438 results
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2. Numerical methods for static shallow shells lying over an obstacle
- Author
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Paolo Piersanti, Xiaoqin Shen, City University of Hong Kong (CityU), and Xi'an Jiaotong University (Xjtu)
- Subjects
Original Paper ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,010103 numerical & computational mathematics ,Bilinear form ,Half-space ,Obstacle problems · Elliptic variational inequalities ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Elliptic variational inequalities ,Non-conforming finite element method ,Enriching operator ,Obstacle problems ,Shallow shell ,Obstacle ,Theory of computation ,Convergence (routing) ,Obstacle problem ,0101 mathematics ,Nonconforming finite element method ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Mathematics - Abstract
In this paper a finite element analysis to approximate the solution of an obstacle problem for a static shallow shell confined in a half space is presented. First, we rigorously prove some estimates for a suitable enriching operator connecting Morley's triangle to Hsieh-Clough-Tocher triangle. Secondly, we establish an estimate for the approximate bilinear form associated with the problem under consideration. Finally, we conduct an error analysis and we prove the convergence of the proposed numerical scheme.
- Published
- 2020
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3. Waveform relaxation of partial differential equations
- Author
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Zhen Miao and Yao-Lin Jiang
- Subjects
Partial differential equation ,Applied Mathematics ,Numerical analysis ,Short paper ,Relaxation (iterative method) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Theory of computation ,Convergence (routing) ,Applied mathematics ,Waveform ,0101 mathematics ,Energy (signal processing) ,Mathematics - Abstract
This short paper concludes a general waveform relaxation (WR) method at the PDE level for semi-linear reaction-diffusion equations. For the case of multiple coupled PDE(s), new Jacobi WR and Gauss-Seidel WR are provided to accelerate the convergence result of classical WR. The convergence conditions are proved based on energy estimate. Numerical experiments are demonstrated with several WR methods in parallel to verify the effectiveness of the general WR method.
- Published
- 2018
4. A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues
- Author
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Sven-Erik Ekström and P. Vassalos
- Subjects
Beräkningsmatematik ,Applied Mathematics ,010102 general mathematics ,Generating function ,Order (ring theory) ,Asymptotic expansion ,Spectral analysis ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Function (mathematics) ,Type (model theory) ,01 natural sciences ,Toeplitz matrix ,Combinatorics ,Computational Mathematics ,Matrix (mathematics) ,Toeplitz matrices ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Structured matrices ,Eigenvalues and eigenvectors ,Mathematics - Abstract
It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.
- Published
- 2021
5. Stability analysis of the method of fundamental solutions with smooth closed pseudo-boundaries for Laplace’s equation: better pseudo-boundaries
- Author
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Li-Ping Zhang, Zi-Cai Li, Ming-Gong Lee, and Hung-Tsai Huang
- Subjects
Laplace's equation ,Polynomial ,Laplace transform ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Bounded function ,Applied mathematics ,Method of fundamental solutions ,0101 mathematics ,Condition number ,Circulant matrix ,Mathematics - Abstract
Consider Laplace’s equation in a bounded simply-connected domain S, and use the method of fundamental solutions (MFS). The error and stability analysis is made for circular/elliptic pseudo-boundaries in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020), and polynomial convergence rates and exponential growth rates of the condition number (Cond) are obtained. General pseudo-boundaries are suggested for more complicated solution domains in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020, Section 5). Since the ill-conditioning is severe, the success in computation by the MFS mainly depends on stability. This paper is devoted to stability analysis for smooth closed pseudo-boundaries of source nodes. Bounds of the Cond are derived, and exponential growth rates are also obtained. This paper is the first time to explore stability analysis of the MFS for non-circular/non-elliptic pseudo-boundaries. Circulant matrices are often employed for stability analysis of the MFS; but the stability analysis in this paper is explored based on new techniques without using circulant matrices as in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020). To pursue better pseudo-boundaries, the sensitivity index is proposed from growth/convergence rates of stability via accuracy. Better pseudo-boundaries in the MFS can be found by trial computations, to develop the study in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020) for the selection of pseudo-boundaries. For highly smooth and singular solutions, better pseudo-boundaries are different; an analysis of the sensitivity index is explored. Circular/elliptic pseudo-boundaries are optimal for highly smooth solutions, but not for singular solutions. In this paper, amoeba-like domains are chosen in computation. Several useful types of pseudo-boundaries are developed and their algorithms are simple without using nonlinear solutions. For singular solutions, numerical comparisons are made for different pseudo-boundaries via the sensitivity index.
- Published
- 2021
6. Weak Galerkin finite element methods with or without stabilizers
- Author
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Xiaoshen Wang, Xiu Ye, and Shangyou Zhang
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010101 applied mathematics ,Applied Mathematics ,Numerical analysis ,Convergence (routing) ,Theory of computation ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Algebra over a field ,Galerkin method ,01 natural sciences ,Finite element method ,Mathematics - Abstract
The purpose of this paper is to investigate the connections between the weak Galerkin (WG) methods with and without stabilizers. The choices of stabilizers directly affect the convergence rates of the corresponding WG methods in general. However, we observed that the convergence rates are independent of the choices of stabilizers for these WG elements with stabilizers being optional. In this paper, we will verify such phenomena theoretically as well as numerically.
- Published
- 2021
7. Highly efficient schemes for time-fractional Allen-Cahn equation using extended SAV approach
- Author
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Chuanju Xu, Hongyi Zhu, Dianming Hou, Institut de Mécanique et d'Ingénierie (I2M), Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Arts et Métiers Sciences et Technologies, and HESAM Université (HESAM)-HESAM Université (HESAM)
- Subjects
Discretization ,Applied Mathematics ,Numerical analysis ,Scalar (physics) ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,[SPI.MAT]Engineering Sciences [physics]/Materials ,010101 applied mathematics ,Nonlinear system ,Theory of computation ,FOS: Mathematics ,Order (group theory) ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Allen–Cahn equation ,Mathematics - Abstract
In this paper, we propose and analyze high-order efficient schemes for the time-fractional Allen-Cahn equation. The proposed schemes are based on the L1 discretization for the time-fractional derivative and the extended scalar auxiliary variable (SAV) approach developed very recently to deal with the nonlinear terms in the equation. The main contributions of the paper consist of (1) constructing first- and higher order unconditionally stable schemes for different mesh types, and proving the unconditional stability of the constructed schemes for the uniform mesh; (2) carrying out numerical experiments to verify the efficiency of the schemes and to investigate the coarsening dynamics governed by the time-fractional Allen-Cahn equation. In particular, the influence of the fractional order on the coarsening behavior is carefully examined. Our numerical evidence shows that the proposed schemes are more robust than the existing methods, and their efficiency is less restricted to particular forms of the nonlinear potentials.
