488 results
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2. An alternative analysis for the local convergence of iterative methods for multiple roots including when the multiplicity is unknown.
- Author
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Alarcón, Diego, Hueso, Jose L., and Martínez, Eulalia
- Subjects
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MULTIPLICITY (Mathematics) , *ITERATIVE methods (Mathematics) , *STOCHASTIC convergence , *NUMERICAL analysis , *RADIUS (Geometry) , *NONLINEAR equations , *ALGEBRAIC equations - Abstract
In this paper we propose an alternative for the study of local convergence radius and the uniqueness radius for some third-order methods for multiple roots whose multiplicity is known. The main goal is to provide an alternative that avoids the use of sophisticated properties of divided differences that are used in already published papers about local convergence for multiple roots. We defined the local study by using a technique taking into consideration a bounding condition for the (m + i) th derivative of the function f (x) with i=1,2. In the case that the method uses first and second derivative in its iterative expression and i=1 in case the method only uses first derivative. Furthermore we implement a numerical analysis in the following sense. Since the radius of local convergence for high-order methods decreases with the order, we must take into account the analysis of ITS behaviour when we introduce a new iterative method. Finally, we have used these iterative methods for multiple roots for the case where the multiplicity m is unknown, so we estimate this factor by different strategies comparing the behaviour of the corresponding estimations and how this fact affect to the original method. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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3. The Stability and Convergence of The Numerical Computation for the Temporal Fractional Black-Scholes Equation.
- Author
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Mesgarani, H., Bakhshandeh, M., and Aghdam, Y. Esmaeelzade
- Subjects
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STOCHASTIC convergence , *BLACK-Scholes model , *NUMERICAL analysis , *VARIANCES , *INTERPOLATION - Abstract
In this paper, the temporal fractional Black-Scholes model (TFBSM) is discussed in the limited specific domain which the time derivative of this template is the Caputo fractional function. The value variance of the associated fractal transmission method was applied to forecast TFBSM. For solving, at first the semi-discrete scheme is obtained by using linear interpolation with a temporally τ2-α order accuracy. Then, the full scheme is collected by approximating the spatial derivative terms with the help of the Chebyshev collocation system focused on the fourth form. Finally, the unconditional stability and convergence order are evaluated by performing the energy process. As an implementation of this method, two examples of the TFBSM were reported to demonstrate the accuracy of the developed scheme. Simulation and comparison show that the suggested strategy is very accurate and effective. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
4. Generalized conditioning based approaches to computing confidence intervals for solutions to stochastic variational inequalities.
- Author
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Lamm, Michael and Lu, Shu
- Subjects
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NUMERICAL analysis , *APPROXIMATION theory , *STOCHASTIC convergence , *VARIATIONAL inequalities (Mathematics) , *SAMPLE average approximation method - Abstract
Stochastic variational inequalities (SVI) provide a unified framework for the study of a general class of nonlinear optimization and Nash-type equilibrium problems with uncertain model data. Often the true solution to an SVI cannot be found directly and must be approximated. This paper considers the use of a sample average approximation (SAA), and proposes a new method to compute confidence intervals for individual components of the true SVI solution based on the asymptotic distribution of SAA solutions. We estimate the asymptotic distribution based on one SAA solution instead of generating multiple SAA solutions, and can handle inequality constraints without requiring the strict complementarity condition in the standard nonlinear programming setting. The method in this paper uses the confidence regions to guide the selection of a single piece of a piecewise linear function that governs the asymptotic distribution of SAA solutions, and does not rely on convergence rates of the SAA solutions in probability. It also provides options to control the computation procedure and investigate effects of certain key estimates on the intervals. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
5. A survey on the high convergence orders and computational convergence orders of sequences.
- Author
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Cătinaş, Emil
- Subjects
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STOCHASTIC convergence , *COMPUTATIONAL mathematics , *ASYMPTOTIC distribution , *NONLINEAR equations , *ITERATIVE methods (Mathematics) , *NUMERICAL analysis - Abstract
Abstract Twenty years after the classical book of Ortega and Rheinboldt was published, five definitions for the Q -convergence orders of sequences were independently and rigorously studied (i.e., some orders characterized in terms of others), by Potra (1989), resp. Beyer, Ebanks and Qualls (1990). The relationship between all the five definitions (only partially analyzed in each of the two papers) was not subsequently followed and, moreover, the second paper slept from the readers attention. The main aim of this paper is to provide a rigorous, selfcontained, and, as much as possible, a comprehensive picture of the theoretical aspects of this topic, as the current literature has taken away the credit from authors who obtained important results long ago. Moreover, this paper provides rigorous support for the numerical examples recently presented in an increasing number of papers, where the authors check the convergence orders of different iterative methods for solving nonlinear (systems of) equations. Tight connections between some asymptotic quantities defined by theoretical and computational elements are shown to hold. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
6. The genesis and early developments of Aitken's process, Shanks' transformation, the ε-algorithm, and related fixed point methods.
- Author
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Brezinski, Claude and Redivo-Zaglia, Michela
- Subjects
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EXTRAPOLATION , *STOCHASTIC convergence , *NUMERICAL analysis , *VECTORS (Calculus) , *APPLIED mathematics , *MATRICES (Mathematics) - Abstract
In this paper, we trace back the genesis of Aitken's Δ2 process and Shanks' sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks' transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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7. Strong Convergence Theorems for the Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in the Intermediate Sense in Hilbert Spaces.
- Author
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Yan, Qian and Cai, Gang
- Subjects
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STOCHASTIC convergence , *HILBERT space , *NONEXPANSIVE mappings , *NUMERICAL analysis , *PARAMETER estimation - Abstract
The aim of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in the intermediate sense in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also give a numerical example to support our main results. The results obtained in this paper improve and extend many recent ones in this culture. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
8. Strong convergence of the truncated Euler-Maruyama method for stochastic functional differential equations.
- Author
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Zhang, Wei, Song, M. H., and Liu, M. Z.
