational effectiveness and the theoretical result.
基金Project supported by the National Natural Science Foundation of China (No. 11071067)the Hunan Graduate Student Science and Technology Innovation Project (No. CX2011B184)
文摘The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta (RK) method is an important part of the former, and the continuous finite element method (CFEM) belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms --the regular method. Finally, numerical experiments are given to verify the theoretical results.
基金supported by the National Natural Science Foundation of China (Nos. 10772147 and10632030)the Ph. D. Program Foundation of Ministry of Education of China (No. 20070699028)+2 种基金the Natural Science Foundation of Shaanxi Province of China (No. 2006A07)the Open Foundationof State Key Laboratory of Structural Analysis of Industrial Equipment (No. GZ0802)the Foundation for Fundamental Research of Northwestern Polytechnical University
文摘Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations (PDEs) derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.
基金Project supported by the National Natural Science Foundation of China(No.11432010)the Doctoral Program Foundation of Education Ministry of China(No.20126102110023)+2 种基金the 111Project of China(No.B07050)the Fundamental Research Funds for the Central Universities(No.310201401JCQ01001)the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University(No.CX201517)
文摘Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.
文摘The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)+by(t-τ)+cy’(t-τ), t>0, y(t)=g(t), -τ≤t≤0, with a,b,c∈[FK(W+3mm\.3mm][TPP129A,+3mm?3mm,BP], τ>0 and g(t) is a continuous real value function. In this paper we are concerned with the dependence of stability region on a fixed but arbitrary delay τ. In fact, it is one of the N.Guglielmi open problems to investigate the delay dependent stability analysis for NDDEs. The results that the 2,3 stages non natural R-K methods are unstable as Radau IA and Lobatto IIIC are proved. And the s stages Radau IIA methods are unstable, however all Gauss methods are compatible.
文摘An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.
文摘This paper mainly presents Euler method and fourth-order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and practically well suited for solving these problems. In order to verify the ac-curacy, we compare numerical solutions with the exact solutions. The numerical solutions are in good agreement with the exact solutions. Numerical comparisons between Euler method and Runge Kutta method have been presented. Also we compare the performance and the computational effort of such methods. In order to achieve higher accuracy in the solution, the step size needs to be very small. Finally we investigate and compute the errors of the two proposed methods for different step sizes to examine superiority. Several numerical examples are given to demonstrate the reliability and efficiency.
文摘A class of parallel Runge-Kutta Methods for differential-algebraic equations of index 2are constructed for multiprocessor system. This paper gives the order conditions and investigatesthe convergence theory for such methods.
文摘This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods. It is proved that for the suitable selection of the diagonal matrix D, the method based on Radau IIA is δ-convergent, and the estimates for the non-stiff speed and the stiff speed of convergence are given.