- Published
- 2021
8. A nonnegativity preserving algorithm for multilinear systems with nonsingular ${\mathcal M}$-tensors
- Author
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Hongjin He, Guanglu Zhou, Chen Ling, and Xueli Bai
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Sequence ,Multilinear map ,Applied Mathematics ,Numerical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,System of linear equations ,01 natural sciences ,law.invention ,010101 applied mathematics ,Invertible matrix ,law ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Theory of computation ,Tensor ,0101 mathematics ,Algorithm ,Mathematics ,Numerical partial differential equations - Abstract
This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the multilinear system under consideration involves a nonsingular ${\mathscr{M}}$ -tensor and a nonnegative right-hand side vector, it may have multiple nonnegative solutions. In this paper, we propose an algorithm which can always preserve the nonnegativity of solutions. Theoretically, we show that the sequence generated by the proposed algorithm is a nonnegative componentwise nonincreasing sequence and converges to a nonnegative solution of the system. Numerical results further support the novelty of the proposed method.
- Published
- 2020
9. An implicit-explicit preconditioned direct method for pricing options under regime-switching tempered fractional partial differential models
- Author
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Xu Chen, Wenfei Wang, Siu-Long Lei, and Deng Ding
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Partial differential equation ,Direct sum ,Applied Mathematics ,Direct method ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Toeplitz matrix ,010101 applied mathematics ,Valuation of options ,Applied mathematics ,Partial derivative ,0101 mathematics ,Coefficient matrix ,Mathematics - Abstract
Recently, fractional partial differential equations have been widely applied in option pricing problems, which better explains many important empirical facts of financial markets, but rare paper considers the multi-state options pricing problem based on fractional diffusion models. Thus, multi-state European option pricing problem under regime-switching tempered fractional partial differential equation is considered in this paper. Due to the expensive computational cost caused by the implicit finite difference scheme, a novel implicit-explicit finite difference scheme has been developed with consistency, stability, and convergence guarantee. Since the resulting coefficient matrix equals to the direct sum of several Toeplitz matrices, a preconditioned direct method has been proposed with ${\mathcal O}(\bar {S}N\log N+\bar {S}^{2} N)$ operation cost on each time level with adaptability analysis, where $\bar {S}$ is the number of states and N is the number of grid points. Related numerical experiments including an empirical example have been presented to demonstrate the effectiveness and accuracy of the proposed numerical method.
- Published
- 2020
10. Continuous and discrete zeroing dynamics models using JMP function array and design formula for solving time-varying Sylvester-transpose matrix inequality
- Author
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Huanchang Huang, Xiao Liu, Min Yang, Yunong Zhang, and Yihong Ling
- Subjects
Discretization ,Truncation error (numerical integration) ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,010101 applied mathematics ,Matrix (mathematics) ,Transpose ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Mathematics ,Variable (mathematics) - Abstract
Zeroing dynamics (ZD) has shown great performance to solve various time-varying problems. In this paper, the problem of time-varying Sylvester-transpose matrix inequality is first investigated. Since it is difficult to solve a matrix inequality with a matrix variable and its transpose by traditional methods, this paper proposes a continuous ZD (CZD) model by employing ZD design formula and JMP function array to solve this challenging problem. Furthermore, for better implementation on digital computers, three discrete ZD (DZD) models are proposed by using three different discretization formulas with different precision, i.e., the Euler-forward formula, the 6-instant Zhang et al discretization (ZeaD) formula and the 7-instant ZeaD formula. What is more, theoretical truncation error analyses and numerical experiments substantiate the convergence, efficacy and superiority of the DZD models for solving time-varying Sylvester-transpose matrix inequality.
- Published
- 2020
11. On equivalence of three-parameter iterative methods for singular symmetric saddle-point problem
- Author
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M. Tzoumas and Apostolos Hadjidimos
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Iterative method ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Symmetric case ,01 natural sciences ,law.invention ,010101 applied mathematics ,Invertible matrix ,law ,Saddle point ,Theory of computation ,Applied mathematics ,0101 mathematics ,Equivalence (measure theory) ,Saddle ,Mathematics - Abstract
There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of convergence for the parameters are found, while in three of them, optimal parameters are determined, and in one of the latter, many more cases, than in all the others, are distinguished, analyzed, and studied. It turns out that two of the optimal parameters coincide making the optimal three-parameter methods be equivalent to the optimal two-parameter known ones. Our aim in this work is manifold: (i) to show that the iterative methods we present are equivalent, (ii) to slightly change some statements in one of the main papers, (iii) to complete the analysis in another one, (iv) to explain how the transition from any of the methods to the others is made, (v) to extend the iterative method to cover the singular symmetric case, and (vi) to present a number of numerical examples in support of our theory. It would be an omission not to mention that the main material which all researchers in the area have inspired from and used is based on the one of the most cited papers by Bai et al. (Numer. Math. 102:1–38, 2005).
- Published
- 2020
12. Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces
- Author
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Oluwatosin Temitope Mewomo, Timilehin Opeyemi Alakoya, and Adeolu Taiwo
- Subjects
Iterative and incremental development ,Applied Mathematics ,Regular polygon ,Banach space ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Monotone polygon ,Corollary ,Theory of computation ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.