- Subjects
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NUMERICAL analysis , *MATHEMATICAL models , *STOCHASTIC convergence , *FUNCTIONAL differential equations , *LIPSCHITZ spaces - Abstract
In this paper, we establish the truncated Euler-Maruyama (EM) method for stochastic functional differential equation (SFDE)
and consider the strong convergence theory for the numerical solutions of SFDEs under the local Lipschitz condition plus Khasminskii-type condition instead of the linear growth condition. The type of convergence specifically addressed in this paper is strong- convergence for , and p is a parameter in Khasminskii-type condition. We also discussed the rates of -convergence for the truncated EM method. [ABSTRACT FROM AUTHOR] - Published
- 2018
- Full Text
- View/download PDF
9. Superconvergence of kernel-based interpolation.
- Author
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Schaback, Robert
- Subjects
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INTERPOLATION , *SPLINE theory , *STOCHASTIC convergence , *EIGENFUNCTIONS , *NUMERICAL analysis - Abstract
From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization , and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
10. "Time"‐Parallel diffusion‐based correlation operators.
- Author
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Weaver, Anthony T., Gürol, Selime, Tshimanga, Jean, Chrust, Marcin, and Piacentini, Andrea
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STATISTICAL correlation , *COVARIANCE matrices , *NUMERICAL analysis , *STOCHASTIC convergence , *STOCHASTIC processes - Abstract
Correlation operators based on the solution of an implicitly formulated diffusion equation can be implemented numerically using the Chebyshev iteration method. The attractive properties of the algorithm for modelling correlation functions on high‐performance computers have been discussed in a recent paper. The current paper describes a straightforward variant of that algorithm that allows the matrix–vector products involved in the sequential pseudo‐time diffusion process to be performed in parallel. Contrary to the original algorithm, which requires solving, in sequence, linear systems involving a symmetric positive‐definite (SPD) matrix, the "time"‐parallel algorithm requires solving a single linear system involving a non‐symmetric positive‐definite (NSPD) matrix. The key information required by the Chebyshev iteration for solving the NSPD problem is an estimate of the extreme eigenvalues of the NSPD matrix. For the problem under consideration, the extreme eigenvalues of the NSPD matrix are the same as those of the original SPD matrix, and can be pre‐computed using a Lanczos algorithm applied to the latter. The convergence properties of the algorithm are studied from a theoretical perspective and using numerical experiments with a diffusion‐based covariance model in a variational data assimilation system for the global ocean. Results suggest that time‐parallelization can reduce the run‐time of an implicit diffusion‐based correlation operator by greater than a factor of two. It can be implemented practically using a hybrid parallelization approach that combines Message Passing Interface tasks in the spatial domain with Open Multi‐Processing threads spanning the pseudo‐time dimension. The sensitivity of the results to preconditioning, to the choice of first guess and to the stopping criterion is discussed. This paper describes a diffusion‐based correlation model in which the M matrix–vector products involved in the sequential M pseudo‐time‐step implicit diffusion process can be performed in parallel. The "time"‐parallel algorithm requires solving a single linear system involving a non‐symmetric positive definite matrix. An approximate solution can be obtained using the Chebyshev iteration. Results suggest that the "time"‐parallel algorithm can reduce run‐time of the correlation operator by a factor between two and three. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
11. Stochastic linear multistep methods for the simulation of chemical kinetics.
- Author
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Barrio, Manuel, Burrage, Kevin, and Burrage, Pamela
- Subjects
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CHEMICAL kinetics , *STOCHASTIC processes , *TRAPEZOIDS , *STOCHASTIC convergence , *LINEAR systems , *NUMERICAL analysis - Abstract
In this paper, we introduce the Stochastic Adams-Bashforth (SAB) and Stochastic Adams-Moulton (SAM) methods as an extension of the τ-leaping framework to past information. Using the Θ-trapezoidal τ-leap method of weak order two as a starting procedure, we show that the k-step SAB method with k ≥ 3 is order three in the mean and correlation, while a predictor-corrector implementation of the SAM method is weak order three in the mean but only order one in the correlation. These convergence results have been derived analytically for linear problems and successfully tested numerically for both linear and non-linear systems. A series of additional examples have been implemented in order to demonstrate the efficacy of this approach. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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12. Using Majorizing Sequences for the Semi-local Convergence of a High-Order and Multipoint Iterative Method along with Stability Analysis.
- Author
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Moccari, M. and Lotfi, T.
- Subjects
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STOCHASTIC convergence , *ITERATIVE methods (Mathematics) , *NUMERICAL analysis , *ITERATIVE decoding , *SET theory - Abstract
This paper deals with the study of relaxed conditions for semi-local convergence for a general iterative method, k-step Newton's method, using majorizing sequences. Dynamical behavior of the mentioned method is also analyzed via Julia set and basins of attraction. Numerical examples of nonlinear systems of equations will be examined to verify the given theory. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
13. Asymptotic Behaviour of Coupled Systems in Discrete and Continuous Time.
- Author
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Paunonen, Lassi and Seifert, David
- Subjects
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COUPLED mode theory (Wave-motion) , *DISCRETE time filters , *STOCHASTIC convergence , *NUMERICAL analysis , *MATHEMATICAL analysis - Abstract
This paper investigates the asymptotic behaviour of solutions to certain infinite systems of coupled recurrence relations. In particular, we obtain a characterisation of those initial values which lead to a convergent solution, and for initial values satisfying a slightly stronger condition we obtain an optimal estimate on the rate of convergence. By establishing a connection with a related problem in continuous time, we are able to use this optimal estimate to improve the rate of convergence in the continuous setting obtained by the authors in a previous paper. We illustrate the power of the general approach by using it to study several concrete examples, both in continuous and in discrete time. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
