A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified the...A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.展开更多
A class of efficient parallel multivalue hybrid methods for stiff differential equations are presented,which are all extremely stable at infinity, A-stable for orders 1-3 and A(α)-stable for orders 4-8. Eachmethod of...A class of efficient parallel multivalue hybrid methods for stiff differential equations are presented,which are all extremely stable at infinity, A-stable for orders 1-3 and A(α)-stable for orders 4-8. Eachmethod of the class can be performed parallelly using two processors with each processor having almost thesame computational amount per integration step as a backward differentiation formula (BDF) of the same orderwith the same stepsize performed in serial, whereas the former has not only much better stability propertiesbut also a computational accuracy higher than the corresponding BDF. Theoretical analysis and numericalexperiments show that the methods constructed in this paper are superior in many respects not only to BDFsbut also to some other commonly used methods.展开更多
在系统生物学的研究中,由于所研究问题的复杂性和多尺度性,经常会遇到刚性方程的求解.而近年来,神经网络和深度学习的发展为上述问题提供了新的解决思路和方法 .本研究以经典的Belousov-Zhabotinsky(B-Z)反应和Van der Pol(VdP)方程为例...在系统生物学的研究中,由于所研究问题的复杂性和多尺度性,经常会遇到刚性方程的求解.而近年来,神经网络和深度学习的发展为上述问题提供了新的解决思路和方法 .本研究以经典的Belousov-Zhabotinsky(B-Z)反应和Van der Pol(VdP)方程为例,对四类非时序神经网络,包括全连接网络、残差网络、改进的残差网络和深度混合卷积网络,以及三类时序神经网络,包括循环神经网络(RNN)、长短时记忆网络(LSTM)、注意力机制进行了系统比较.实验结果表明:时序神经网络应用于刚性问题的求解精度和计算时间都大幅优于非时序神经网络,而四类非时序神经网络之间的表现并无显著差异.此外还将常微分神经网络(ODE-Net)应用于上述刚性问题,并观察到在极短的计算时间内,该方法能够达到极高的精度.本研究为应用神经网络解决系统生物学中各类刚性问题提供了参考和指导.展开更多
In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improv...In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.展开更多
A new polynomial formulation of variable step size linear multistep methods is pre- sented, where each k-step method is characterized by a fixed set of k - 1 or k parameters. This construction includes all methods of ...A new polynomial formulation of variable step size linear multistep methods is pre- sented, where each k-step method is characterized by a fixed set of k - 1 or k parameters. This construction includes all methods of maximal order （p = k for stiff, and p = k ＋ 1 for nonstiff problems）. Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are imple- mented in MATLAB, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multi- step method construction and implementation compares favorably to existing software, although variable order has not yet been included.展开更多
In this paper, a class of A (a)-contractive second derivative multistep methods for solving stiff ODE’s is constructed, in comparison with the Enright method of the same order, the contractivity properties and the st...In this paper, a class of A (a)-contractive second derivative multistep methods for solving stiff ODE’s is constructed, in comparison with the Enright method of the same order, the contractivity properties and the stability properties of the former are better than of the latter and the former preserves other advantages of the latter.展开更多
A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise. Compared to some other already repor...A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise. Compared to some other already reported split-step balanced methods, the drift increment function of the methods can be taken from any chosen ane-step ordinary differential equations （ODEs） solver. The schemes is proved to be strong convergent with order one. For the mean-square stability analysis, the investigation is confined to two cases. Some numerical experiments are reported to testify the performance and the effectiveness of the methods.展开更多
In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong T...In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong Taylor approximations(including Ito-Taylor and Stratonovich-Taylor schemes)with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations(SSDE)by employing truncations of backward stochastic Taylor expansions.We demonstrate that these schemes will converge strongly with corresponding order 1,2,3,....Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order 2,and it has larger meansquare stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order 2.We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes.The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms.Our numerical experiment show these points.展开更多
B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations (VFDEs) are established which provide unified theoretical foundation for the studyof Runge-...B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations (VFDEs) are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differentialequations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs ofother type which appear in practice.展开更多
基金This work was supported by the National High-Tech ICF Committee in Chinathe National Natural Science Foundation of China(Grant No.10271100).
文摘A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.
基金This work was supported by the National High-Tech ICF Committee in China and the National Natural Science Foundation of China (Grant No. 19871070).
文摘A class of efficient parallel multivalue hybrid methods for stiff differential equations are presented,which are all extremely stable at infinity, A-stable for orders 1-3 and A(α)-stable for orders 4-8. Eachmethod of the class can be performed parallelly using two processors with each processor having almost thesame computational amount per integration step as a backward differentiation formula (BDF) of the same orderwith the same stepsize performed in serial, whereas the former has not only much better stability propertiesbut also a computational accuracy higher than the corresponding BDF. Theoretical analysis and numericalexperiments show that the methods constructed in this paper are superior in many respects not only to BDFsbut also to some other commonly used methods.
文摘在系统生物学的研究中,由于所研究问题的复杂性和多尺度性,经常会遇到刚性方程的求解.而近年来,神经网络和深度学习的发展为上述问题提供了新的解决思路和方法 .本研究以经典的Belousov-Zhabotinsky(B-Z)反应和Van der Pol(VdP)方程为例,对四类非时序神经网络,包括全连接网络、残差网络、改进的残差网络和深度混合卷积网络,以及三类时序神经网络,包括循环神经网络(RNN)、长短时记忆网络(LSTM)、注意力机制进行了系统比较.实验结果表明:时序神经网络应用于刚性问题的求解精度和计算时间都大幅优于非时序神经网络,而四类非时序神经网络之间的表现并无显著差异.此外还将常微分神经网络(ODE-Net)应用于上述刚性问题,并观察到在极短的计算时间内,该方法能够达到极高的精度.本研究为应用神经网络解决系统生物学中各类刚性问题提供了参考和指导.
文摘In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.
文摘A new polynomial formulation of variable step size linear multistep methods is pre- sented, where each k-step method is characterized by a fixed set of k - 1 or k parameters. This construction includes all methods of maximal order （p = k for stiff, and p = k ＋ 1 for nonstiff problems）. Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are imple- mented in MATLAB, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multi- step method construction and implementation compares favorably to existing software, although variable order has not yet been included.
文摘In this paper, a class of A (a)-contractive second derivative multistep methods for solving stiff ODE’s is constructed, in comparison with the Enright method of the same order, the contractivity properties and the stability properties of the former are better than of the latter and the former preserves other advantages of the latter.
基金National Natural Science Foundation of China(No.11171352)
文摘A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise. Compared to some other already reported split-step balanced methods, the drift increment function of the methods can be taken from any chosen ane-step ordinary differential equations （ODEs） solver. The schemes is proved to be strong convergent with order one. For the mean-square stability analysis, the investigation is confined to two cases. Some numerical experiments are reported to testify the performance and the effectiveness of the methods.
基金supported by the Fundamental Research Funds for the Central Universities of China,and the second author is supported by the National Natural Fund Projects of China(Nos.11771100,12071332).
文摘In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong Taylor approximations(including Ito-Taylor and Stratonovich-Taylor schemes)with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations(SSDE)by employing truncations of backward stochastic Taylor expansions.We demonstrate that these schemes will converge strongly with corresponding order 1,2,3,....Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order 2,and it has larger meansquare stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order 2.We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes.The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms.Our numerical experiment show these points.
文摘B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations (VFDEs) are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differentialequations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs ofother type which appear in practice.