- Published
- 2020
13. Superconvergence in H1-norm of a difference finite element method for the heat equation in a 3D spatial domain with almost-uniform mesh
- Author
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Ruijian He, Zhangxin Chen, and Xinlong Feng
- Subjects
Backward differentiation formula ,Computational complexity theory ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,010103 numerical & computational mathematics ,Superconvergence ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Norm (mathematics) ,Bounded function ,Heat equation ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose a novel difference finite element (DFE) method based on the P1-element for the 3D heat equation on a 3D bounded domain. One of the novel ideas of this paper is to use the second-order backward difference formula (BDF) combining DFE method to overcome the computational complexity of conventional finite element (FE) method for the high-dimensional parabolic problem. First, we design a fully discrete difference FE solution ${u^{n}_{h}}$ by the second-order backward difference formula in the temporal t-direction, the center difference scheme in the spatial z-direction, and the P1-element on a almost-uniform mesh Jh in the spatial (x, y)-direction. Next, the H1-stability of ${u_{h}^{n}}$ and the second-order H1-convergence of the interpolation post-processing function on ${u_{h}^{n}}$ with respect to u(tn) are provided. Finally, numerical tests are presented to show the second-order H1-convergence results of the proposed DFE method for the heat equation in a 3D spatial domain.
- Published
- 2020
14. Analysis and application of the interpolating element-free Galerkin method for extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling
- Author
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Mohammad Ilati
- Subjects
Partial differential equation ,Applied Mathematics ,Numerical analysis ,Finite difference ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Convergence (routing) ,Biharmonic equation ,Fisher–Kolmogorov equation ,Applied mathematics ,0101 mathematics ,Galerkin method ,Mathematics - Abstract
In this paper, the interpolating element-free Galerkin method is applied for solving the nonlinear biharmonic extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. At first, a finite difference formula is utilized for obtaining a time-discrete scheme. The unconditional stability and convergence of the time-discrete method are proved by the energy method. Then, we use the interpolating element-free Galerkin method to approximate the spatial derivatives. An error analysis of the interpolating element-free Galerkin method is proposed for this nonlinear equation. Moreover, this method is compared with some other meshless local weak-form techniques. The main aim of this paper is to show that the interpolating element-free Galerkin is a suitable technique for solving the nonlinear fourth-order partial differential equations especially extended Fisher–Kolmogorov equation. The numerical experiments confirm the analytical results and show the good efficiency of the interpolating element-free Galerkin method for solving this nonlinear biharmonic equation.
- Published
- 2019
15. Multiscale radial kernels with high-order generalized Strang-Fix conditions
- Author
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Wenwu Gao and Xuan Zhou
- Subjects
Polynomial ,Applied Mathematics ,Numerical analysis ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Convolution ,010101 applied mathematics ,Moment (mathematics) ,Linear form ,Theory of computation ,Applied mathematics ,0101 mathematics ,ComputingMethodologies_COMPUTERGRAPHICS ,Variable (mathematics) ,Mathematics - Abstract
The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders.
- Published
- 2019
16. An adaptive local discontinuous Galerkin method for nonlinear two-point boundary-value problems
- Author
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Mahboub Baccouch
- Subjects
Discretization ,Adaptive mesh refinement ,Applied Mathematics ,Estimator ,010103 numerical & computational mathematics ,Superconvergence ,01 natural sciences ,Prime (order theory) ,010101 applied mathematics ,Rate of convergence ,Discontinuous Galerkin method ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose an adaptive mesh refinement (AMR) strategy based on a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form $u^{\prime \prime }=f(x,u),\ x\in [a,b]$ subject to some suitable boundary conditions at the endpoint of the interval [a, b]. We first use the superconvergence results proved in the first part of this paper as reported by Baccouch (Numer. Algorithm. 79(3), 697–718 2018) to show that the significant parts of the local discretization errors are proportional to (p + 1)-degree Radau polynomials, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimators which are obtained by solving a local residual problem with no boundary conditions on each element. The proposed error estimates are efficient, reliable, and asymptotically exact. We prove that, for smooth solutions, the proposed a posteriori error estimates converge to the exact errors in the L2-norm with order of convergence p + 3/2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p ≥ 1. Several numerical results are presented to validate the theoretical results and to show the efficiency of the grid refinement strategy.
- Published
- 2019
17. Family weak conjugate gradient algorithms and their convergence analysis for nonconvex functions
- Author
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Zhou Sheng, Gonglin Yuan, and Xiaoliang Wang
- Subjects
Trust region ,Line search ,Applied Mathematics ,Numerical analysis ,Structure (category theory) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Simple (abstract algebra) ,Conjugate gradient method ,Convergence (routing) ,Theory of computation ,0101 mathematics ,Algorithm ,Mathematics - Abstract
It is well-known that conjugate gradient algorithms are widely applied in many practical fields, for instance, engineering problems and finance models, as they are straightforward and characterized by a simple structure and low storage. However, challenging problems remain, such as the convergence of the PRP algorithms for nonconvexity under an inexact line search, obtaining a sufficient descent for all conjugate gradient methods, and other theory properties regarding global convergence and the trust region feature for nonconvex functions. This paper studies family conjugate gradient formulas based on the six classic formulas, PRP, HS, CD, FR, LS, and DY, where the family conjugate gradient algorithms have better theory properties than those of the formulas by themselves. Furthermore, this technique of the presented conjugate gradient formulas can be extended to any two-term conjugate gradient formula. This paper designs family conjugate gradient algorithms for nonconvex functions, which have the following features without other conditions: (i) the sufficient descent property holds, (ii) the trust region feature is true, and (iii) the global convergence holds under normal assumptions. Numerical results show that the given conjugate gradient algorithms are competitive with those of normal methods.
- Published
- 2019
18. Interval methods of Adams-Bashforth type with variable step sizes
- Author
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Andrzej Marciniak and Malgorzata A. Jankowska
- Subjects
Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Exact solutions in general relativity ,Theory of computation ,Initial value problem ,Interval (graph theory) ,Applied mathematics ,0101 mathematics ,Constant (mathematics) ,Mathematics ,Variable (mathematics) ,Linear multistep method - Abstract
In a number of our previous papers, we have proposed interval versions of multistep methods (explicit and implicit), including interval predictor-corrector methods, in which the step size was constant. In this paper, we present interval versions of Adams-Bashforth methods with a possibility to change step sizes. This possibility can be used to obtain interval enclosures of the exact solution with a width given beforehand.