14. Robust finite-time guidance against maneuverable targets with unpredictable evasive strategies.
- Author
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Zhang, Ran, Wang, Jiawei, Li, Huifeng, Li, Zhenhong, and Ding, Zhengtao
- Subjects
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ROBUST control , *PROBLEM solving , *STABILITY theory , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
This paper presents a robust finite-time guidance (RFTG) law to a short-range interception problem. The main challenge is that the evasive strategy of the target is unpredictable because it is determined not only by the states of both the interceptor and the target, but also by external un-modeled factors. By robustly stabilizing a line-of-sight rate, this paper proposes an integrated continuous finite-time disturbance observer/bounded continuous finite-time stabilizer strategy. The design of this integrated strategy has two points: 1) effect of a target maneuver is modeled as disturbance and then is estimated by the second-order homogeneous observer; 2) the finite-time stabilizer is actively coupled with the observer. Based on homogeneity technique, the local finite-time input-to-state stability is established for the closed-loop guidance system, thus implying the proposed RFTG law can quickly render the LOS rate within a bounded error throughout intercept. Moreover, convergence properties of the LOS rate in the presence of control saturation are discussed. Numerical comparison studies demonstrate the guidance performance. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
15. Evaluating non-analytic functions of matrices.
- Author
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Sharon, Nir and Shkolnisky, Yoel
- Subjects
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STOCHASTIC convergence , *MATRICES (Mathematics) , *POLYNOMIALS , *MATHEMATICAL bounds , *NUMERICAL analysis - Abstract
The paper revisits the classical problem of evaluating f ( A ) for a real function f and a matrix A with real spectrum. The evaluation is based on expanding f in Chebyshev polynomials, and the focus of the paper is to study the convergence rates of these expansions. In particular, we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix A . We present several numerical examples to illustrate our analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
16. New extragradient method for a class of equilibrium problems in Hilbert spaces.
- Author
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Hieu, Dang Van
- Subjects
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HILBERT space , *LIPSCHITZ spaces , *ALGORITHMS , *STOCHASTIC convergence , *SET theory , *PROBLEM solving , *NUMERICAL analysis - Abstract
The paper proposes a new extragradient algorithm for solving strongly pseudomonotone equilibrium problems which satisfy a Lipschitz-type condition recently introduced by Mastroeni in auxiliary problem principle. The main novelty of the paper is that the algorithm generates the strongly convergent sequences in Hilbert spaces
without the prior knowledge of Lipschitz-type constants and any hybrid method. Several numerical experiments on a test problem are also presented to illustrate the convergence of the algorithm. [ABSTRACT FROM AUTHOR]- Published
- 2018
- Full Text
- View/download PDF
17. A new modified weak Galerkin finite element scheme for solving the stationary Stokes equations.
- Author
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Tian, Tian, Zhai, Qilong, and Zhang, Ran
- Subjects
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DEGREES of freedom , *GALERKIN methods , *STOKES equations , *NUMERICAL analysis , *STOCHASTIC convergence - Abstract
In this paper, a modified weak Galerkin method is proposed for the Stokes problem. The numerical scheme is based on a novel variational form of the Stokes problem. The degree of freedoms in the modified weak Galerkin method is less than that in the original weak Galerkin method, while the accuracy stays the same. In this paper, the optimal convergence orders are given and some numerical experiments are presented to verify the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
18. Convergence conditions and numerical comparison of global optimization methods based on dimensionality reduction schemes.
- Author
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Grishagin, Vladimir, Israfilov, Ruslan, and Sergeyev, Yaroslav
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STOCHASTIC convergence , *NUMERICAL analysis , *GLOBAL optimization , *COMPUTER algorithms , *MATHEMATICAL domains - Abstract
This paper is devoted to numerical global optimization algorithms applying several ideas to reduce the problem dimension. Two approaches to the dimensionality reduction are considered. The first one is based on the nested optimization scheme that reduces the multidimensional problem to a family of one-dimensional subproblems connected in a recursive way. The second approach as a reduction scheme uses Peano-type space-filling curves mapping multidimensional domains onto one-dimensional intervals. In the frameworks of both the approaches, several univariate algorithms belonging to the characteristical class of optimization techniques are used for carrying out the one-dimensional optimization. Theoretical part of the paper contains a substantiation of global convergence for the considered methods. The efficiency of the compared global search methods is evaluated experimentally on the well-known GKLS test class generator used broadly for testing global optimization algorithms. Results for representative problem sets of different dimensions demonstrate a convincing advantage of the adaptive nested optimization scheme with respect to other tested methods. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
19. Equivalent conditions of complete convergence and complete moment convergence for END random variables.
- Author
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Shen, Aiting, Yao, Mei, and Xiao, Benqiong
- Subjects
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STOCHASTIC convergence , *RANDOM variables , *MATHEMATICAL equivalence , *NUMERICAL analysis , *EXTRAPOLATION - Abstract
In this paper, the complete convergence and the complete moment convergence for extended negatively dependent (END, in short) random variables without identical distribution are investigated. Under some suitable conditions, the equivalence between the moment of random variables and the complete convergence is established. In addition, the equivalence between the moment of random variables and the complete moment convergence is also proved. As applications, the Marcinkiewicz-Zygmund-type strong law of large numbers and the Baum-Katz-type result for END random variables are established. The results obtained in this paper extend the corresponding ones for independent random variables and some dependent random variables. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
20. A new fast direct solver for the boundary element method.
- Author
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Huang, S. and Liu, Y.