- Published
- 2019
19. A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method
- Author
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Sławomir Milewski
- Subjects
Discretization ,business.industry ,Applied Mathematics ,Numerical analysis ,Monte Carlo method ,010103 numerical & computational mathematics ,System of linear equations ,Random walk ,01 natural sciences ,010101 applied mathematics ,Software ,Applied mathematics ,Meshfree methods ,Boundary value problem ,0101 mathematics ,business ,Mathematics - Abstract
This paper is devoted to the development of an innovative Matlab software, dedicated to the numerical analysis of two-dimensional elliptic problems, by means of the probabilistic approach. This approach combines features of the Monte Carlo random walk method with discretization and approximation techniques, typical for meshless methods. It allows for determination of an approximate solution of elliptic equations at the specified point (or group of points), without a necessity to generate large system of equations for the entire problem domain. While the procedure is simple and fast, the final solution may suffer from both stochastic and discretization errors. The attached Matlab software is based on several original author’s concepts. It permits the use of arbitrarily irregular clouds of nodes, non-homogeneous right-hand side functions, mixed type of boundary conditions as well as variable material coefficients (of anisotropic materials). The paper is illustrated with results of analysis of selected elliptic problems, obtained by means of this software.
- Published
- 2019
20. Unconditional optimal error estimate of the projection/Lagrange-Galerkin finite element method for the Boussinesq equations
- Author
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Zhiyong Si, Zhang Tong, and Yanfang Lei
- Subjects
Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Error function ,Exact solutions in general relativity ,Convergence (routing) ,Projection method ,Applied mathematics ,0101 mathematics ,Constant (mathematics) ,Projection (set theory) ,Mathematics - Abstract
This paper provides an unconditional optimal convergence of a fractional-step method for solving the Boussinesq equations. In this method, the convection is treated by the Lagrange-Galerkin technique, whereas the diffusion and the incompressibility are treated by the projection method. There are lots of authors who worked on this method, and some authors gave the error estimate of this method. But, to our best knowledge, the error estimate for this method is under certain time-step restrictions. In this paper, we prove that the methods are stable almost unconditionally, i.e., when τ and h are smaller than a given constant. The basic idea of our analysis is splitting the error function into three terms, one term between the finite element solution and the projection, the other term between the projection and the time-discrete solution, the third term between the time-discrete solution and the exact solution, and giving the error estimates for each term respectively. Then, we obtain the optimal error estimates in L2 and H1-norm for the velocity and L2-norm for the pressure. In order to show the efficiency of our method, some numerical results are presented.
- Published
- 2019
21. Optimal parameter selections for a general Halpern iteration
- Author
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Tao Wu, Songnian He, Themistocles M. Rassias, and Yeol Je Cho
- Subjects
Sequence ,Applied Mathematics ,Numerical analysis ,Hilbert space ,Field (mathematics) ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Rate of convergence ,Variational inequality ,symbols ,0101 mathematics ,Real number ,Mathematics - Abstract
Let C be a closed affine subset of a real Hilbert space H and $T:C \rightarrow C$ be a nonexpansive mapping. In this paper, for any fixed u ∈ C, a general Halpern iteration process: $$\left\{\begin{array}{ll} x_{0} \in C,\\ x_{n + 1}=t_{n}u+(1-t_{n})Tx_{n},n\geq 0, \end{array}\right. $$ is considered for finding a fixed point of T nearest to u, where the parameter sequence {tn} is selected in the real number field, $\mathbb {R}$. The core problem to be addressed in this paper is to find the optimal parameter sequence so that this iteration process has the optimal convergence rate and to give some numerical results showing advantages of our algorithms. Also, we study the problem of selecting the optimal parameters for a general viscosity approximation method and apply the results obtained from this study to solve a class of variational inequalities.
- Published
- 2019
22. A bias-compensated fractional order normalized least mean square algorithm with noisy inputs
- Author
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Jianmei Shuai, Yiheng Wei, Weidi Yin, Songsong Cheng, and Yong Wang
- Subjects
Applied Mathematics ,Numerical analysis ,Stability (learning theory) ,010103 numerical & computational mathematics ,Variance (accounting) ,01 natural sciences ,Noise (electronics) ,010101 applied mathematics ,Least mean squares filter ,Convergence (routing) ,Theory of computation ,0101 mathematics ,Root-mean-square deviation ,Algorithm ,Mathematics - Abstract
This paper comes up with a stable bias-compensated fractional order normalized least mean square (BC-FONLMS) algorithm with noisy inputs. This kind of bias-compensated algorithm needs the estimation of input noise variance to avoid the bias caused by noisy inputs. Yet, existing algorithms either cause instability because of the method used to estimate input noise variance, or surmount the instability problems at the price of performance diminishment. This paper introduces fractional order calculus into LMS algorithm to be a new BC-FONLMS algorithm. Then, analyze the stability of the BC-FONLMS algorithm through probing the recursive equations of mean deviation (MD) and mean square deviation (MSD). On the basis of the stability analysis, methods to estimate input noise variance and to adjust step size are suggested to stabilize the algorithm and likewise to enhance the performance such as convergence speed and steady-state error. Numerical simulations are given at last, whose results show that the proposed BC-FONLMS algorithm performs well.
- Published
- 2018
23. Weighted and deflated global GMRES algorithms for solving large Sylvester matrix equations
- Author
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Gang Wu, Najmeh Azizi Zadeh, and Azita Tajaddini
- Subjects
Sylvester matrix ,Applied Mathematics ,Numerical analysis ,Linear system ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,Krylov subspace ,Computer Science::Numerical Analysis ,01 natural sciences ,Generalized minimal residual method ,Mathematics::Numerical Analysis ,Weighting ,010101 applied mathematics ,Matrix (mathematics) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0101 mathematics ,Algorithm ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The solution of a large-scale Sylvester matrix equation plays an important role in control and large scientific computations. In this paper, we are interested in the large Sylvester matrix equation with large dimensionA and small dimension B, and a popular approach is to use the global Krylov subspace method. In this paper, we propose three new algorithms for this problem. We first consider the global GMRES algorithm with weighting strategy, which can be viewed as a precondition method. We present three new schemes to update the weighting matrix during iterations. Due to the growth of memory requirements and computational cost, it is necessary to restart the algorithm effectively. The deflation strategy is efficient for the solution of large linear systems and large eigenvalue problems; to the best of our knowledge, little work is done on applying deflation to the (weighted) global GMRES algorithm for large Sylvester matrix equations. We then consider how to combine the weighting strategy with deflated restarting, and propose a weighted global GMRES algorithm with deflation for solving large Sylvester matrix equations. In particular, we are interested in the global GMRES algorithm with deflation, which can be viewed as a special case when the weighted matrix is chosen as the identity. Theoretical analysis is given to show rationality of the new algorithms. Numerical experiments illustrate the numerical behavior of the proposed algorithms.