- Subjects
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BOUNDARY element methods , *MATRICES (Mathematics) , *COMPUTATIONAL complexity , *NUMERICAL analysis , *STOCHASTIC convergence , *LINEAR equations - Abstract
A new fast direct linear equation solver for the boundary element method (BEM) is presented in this paper. The idea of the new fast direct solver stems from the concept of the hierarchical off-diagonal low-rank matrix. The hierarchical off-diagonal low-rank matrix can be decomposed into the multiplication of several diagonal block matrices. The inverse of the hierarchical off-diagonal low-rank matrix can be calculated efficiently with the Sherman-Morrison-Woodbury formula. In this paper, a more general and efficient approach to approximate the coefficient matrix of the BEM with the hierarchical off-diagonal low-rank matrix is proposed. Compared to the current fast direct solver based on the hierarchical off-diagonal low-rank matrix, the proposed method is suitable for solving general 3-D boundary element models. Several numerical examples of 3-D potential problems with the total number of unknowns up to above 200,000 are presented. The results show that the new fast direct solver can be applied to solve large 3-D BEM models accurately and with better efficiency compared with the conventional BEM. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
21. Linear Stochastic Approximation Algorithms and Group Consensus Over Random Signed Networks.
- Author
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Chen, Ge, Duan, Xiaoming, Mei, Wenjun, and Bullo, Francesco
- Subjects
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STOCHASTIC convergence , *NUMERICAL analysis , *MULTIAGENT systems , *ALGORITHMS , *LINEAR algebra - Abstract
This paper studies linear stochastic approximation (SA) algorithms and their application to multiagent systems in engineering and sociology. As main contribution, we provide necessary and sufficient conditions for convergence of linear SA algorithms to a deterministic or random final vector. We also characterize the system convergence rate, when the system is convergent. Moreover, differing from non-negative gain functions in traditional SA algorithms, this paper considers also the case when the gain functions are allowed to take arbitrary real numbers. Using our general treatment, we provide necessary and sufficient conditions to reach consensus and group consensus for first-order discrete-time multiagent system over random signed networks and with state-dependent noise. Finally, we extend our results to the setting of multidimensional linear SA algorithms and characterize the behavior of the multidimensional Friedkin–Johnsen model over random interaction networks. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
22. A New Modification of Conjugate Gradient Method with Global Convergence Properties.
- Author
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Dawahdeh, Mahmoud, Mamat, Mustafa, Alhawarat, Ahmad, and Rivaie, Mohd
- Subjects
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CONJUGATE gradient methods , *MATHEMATICAL optimization , *STOCHASTIC convergence , *ITERATIVE methods (Mathematics) , *NUMERICAL analysis - Abstract
The most well-known technique or method in unconstrained optimization is the conjugate gradient method. It is used to get the greatest solution for the unconstrained optimization problems. This method is used in many fields especially, computer science, and engineering due to its convergence speed, simplicity, and the low memory requirements. A new modified conjugate gradient method is presented in this paper. This method is proved with the strong Wolfe-Powell (SWP) line search that it possesses sufficient descent property, and is globally convergent. Numerical results for a set of 141 test problems show the outperformance of the new proposed formula comparing with other methods using the same line search. The performance of this method is more efficient and better than the others. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
23. A VARIANT OF CHEBYSHEV'S METHOD.
- Author
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Sri, Ramadevi, Vatti, V. B. Kumar, and Mylapalli, M. S. Kumar
- Subjects
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CHEBYSHEV systems , *ITERATIVE methods (Mathematics) , *NONLINEAR equations , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
In this paper, we present a new two step iterative method to solve the nonlinear equation f (x) - 0 and discuss about its convergence. Few numerical examples are considered to show the efficiency of the new method in comparison with the other methods considered in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
24. Finite-Time Bounded Synchronization of the Growing Complex Network with Nondelayed and Delayed Coupling.
- Author
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Xu, Yuhua, Zhang, Jincheng, Zhou, Wuneng, and Tong, Dongbing
- Subjects
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FINITE element method , *SYNCHRONIZATION , *DISCRETE systems , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
The objective of this paper is to discuss finite-time bounded synchronization for a class of the growing complex network with nondelayed and delayed coupling. In order to realize finite-time synchronization of complex networks, a new finite-time stable theory is proposed; effective criteria are developed to realize synchronization of the growing complex dynamical network in finite time. Moreover, the error of two growing networks is bounded simultaneously in the process of finite-time synchronization. Finally, some numerical examples are provided to verify the theoretical results established in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
25. A Nonmonotone Projection Method for Constrained System of Nonlinear Equations.
- Author
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Dang, Yazheng and Liu, Wenwen
- Subjects
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NONMONOTONIC logic , *NONLINEAR equations , *STOCHASTIC convergence , *COMPUTER algorithms , *NUMERICAL analysis - Abstract
This paper deals with the nonmonotone projection algorithm for constrained nonlinear equations. For some starting points, the previous projection algorithms for the problem may encounter slow convergence which is related to the monotone behavior of the iterative sequence as well as the iterative direction. To circumvent this situation, we adopt the nonmonotone technique introduced by Dang to develop a nonmonotone projection algorithm. After constructing the nonmonotone projection algorithm, we show its convergence under some suitable condition. Preliminary numerical experiment is reported at the end of this paper, from which we can see that the algorithm we propose converges more quickly than that of the usual projection algorithm for some starting points. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
26. Convergence and error estimates of viscosity-splitting finite-element schemes for the primitive equations.
- Author
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Guillén-González, F. and Redondo-Neble, M.V.