- Published
- 2018
24. A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses, matrix inverses, and scalar reciprocals
- Author
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Mingzhi Mao, Jian Li, Yunong Zhang, and Frank Uhlig
- Subjects
Discretization ,Applied Mathematics ,Numerical analysis ,Scalar (mathematics) ,Finite difference ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Discrete time and continuous time ,Theory of computation ,Euler's formula ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula.
- Published
- 2018
25. A two-grid parallel partition of unity finite element scheme
- Author
-
Guangzhi Du and Liyun Zuo
- Subjects
Two grid ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Grid ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Partition of unity ,Scheme (mathematics) ,Theory of computation ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
A two-grid parallel partition of unity finite element scheme is proposed and analyzed in this paper for linear elliptic boundary value problems. The interesting features of this scheme contain the following: (1) a partition of unity is constructed to derive the globally continuous finite element solution; (2) errors decay exponentially with patches of diameter kH increase; and (3) a global coarse grid correction is done to improve the L2 − accuracy of the approximation. Numerical experiments are presented at the end of the paper to support our analysis.
- Published
- 2018
26. Convergence of discrete time waveform relaxation methods
- Author
-
Zhencheng Fan
- Subjects
Discretization ,Applied Mathematics ,Numerical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Relaxation (iterative method) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Rate of convergence ,Discrete time and continuous time ,Ordinary differential equation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Applied mathematics ,Waveform ,0101 mathematics ,Interpolation ,Mathematics - Abstract
This paper concerns the discrete time waveform relaxation (DWR) methods for ordinary differential equations (ODEs). We present a general algorithm of constructing the DWR methods with any order of convergence, which applies any numerical methods of ODEs to the perturbed equations of iterative schemes of continuous time waveform relaxation methods. It is demonstrated that the DWR method presented in this paper has the same convergent order as the numerical method used to discretize perturbed equations. Two classes of interpolation polynomials are given to generate perturbed equations. Finally, numerical experiments are presented in order to check against results obtained.
- Published
- 2018
27. On the eigenvalues of the saddle point matrices discretized from Navier–Stokes equations
- Author
-
Na Huang and Changfeng Ma
- Subjects
Discretization ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,010103 numerical & computational mathematics ,Positive-definite matrix ,Computer Science::Numerical Analysis ,01 natural sciences ,Generalized minimal residual method ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Linearization ,Saddle point ,0101 mathematics ,Navier–Stokes equations ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we study the spectral distributions of the saddle point matrices arising from the discretization and linearization of the Navier–Stokes equations, where the (1,1) block is nonsymmetric positive definite. In this paper, we derive the lower and upper bounds of the real and imaginary parts of all the eigenvalues of the saddle point matrices. We then propose a new class of block triangle preconditioners for solving the saddle point problems, and analyze the spectral properties of the preconditioned systems. Some numerical experiments with the preconditioned restarted generalized minimal residual method are reported to demonstrate the effectiveness and feasibility of these block triangle preconditioners.
- Published
- 2017
28. Finite-difference method for singular nonlinear systems
- Author
-
Sanja Rapajić, Tatjana Grbic, Sandra Buhmiler, and Slavica Medic
- Subjects
Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Finite difference method ,010103 numerical & computational mathematics ,01 natural sciences ,Local convergence ,010101 applied mathematics ,Nonlinear system ,Singularity ,Rate of convergence ,Singular solution ,0101 mathematics ,Local algorithm ,Mathematics - Abstract
© 2017, Springer Science+Business Media, LLC. This paper presents a method for solving nonlinear system with singular Jacobian at the solution. The convergence rate in the case of singularity deteriorates and one way to accelerate convergence is to form bordered system. A local algorithm, with finite-difference approximations, for forming and solving such system is proposed in this paper. To overcome the need that initial approximation has to be very close to the solution, we also propose a method which is a combination of descent method with finite-differences and local algorithm. Some numerical results obtained on relevant examples are presented.
- Published
- 2017
29. Semi-convergence analysis of preconditioned deteriorated PSS iteration method for singular saddle point problems
- Author
-
Zhao-Zheng Liang and Guo-Feng Zhang
- Subjects
Preconditioner ,Iterative method ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Krylov subspace ,01 natural sciences ,Generalized minimal residual method ,010101 applied mathematics ,Arnoldi iteration ,Fixed-point iteration ,Power iteration ,Saddle point ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.
- Published
- 2017
30. Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems
- Author
-
Dang Van Hieu
- Subjects
Sequence ,Mathematical optimization ,021103 operations research ,Iterative method ,Applied Mathematics ,Numerical analysis ,0211 other engineering and technologies ,Hilbert space ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,Theory of computation ,Convergence (routing) ,symbols ,Equilibrium problem ,0101 mathematics ,Algebra over a field ,Algorithm ,Mathematics - Abstract
The paper introduces and analyzes the convergence of a new iterative algorithm for approximating solutions of equilibrium problems involving strongly pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. The algorithm uses a stepsize sequence which is non-increasing, diminishing, and non-summable. This leads to the main advantage of the algorithm, namely that the construction of solution approximations and the proof of its convergence are done without the prior knowledge of the modulus of strong pseudomonotonicity and Lipschitz-type constants of bifunctions. The strongly convergent theorem is established under suitable assumptions. The paper also discusses the assumptions used in the formulation of the convergent theorem. Several numerical results are reported to illustrate the behavior of the algorithm with different sequences of stepsizes and also to compare it with others.
- Published
- 2017
31. A feasible and effective technique in constructing ERKN methods for multi-frequency multidimensional oscillators in scientific computation
- Author
-
Xianyang Zeng and Hongli Yang
- Subjects
010101 applied mathematics ,Mathematical optimization ,Applied Mathematics ,Ordinary differential equation ,Integrator ,Numerical analysis ,Theory of computation ,010103 numerical & computational mathematics ,0101 mathematics ,Algebra over a field ,01 natural sciences ,Computational science ,Mathematics - Abstract
In last few years, many ERKN methods have been investigated for solving multi-frequency multidimensional second-order ordinary differential equations, and the numerical efficiency has been checked strongly in scientific computation. But in the constructions of (especially high-order) new ERKN methods, lots of time and effort are costed in presenting the practical order conditions firstly and then in adding some reasonable assumptions to get the coefficient functions finally. In this paper, a feasible and effective technique is given which makes the construction of ERKN methods finished in a few seconds or a few minutes, even for high-order integrators. Moreover, this technique does not need any more information and knowledge except the classical RKN method. And this paper also gives the theoretical explanation to guarantee that the ERKN method obtained from this technique has the same order and the same properties as the underlying RKN method.
- Published
- 2017
32. Fast multipole methods for approximating a function from sampling values
- Author
-
Guidong Liu and Shuhuang Xiang
- Subjects
Discrete mathematics ,Applied Mathematics ,Fast multipole method ,Numerical analysis ,Lagrange polynomial ,Sampling (statistics) ,010103 numerical & computational mathematics ,Function (mathematics) ,Barycentric coordinate system ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,symbols ,0101 mathematics ,Multipole expansion ,Mathematics ,Interpolation - Abstract
Both barycentric Lagrange interpolation and barycentric rational interpolation are thought to be stable and effective methods for approximating a given function on some special point sets. A direct evaluation of these interpolants due to N interpolation points at M sampling points requires \(\mathcal {O}(NM)\) arithmetic operations. In this paper, we introduce two fast multipole methods to reduce the complexity to \(\mathcal {O}(\max \left \{N,M\right \})\). The convergence analysis is also presented in this paper.
- Published
- 2017
33. Viscosity iterative algorithms for fixed point problems of asymptotically nonexpansive mappings in the intermediate sense and variational inequality problems in Banach spaces
- Author
-
Olaniyi S. Iyiola, Gang Cai, and Yekini Shehu
- Subjects
Applied Mathematics ,Numerical analysis ,010102 general mathematics ,Regular polygon ,Banach space ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,Monotone polygon ,Convergence (routing) ,Variational inequality ,Theory of computation ,0101 mathematics ,Algorithm ,Mathematics - Abstract
In this paper, we introduce a generalized viscosity algorithm for finding a fixed point of an asymptotically nonexpansive mapping in the intermediate sense which is also a solution to a variational inequality problem of two inverse-strongly monotone operators in 2-uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are given under suitable assumptions imposed on the parameters. The results obtained in this paper improve and extend many recent ones in the literature. Three numerical examples are also given to show the efficiency and implementation of our results.
- Published
- 2017
34. An improved tri-coloured rooted-tree theory and order conditions for ERKN methods for general multi-frequency oscillatory systems
- Author
-
Xinyuan Wu, Hongli Yang, and Xianyang Zeng
- Subjects
Applied Mathematics ,Order up to ,Numerical analysis ,Order (ring theory) ,010103 numerical & computational mathematics ,01 natural sciences ,Bottleneck ,010101 applied mathematics ,Set (abstract data type) ,Homogeneous space ,Theory of computation ,Tree (set theory) ,0101 mathematics ,Algorithm ,Mathematics - Abstract
This paper develops an improved tri-coloured rooted-tree theory for the order conditions for ERKN methods solving general multi-frequency and multidimensional second-order oscillatory systems. The bottleneck of the original tricoloured rooted-tree theory is the existence of numerous redundant trees. In light of the fact that the sum of the products of the symmetries and the elementary differentials is meaningful, this paper naturally introduces the so-called extended elementary differential mappings. Then, the new improved tri-coloured rooted tree theory is established based on a subset of the original tri-coloured rooted-tree set. This new theory makes all redundant trees disappear, and thus, the order conditions of ERKN methods for general multi-frequency and multidimensional second-order oscillatory systems are reduced greatly. Furthermore, with this new theory, we present some new ERKN methods of order up to four. Numerical experiments are implemented and the results show that ERKN methods can be competitive with other existing methods in the scientific literature, especially when comparatively large stepsizes are used.
- Published
- 2016
35. An arbitrary band structure construction of totally nonnegative matrices with prescribed eigenvalues
- Author
-
Masashi Iwasaki, Kanae Akaiwa, Akira Yoshida, Koichi Kondo, and Yoshimasa Nakamura
- Subjects
Integrable system ,Applied Mathematics ,010102 general mathematics ,Diagonal ,Triangular matrix ,Inverse ,010103 numerical & computational mathematics ,Lambda ,01 natural sciences ,Square matrix ,Combinatorics ,Hadamard transform ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The construction of totally nonnegative (TN) matrices with prescribed eigenvalues is an important topic in real-valued nonnegative inverse eigenvalue problems. TN matrices are square matrices whose minors are all nonnegative. Our previous paper (Numer. Algor. 70, 469–484, ??2015) presented a finite-step construction of TN matrices limited to upper or lower Hessenberg forms with prescribed eigenvalues, based on the discrete hungry Toda (dhToda) equation which is derived from the study of integrable systems. Building on our previous paper, we produce the construction of banded TN matrices with an arbitrary number of diagonals in both lower and upper triangular parts and prescribed eigenvalues, involving upper Hessenberg, lower Hessenberg, and dense TN matrices with prescribed eigenvalues. We first prepare an infinite sequence associated with distinct eigenvalues \(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}\) and two integers M and N which determine the upper and lower bandwidths of m-by-m banded matrices, respectively. Both M and N play a key role for achieving our purpose. The study follows similar lines to our previous paper, but is complicated by the introduction of N. We next consider extended Hankel determinants and extended Hadamard polynomials involving elements of the infinite sequence and then derive their relationships. These relationships help us understand banded TN matrices with eigenvalues \(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}\) from the viewpoint of an extension of the dhToda equation. Finally, we propose a finite-step procedure for constructing TN matrices with an arbitrary upper and lower bandwidths and prescribed eigenvalues and also give illustrative examples.