- Subjects
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STOCHASTIC convergence , *ERROR analysis in mathematics , *FINITE element method , *NUMERICAL analysis , *NONLINEAR differential equations - Abstract
This paper is devoted to the numerical analysis of a first order fractional-step time-scheme (via decomposition of the viscosity) and “inf-sup” stable finite-element spatial approximations applied to the Primitive Equations of the Ocean. The aim of the paper is twofold. First, we prove that the scheme is unconditionally stable and convergent towards weak solutions of the Primitive Equations. Second, optimal error estimates for velocity and pressure are provided of order O ( k + h l ) for l = 1 or l = 2 considering either first or second order finite-element approximations ( k and h being the time step and the mesh size, respectively). In both cases, these error estimates are obtained under the same constraint k ≤ C h 2 . [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
27. Consensus analysis of directed multi-agent networks with singular configurations.
- Author
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Wu, Yonghong and Guan, Zhi-Hong
- Subjects
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ARTIFICIAL neural networks , *CONTINUOUS time systems , *MULTIAGENT systems , *NUMERICAL analysis , *DERIVATIVES (Mathematics) , *STOCHASTIC convergence - Abstract
This paper discusses the continuous-time consensus problems for directed multi-agent networks under certain coupling or control. Since agents are driven not only by their neighbors' states but also by their derivatives in many realistic situations, dynamical networks described by singular systems are appropriate to study. Consensus problems for such multi-agent networks are considered when the agents communicate in the presence or absence of time delays. The maximum tolerated time-delay is obtained when the multi-agent network asymptotically reaches consensus. The results of this paper indicate that such multi-agent networks can achieve consensus with a demanding convergent speed through agents' interactions. Numerical examples are given to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
28. Convergence analysis of iterative methods for nonsmooth convex optimization over fixed point sets of quasi-nonexpansive mappings.
- Author
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Iiduka, Hideaki
- Subjects
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STOCHASTIC convergence , *NONSMOOTH optimization , *MATHEMATICAL functions , *CONVEX functions , *NUMERICAL analysis - Abstract
This paper considers a networked system with a finite number of users and supposes that each user tries to minimize its own private objective function over its own private constraint set. It is assumed that each user's constraint set can be expressed as a fixed point set of a certain quasi-nonexpansive mapping. This enables us to consider the case in which the projection onto the constraint set cannot be computed efficiently. This paper proposes two methods for solving the problem of minimizing the sum of their nondifferentiable, convex objective functions over the intersection of their fixed point sets of quasi-nonexpansive mappings in a real Hilbert space. One method is a parallel subgradient method that can be implemented under the assumption that each user can communicate with other users. The other is an incremental subgradient method that can be implemented under the assumption that each user can communicate with its neighbors. Investigation of the two methods' convergence properties for a constant step size reveals that, with a small constant step size, they approximate a solution to the problem. Consideration of the case in which the step-size sequence is diminishing demonstrates that the sequence generated by each of the two methods strongly converges to the solution to the problem under certain assumptions. Convergence rate analysis of the two methods under certain situations is provided to illustrate the two methods' efficiency. This paper also discusses nonsmooth convex optimization over sublevel sets of convex functions and provides numerical comparisons that demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
29. A family of second-order methods for convex $$\ell _1$$ -regularized optimization.
- Author
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Byrd, Richard, Chin, Gillian, Nocedal, Jorge, and Oztoprak, Figen
- Subjects
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CONVEX functions , *MATHEMATICAL optimization , *STOCHASTIC convergence , *NUMERICAL analysis , *ESTIMATES - Abstract
This paper is concerned with the minimization of an objective that is the sum of a convex function f and an $$\ell _1$$ regularization term. Our interest is in active-set methods that incorporate second-order information about the function f to accelerate convergence. We describe a semismooth Newton framework that can be used to generate a variety of second-order methods, including block active set methods, orthant-based methods and a second-order iterative soft-thresholding method. The paper proposes a new active set method that performs multiple changes in the active manifold estimate at every iteration, and employs a mechanism for correcting these estimates, when needed. This corrective mechanism is also evaluated in an orthant-based method. Numerical tests comparing the performance of three active set methods are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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30. Theoretical and numerical analysis of a non-local dispersion model for light interaction with metallic nanostructures.
- Author
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Huang, Yunqing, Li, Jichun, and Yang, Wei
- Subjects
- *
NUMERICAL analysis , *NANOSTRUCTURES , *TIME-domain analysis , *MAXWELL equations , *PARTIAL differential equations , *FINITE element method , *STOCHASTIC convergence - Abstract
In this paper, we discuss the time-domain Maxwell’s equations coupled to another partial differential equation, which arises from modeling of light and structure interaction at the nanoscale. One major contribution of this paper is that the well-posedness is rigorously justified for the first time. Then we propose a fully-discrete finite element method to solve this model. It is interesting to note that we need use curl conforming, divergence conforming, and L 2 finite elements for this model. Numerical stability and optimal error estimate of the scheme are proved. Numerical results justifying our theoretical convergence rate are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
31. A general implicit direct forcing immersed boundary method for rigid particles.
- Author
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Tschisgale, Silvio, Kempe, Tobias, and Fröhlich, Jochen
- Subjects
- *
PARTICULATE matter , *MULTIPHASE flow , *NUMERICAL analysis , *STOCHASTIC convergence , *FLUID dynamics - Abstract
This paper presents a new immersed boundary method for rigid particles of arbitrary shape and arbitrary density, which can be exactly zero. Especially in the latter case, the coupling of the fluid and the solid part requires special numerical techniques to obtain stability. Exploiting the direct forcing approach an algorithm with strong coupling between fluid and particles is developed, which is exempt from any global iteration between the fluid part and the solid part within a single time step. Starting point is a previously proposed method restricted to spherical particles [37]. It is proved that the coupling concept can be generalized to rigid particles with complex shapes, which is not obvious a priori . The extension requires various non-trivial methodological extensions, especially with respect to the angular motion of the particle. Using analytical techniques it is demonstrated that the implicit treatment of the coupling force in the equations of motion results in additional terms related to a surrounding numerical fluid layer. As a main improvement over other non-iterative methods the proposed scheme is unconditionally stable for the entire range of density ratios and particle shapes allowing large time steps, with Courant numbers around unity. To date, no other non-iterative coupling approach offers such a generality regarding fluid-particle interactions. In addition to the detailed description of the underlying methodology and its differences from other methods, the paper provides all modifications required to improve immersed boundary methods for particulate flows from weak to strong coupling. Furthermore, an extensive validation of the scheme for particles of different density ratios and shapes is presented. The accuracy of the method as well as the convergence behavior are assessed by systematic studies. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
32. Convergence Orders in Length Estimation for Exponential Parameterization and ε-Uniform Samplings.
- Author
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Kozera, R., Noakes, L., and Szmielew, P.
- Subjects
- *
STOCHASTIC convergence , *ESTIMATION theory , *EXPONENTIAL functions , *PARAMETERIZATION , *STATISTICAL sampling - Abstract
We discuss the problem of approximating the length of a curve γ in arbitrary euclidean space En, from ε -uniformly sampled reduced data Qm ={qi}mi =0, where γ (ti)=qi. The interpolation knots {ti}mi=0 are assumed in this paper to be unknown (the so-called non-parametric interpolation). We fit Qm with the piecewise-quadratic interpolant γ2 based on the so-called exponential parameterization (depending on parameter λ ∈ [0,1]) which estimates the missing knots {ti}m i=0 ≈ {ti}mi=0. The asymptotic orders βε (λ) for length estimation d(γ) ≈ d(y2) in case of λ = 0 (uniformly guessed knots) read as βε (0) = min{4,4ε} (for ε > 0) - see [1]. On the other hand λ = 1 (cumulative chords) yields βε (1) = min{4,3+ε} (see [2]). This paper establishes a more general result holding for all λ ∈ [0,1] and ε -uniform samplings - see Th. 5. More specifically the respective convergence orders amount to βε (λ) = min{4,4ε} for λ ∈ [0,1). Consequently βε (λ) are independent of λ ∈ [0,1) and the discontinuity in asymptotic orders βε (λ) at λ = 1 occurs, for all ε ∈ (0,1). The full proof of Th. 5 is presented in ICNAAM'14 post-conference journal publication together with more exhaustive relevant experimentation. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
33. A Numerical Approach for Solving of Fractional Emden-Fowler Type Equations.
- Author
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Rebenda, Josef and Šmarda, Zdeněk
- Subjects
- *
FRACTIONAL calculus , *NUMERICAL analysis , *PROBLEM solving , *INITIAL value problems , *STOCHASTIC convergence - Abstract
In the paper, we utilize the fractional differential transformation (FDT) to solving singular initial value problem of fractional Emden-Fowler type differential equations. The solutions of our model equations are calculated in the form of convergent series with fast computable components. The numerical results show that the approach is correct, accurate and easy to implement when applied to fractional differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
34. Parametric Quintic Spline Approach for Two-dimensional Fractional Sub-diffusion Equation.
- Author
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Xuhao Li and Wong, Patricia J. Y.
- Subjects
- *
FRACTIONAL calculus , *SPLINE theory , *STOCHASTIC convergence , *NUMERICAL analysis , *HEAT equation - Abstract
In this paper, we shall tackle the numerical treatment of two-dimensional fractional sub-diffusion equations using parametric quintic spline. It is shown that this numerical scheme is solvable, stable and convergent with high accuracy which improves some earlier work. Finally, we carry out an experiment to demonstrate the efficiency of our numerical scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
35. Numerical Solution of Fourth-order Fractional Diffusion Wave Model.
- Author
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Xuhao Li and Wong, Patricia J. Y.
- Subjects
- *
STOCHASTIC convergence , *NUMERICAL analysis , *SIMULATION methods & models , *FRACTIONAL calculus , *STABILITY theory - Abstract
In this paper, we shall construct a new numerical scheme for fourth-order fractional diffusion wave model. The solvability, stability and convergence of proposed method are established in l2 norm and it is shown that the numerical scheme improves the earlier work done. Simulation is carried out to verity the accuracy and efficiency of the numerical scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
36. Convergence, error estimation and adaptivity in non-elliptic coupled electro-mechanical problems.
- Author
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Zboiński, Grzegorz
- Subjects
- *
MONOTONIC functions , *REAL variables , *ERROR analysis in mathematics , *NUMERICAL analysis , *STOCHASTIC convergence - Abstract
This paper presents the influence of the lack of ellipticity property on the solution convergence of the coupled electro-mechanical problems. This influence consists in the non-monotonic convergence which can hardly be described analytically. We recall our previous unpublished research where we demonstrate that the non-monotonicity depends very much on the energy level of the two component parts of the energy related to the coupled fields of mechanical and electric character. We further investigate the influence of this non-monotonic character of the convergence on the error estimation via equilibrated residual method. We also assess the influence of such convergence on the three-step error-controlled adaptive algorithms. We indicate the methods of practical overcoming the mentioned problems related to the lack of ellipticity. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