- Published
- 2016
36. Truncation error estimates for generalized Hermite sampling
- Author
-
R. M. Asharabi and H. S. Al-Abbas
- Subjects
Truncation error ,Hermite polynomials ,Applied Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Sampling (statistics) ,010103 numerical & computational mathematics ,Function (mathematics) ,Space (mathematics) ,01 natural sciences ,Domain (mathematical analysis) ,Theory of computation ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
The generalized Hermite sampling uses samples from the function itself and its derivatives up to order r. In this paper, we investigate truncation error estimates for the generalized Hermite sampling series on a complex domain for functions from Bernstein space. We will extend some known techniques to derive those estimates and the bounds of Jagerman (SIAM J. Appl. Math. 14, 714---723 1966), Li (J. Approx. Theory 93, 100---113 1998), Annaby-Asharabi (J. Korean Math. Soc. 47, 1299---1316 2010), and Ye and Song (Appl. Math. J. Chinese Univ. 27, 412---418 2012) will be special cases for our results. Some examples with tables and figures are given at the end of the paper.
- Published
- 2016
37. Almost sure stability of the Euler–Maruyama method with random variable stepsize for stochastic differential equations
- Author
-
Wei Liu and Xuerong Mao
- Subjects
Applied Mathematics ,Numerical analysis ,Mathematical analysis ,010103 numerical & computational mathematics ,Adaptive stepsize ,01 natural sciences ,Stability (probability) ,Euler–Maruyama method ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Stochastic differential equation ,Semimartingale ,Stopping time ,Computer Science::Multimedia ,0101 mathematics ,QA ,Random variable ,Mathematics - Abstract
In this paper, the Euler---Maruyama (EM) method with random variable stepsize is studied to reproduce the almost sure stability of the true solutions of stochastic differential equations. Since the choice of the time step is based on the current state of the solution, the time variable is proved to be a stopping time. Then the semimartingale convergence theory is employed to obtain the almost sure stability of the random variable stepsize EM solution. To our best knowledge, this is the first paper to apply the random variable stepsize (with clear proof of the stopping time) to the analysis of the almost sure stability of the EM method.
- Published
- 2016
38. A Legendre-Galerkin method for solving general Volterra functional integral equations
- Author
-
Jiafei Qi and Haotao Cai
- Subjects
Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Order (ring theory) ,010103 numerical & computational mathematics ,01 natural sciences ,Volterra integral equation ,Integral equation ,010101 applied mathematics ,symbols.namesake ,symbols ,Uniform boundedness ,0101 mathematics ,Galerkin method ,Coefficient matrix ,Legendre polynomials ,Mathematics - Abstract
We propose in this paper a fully discrete Legendre-Galerkin method for solving general Volterra functional integral equations. The focus of this paper is the stability analysis of this method. Based on this stability result, we prove that the approximation equation has a unique solution, and then show that the Legendre-Galerkin method gives the optimal convergence order O(nźm)$\mathcal {O}(n^{-m})$, where m denotes the degree of the regularity of the exact solution and n+1 denotes the dimensional number of the approximation space. Moreover, we establish that the spectral condition constant of the coefficient matrix relative to the corresponding linear system is uniformly bounded for sufficiently large n. Finally, we use numerical examples to confirm the theoretical prediction.
- Published
- 2016
39. Iterative algorithms for solving variational inequalities and fixed point problems for asymptotically nonexpansive mappings in Banach spaces
- Author
-
Olaniyi S. Iyiola, Yekini Shehu, and Gang Cai
- Subjects
Applied Mathematics ,Numerical analysis ,010102 general mathematics ,Regular polygon ,Banach space ,Uniformly convex space ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,Variational inequality ,Theory of computation ,Convergence (routing) ,0101 mathematics ,Algorithm ,Mathematics - Abstract
The purpose of this paper is to study some iterative algorithms for finding a common element of the set of solutions of systems of variational inequalities for inverse-strongly accretive mappings and the set of fixed points of an asymptotically nonexpansive mapping in uniformly convex and 2-uniformly smooth Banach space or uniformly convex and q-uniformly smooth Banach space. Strong convergence theorems are obtained under suitable conditions. We also give some numerical examples to support our main results. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.
- Published
- 2016
40. Comments on direct transcription solution of DAE constrained optimal control problems with two discretization approaches
- Author
-
John T. Betts and Stephen L. Campbell
- Subjects
Mathematical optimization ,Discretization ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Optimal control ,01 natural sciences ,Regularization (mathematics) ,010101 applied mathematics ,Theory of computation ,0101 mathematics ,Transcription (software) ,Differential algebraic equation ,Mathematics - Abstract
There have been a number of results in the literature showing that a direct transcription numerical approach to optimal control problems could have a number of surprising and desirable behaviors that were not always predicted by the existing theory especially when differential algebraic equations are involved. Most of these results were developed with an implicit Runge-Kutta (IRK) discretization being used. It is important to know which of these observations hold for direct transcription software using different types of discretizations and which are discretization specific. This paper reexamines several of these questions but using an hp-pseudospectral code. It is seen that while philosophically the results are often similar to the IRK results, there are some differences that should be understood by users solving constrained optimal control problems. This paper also discusses a type of regularization for higher index problems that does not reduce the index.
- Published
- 2016
41. Analysis of a meshless method for the time fractional diffusion-wave equation
- Author
-
Akbar Mohebbi, Mostafa Abbaszadeh, and Mehdi Dehghan
- Subjects
Regularized meshless method ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Mixed boundary condition ,Singular boundary method ,01 natural sciences ,Robin boundary condition ,010101 applied mathematics ,symbols.namesake ,Dirichlet boundary condition ,symbols ,Gaussian quadrature ,Radial basis function ,0101 mathematics ,Galerkin method ,Mathematics - Abstract
In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is O(ź3źź)$\mathcal {O}(\tau ^{3-\alpha })$. Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when βź+ź$\beta \rightarrow +\infty $ solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is O(h)$\mathcal {O}(h)$. In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.