37. An incremental pressure correction finite element method for the time-dependent Oldroyd flows.
- Author
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Liu, Cui and Si, Zhiyong
- Subjects
- *
FINITE element method , *DISCRETE systems , *ERROR analysis in mathematics , *STOCHASTIC convergence , *STABILITY theory - Abstract
Abstract In this paper, we present an incremental pressure correction finite element method for the time-dependent Oldroyd flows. This method is a fully discrete projection method. As we all know, most projection methods have been studied without space discretization. Then the ensuing analysis may not extend to this case. We also give the stability analysis and the optimal error analysis. The analysis is based on a time discrete error and a spatial discrete error. In order to show the effectiveness of the method, we also present some numerical results. The numerical results confirm our analysis and show clearly the stability and optimal convergence of the incremental pressure correction finite element method for the time-dependent Oldroyd flows. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
38. ANALYSIS OF HIGHER ORDER DIFFERENCE METHOD FOR A PSEUDO-PARABOLIC EQUATION WITH DELAY.
- Author
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AMIRALI, ILHAME
- Subjects
- *
DELAY differential equations , *INITIAL value problems , *DERIVATIVES (Mathematics) , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
In this paper, the author considers the one dimensional initial-boundary problem for a pseudo-parabolic equation with time delay in second spatial derivative. To solve this problem numerically, the author constructs higher order difference method and obtain the error estimate for its solution. Based on the method of energy estimates the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Some numerical examples illustrate the convergence and effectiveness of the numerical method. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
39. Parameter-uniform convergence of a numerical method for a coupled system of singularly perturbed semilinear reaction–diffusion equations with boundary and interior layers.
- Author
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Rao, S. Chandra Sekhara and Chawla, Sheetal
- Subjects
- *
REACTION-diffusion equations , *BOUNDARY layer equations , *COUPLED mode theory (Wave-motion) , *NUMERICAL analysis , *STOCHASTIC convergence , *PERTURBATION theory - Abstract
Abstract In this paper, we consider a coupled system of m (≥ 2) singularly perturbed semilinear reaction–diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain. The diffusion term of each equation is multiplied by small singular perturbation parameter, but these parameters are assumed to be different in magnitude. A numerical method is constructed on a variant of Shishkin mesh. The approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters. Numerical results are in support of the theoretical results. Highlights • A coupled system of m (≥ 2) singularly perturbed semilinear reaction-diffusion equations is considered. • The source term is having a discontinuity at a point in the interior of the domain. • The diffusion term of each equation is multiplied by a small perturbation parameter. • All the perturbation parameters are different in magnitude. • The constructed numerical method has almost second parameter uniform convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
40. Numerical efficiency of some exponential methods for an advection-diffusion equation.
- Author
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Macías-Díaz, Jorge Eduardo and İnan, Bilge
- Subjects
- *
ADVECTION-diffusion equations , *NUMERICAL analysis , *STOCHASTIC convergence , *APPROXIMATION theory , *BURGERS' equation - Abstract
In this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Padé approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value problems for different Reynolds numbers show a good agreement between them. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
41. One-leg methods for nonlinear stiff fractional differential equations with Caputo derivatives.
- Author
-
Zhou, Yongtao and Zhang, Chengjian
- Subjects
- *
NONLINEAR differential equations , *FRACTIONAL differential equations , *STOCHASTIC convergence , *DERIVATIVES (Mathematics) , *NUMERICAL analysis - Abstract
Highlights • A type of extended one-leg methods are constructed for a class of nonlinear stiff fractional differential equations. • Under some suitable conditions, the extended one-leg methods are proved to be stable and convergent of order min { p , 2 − γ }. • Several interesting numerical examples are presented to illustrate the computational efficiency and accuracy of the extended one-leg methods. Abstract This paper is concerned with numerical solutions for a class of nonlinear stiff fractional differential equations (SFDEs). By combining the underlying one-leg methods with piecewise linear interpolation, a type of extended one-leg methods for nonlinear SFDEs with γ -order (0 < γ < 1) Caputo derivatives are constructed. It is proved under some suitable conditions that the extended one-leg methods are stable and convergent of order min { p , 2 − γ } , where p is the consistency order of the underlying one-leg methods. Several numerical examples are given to illustrate the computational efficiency and accuracy of the methods. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
42. A preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problem.
- Author
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Dai, Ping-Fan, Li, Jicheng, Bai, Jianchao, and Qiu, Jinming
- Subjects
- *
ITERATIVE methods (Mathematics) , *LINEAR complementarity problem , *STOCHASTIC convergence , *NUMERICAL analysis , *PROBLEM solving - Abstract
Abstract In this paper, a preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problems associated with an M -matrix is proposed. The convergence analysis of the presented method is given. In particular, we provide a comparison theorem between preconditioned two-step modulus-based Gauss–Seidel (PTMGS) iteration method and two-step modulus-based Gauss–Seidel (TMGS) iteration method, which shows that PTMGS method improves the convergence rate of original TMGS method for linear complementarity problem. Numerical tested examples are used to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
43. Modified alternately linearized implicit iteration method for M-matrix algebraic Riccati equations.
- Author
-
Guan, Jinrui
- Subjects
- *
LINEAR systems , *ITERATIVE methods (Mathematics) , *RICCATI equation , *MATRICES (Mathematics) , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
Abstract Research on the theories and efficient numerical methods of M-matrix algebraic Riccati equation (MARE) has become a hot topic in recent years. In this paper, we consider numerical solution of M-matrix algebraic Riccati equation and propose a modified alternately linearized implicit iteration method (MALI) for computing the minimal nonnegative solution of MARE. Convergence of the MALI method is proved by choosing proper parameters for the nonsingular M-matrix or irreducible singular M-matrix. Theoretical analysis and numerical experiments show that the MALI method is effective and efficient in some cases. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
44. A new conjugate gradient method based on Quasi-Newton equation for unconstrained optimization.
- Author
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Li, Xiangli, Shi, Juanjuan, Dong, Xiaoliang, and Yu, Jianglan
- Subjects
- *
CONJUGATE gradient methods , *MATHEMATICAL optimization , *QUASI-Newton methods , *STOCHASTIC convergence , *NUMERICAL analysis , *SPECTRAL geometry - Abstract
Abstract The spectral conjugate gradient method is an effective method for large-scale unconstrained optimization problems. In this paper, based on Quasi-Newton direction and Quasi-Newton equation, a new spectral conjugate gradient method is proposed. This method is motivated by the three-term modified Polak-Ribiyre-Polyak (PRP) method and spectral parameters. The global convergence of algorithm is proved for general functions under a strong Wolfe line search. Numerical results show that the new algorithm is superior to the three-term modified PRP method. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
45. Implicit numerical solutions to neutral-type stochastic systems with superlinearly growing coefficients.
- Author
-
Zhou, Shaobo and Jin, Hai
- Subjects
- *
STOCHASTIC convergence , *APPROXIMATION theory , *NUMERICAL analysis , *STOCHASTIC differential equations , *EULER characteristic - Abstract
Abstract In this paper, our main aim is to investigate the stability and strong convergence of an implicit numerical approximations for neutral-type stochastic differential equations with superlinearly growing coefficients. After providing moment boundedness and exponential stability for the exact solutions, we show that the backward Euler–Maruyama numerical method preserves stability and boundedness of moments, and the numerical approximations converge strongly to the true solutions for sufficiently small step size. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
46. A detail preserving variational model for image Retinex.
- Author
-
Gu, Zhihao, Li, Fang, and Lv, Xiao-Guang
- Subjects
- *
PROBLEM solving , *MATHEMATICAL variables , *STOCHASTIC convergence , *MATHEMATICAL regularization , *DECOMPOSITION method , *NUMERICAL analysis - Abstract
Highlights • A new variational model for image retinex is proposed. • The existence of minimizer for the variational model is proved. • An efficient algorithm is derived to solve the model by adding auxiliary variables. • The convergence of the numerical algorithm is proved under some assumptions. • The proposed method can be extended to other regularization and fidelity terms. Abstract In this paper, we propose a detail preserving variational model for Retinex to simultaneously estimate the illumination and the reflectance from an observed image. Most previous models use the log-transform as pretreatment which results in loss of details in reflectance. From this observation, a detail preserving variational method is proposed for better decomposition. Different from the log-transform based models, the proposed model performs the decomposition directly in the image domain. Mathematically, we prove the existence of a solution for the proposed model. Numerically, we derive an efficient iterative algorithm by utilizing alternating direction method of multipliers (ADMM) method. Experimental results demonstrate the effectiveness of the proposed method. Compared with other closely related Retinex methods, the proposed method achieves competitive results on both subjective and objective assessments. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
47. Unconditional L∞ convergence of a conservative compact finite difference scheme for the N-coupled Schrödinger–Boussinesq equations.
- Author
-
Liao, Feng, Zhang, Luming, and Wang, Tingchun
- Subjects
- *
STOCHASTIC convergence , *FINITE differences , *SCHRODINGER equation , *BOUSSINESQ equations , *NUMERICAL analysis - Abstract
Abstract In this paper, a conservative compact finite difference scheme is presented for solving the N-coupled nonlinear Schrödinger–Boussinesq equations. By using the discrete energy method, it is proved that our scheme is unconditionally convergent in the maximum norm and the convergent rate is at O (τ 2 + h 4) with time step τ and mesh size h. Numerical results including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of the method and to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
48. Stability and superconvergence of efficient MAC schemes for fractional Stokes equation on non-uniform grids.
- Author
-
Li, Xiaoli, Rui, Hongxing, and Chen, Shuangshuang
- Subjects
- *
STOCHASTIC convergence , *STOKES equations , *GRIDS (Cartography) , *CAPUTO fractional derivatives , *NUMERICAL analysis - Abstract
Abstract In this paper, the two MAC schemes are introduced and analyzed to solve the time fractional Stokes equation on non-uniform grids. One is the standard MAC scheme and another is the efficient MAC scheme, where the fast evaluation of the Caputo fractional derivative is used. The stability results are derived. We obtain the second order superconvergence in discrete L 2 norm for both velocity and pressure. We also obtain the second order superconvergence for some terms of the H 1 norm of the velocity on non-uniform grids. Besides, the efficient algorithm for the evaluation of the Caputo fractional derivative is used to save the storage and computation cost greatly. Finally, some numerical experiments are presented to show the efficiency and accuracy of MAC schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
49. New Uzawa-type method for nonsymmetric saddle point problems.
- Author
-
Shu-Xin Miao and Li, Juan
- Subjects
- *
SADDLEPOINT approximations , *ITERATIVE methods (Mathematics) , *STOCHASTIC convergence , *NUMERICAL analysis , *STOCHASTIC processes - Abstract
In this paper, based on the Hermitian and skew-Hermitian splitting of the non-Hermitian positive definite (1, 1)-block of the saddle point matrix, a new Uzawa-type iteration method is proposed for solving a class of nonsymmetric saddle point problems. The convergence properties of this iteration method are analyzed. Numerical results verify the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2019
50. On a second order scheme for space fractional diffusion equations with variable coefficients.
- Author
-
Vong, Seakweng and Lyu, Pin
- Subjects
- *
FRACTIONAL differential equations , *COEFFICIENTS (Statistics) , *MATHEMATICAL physics , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
Abstract We study a second order scheme for spatial fractional differential equations with variable coefficients. Previous results mainly concentrate on equations with diffusion coefficients that are proportional to each other. In this paper, by further study on the generating function of the discretization matrix, second order convergence of the scheme is proved for diffusion coefficients satisfying a certain condition but are not necessary to be proportional. The theoretical results are justified by numerical tests. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
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