- Published
- 2016
42. On the numerical solution of the quadratic eigenvalue complementarity problem
- Author
-
Valentina Sessa, Alfredo N. Iusem, Hanif D. Sherali, and Joaquim J. Júdice
- Subjects
Mathematical optimization ,021103 operations research ,Applied Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Nonlinear programming ,Quadratic equation ,Complementarity theory ,Theory of computation ,0101 mathematics ,Mixed complementarity problem ,Global optimization ,Equivalence (measure theory) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The Quadratic Eigenvalue Complementarity Problem (QEiCP) is an extension of the Eigenvalue Complementarity Problem (EiCP) that has been introduced recently. Similar to the EiCP, the QEiCP always has a solution under reasonable hypotheses on the matrices included in its definition. This has been established in a previous paper by reducing a QEiCP of dimension n to a special 2n-order EiCP. In this paper we propose an enumerative algorithm for solving the QEiCP by exploiting this equivalence with an EiCP. The algorithm seeks a global minimum of a special Nonlinear Programming Problem (NLP) with a known global optimal value. The algorithm is shown to perform very well in practice but in some cases terminates with only an approximate optimal solution to NLP. Hence, we propose a hybrid method that combines the enumerative method with a fast and local semi-smooth method to overcome the latter drawback. This algorithm is also shown to be useful for computing a positive eigenvalue for an EiCP under similar assumptions. Computational experience is reported to demonstrate the efficacy and efficiency of the hybrid enumerative method for solving the QEiCP.
- Published
- 2015
43. A new convergence theorem of a projection algorithm with variable steps for variational inequalities
- Author
-
Haiyun Zhou, Jianjun Zhou, and Peiyuan Wang
- Subjects
Dominated convergence theorem ,Factor theorem ,Mathematical optimization ,Picard–Lindelöf theorem ,Applied Mathematics ,Fixed-point theorem ,010103 numerical & computational mathematics ,01 natural sciences ,GeneralLiterature_MISCELLANEOUS ,010101 applied mathematics ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,symbols ,Applied mathematics ,Danskin's theorem ,0101 mathematics ,Brouwer fixed-point theorem ,Mean value theorem ,Mathematics ,Bolzano–Weierstrass theorem - Abstract
In this paper, we improve the convergence theorem in the paper by Yang (Journal of Industrial and Management Optimization 1, 211---217, 2005), and propose a new modified convergence theorem. The theorem and the proof presented in the present paper are interesting improvements on the convergence theorem of Yang.
- Published
- 2015
44. Image deblurring by sparsity constraint on the Fourier coefficients
- Author
-
Mariarosa Mazza, Debora Sesana, Marco Donatelli, and Thomas Huckle
- Subjects
Discrete-time Fourier transform ,Fourier coefficients ,Image deblurring ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Tikhonov regularization ,symbols.namesake ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,Fourier series ,Mathematics ,Filtering methods ,Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Sparse reconstruction ,020206 networking & telecommunications ,Backus–Gilbert method ,Fourier analysis ,Phase correlation ,symbols ,Deconvolution ,Algorithm - Abstract
This paper is concerned with the image deconvolution problem. For the basic model, where the convolution matrix can be diagonalized by discrete Fourier transform, the Tikhonov regularization method is computationally attractive since the associated linear system can be easily solved by fast Fourier transforms. On the other hand, the provided solutions are usually oversmoothed and other regularization terms are often employed to improve the quality of the restoration. Of course, this weighs down on the computational cost of the regularization method. Starting from the fact that images have sparse representations in the Fourier and wavelet domains, many deconvolution methods have been recently proposed with the aim of minimizing the l1-norm of these transformed coefficients. This paper uses the iteratively reweighted least squares strategy to introduce a diagonal weighting matrix in the Fourier domain. The resulting linear system is diagonal and hence the regularization parameter can be easily estimated, for instance by the generalized cross validation. The method benefits from a proper initial approximation that can be the observed image or the Tikhonov approximation. Therefore, embedding this method in an outer iteration may yield further improvement of the solution. Finally, since some properties of the observed image, like continuity or sparsity, are obviously changed when working in the Fourier domain, we introduce a filtering factor which keeps unchanged the large singular values and preserves the jumps in the Fourier coefficients related to the low frequencies. Numerical examples are given in order to show the effectiveness of the proposed method.
- Published
- 2015
45. A modular string averaging procedure for solving the common fixed point problem for quasi-nonexpansive mappings in Hilbert space
- Author
-
Rafał Zalas and Simeon Reich
- Subjects
Discrete mathematics ,021103 operations research ,Iterative method ,Applied Mathematics ,0211 other engineering and technologies ,Block (permutation group theory) ,Hilbert space ,Relaxation (iterative method) ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Combinatorics ,symbols.namesake ,Operator (computer programming) ,Bounded function ,C++ string handling ,symbols ,Convex combination ,0101 mathematics ,Mathematics - Abstract
In this paper we consider the common fixed point problem for a finite family of quasi-nonexpansive mappings Ui:???$U_{i}\colon \mathcal H\rightarrow \mathcal H$, where i ? I := {1,?, M}, M?1, and ?$\mathcal H$ is a real Hilbert space. This problem is defined as follows: find x??i?IFixUi??$x\in \bigcap _{i\in I}\text {Fix} U_{i}\neq \emptyset $, where FixUi:={z???z=Uiz}$U_{i}:=\{z\in \mathcal H\mid z=U_{i}z\}$. We propose the following iterative method: x0??,xk+1:=Tkxk,$$ x^{0}\in\mathcal H,\quad x^{k+1}:=T_{k} x^{k}, $$ where for each k=0,1,2,?, the operator Tk is defined by a certain amalgamation procedure called modular string averaging. The main idea of this procedure is to combine repeatedly and recursively three primal operations: relaxation, convex combination and composition of the given operators Ui. The modular string averaging procedure, when combined with the above iterative method, provides a very flexible framework which covers and fills the gap between different algorithmic approaches such as string averaging and block iterative schemes. Moreover, our framework enables us to construct many algorithmic schemes, the convergence of which has not been investigated so far. The aim of this paper is to establish both weak and strong convergence results for the above iterative method. Moreover, in the case of firmly nonexpansive Ui's, we show that convergence is preserved in the presence of inexact computations. In particular, this implies that the iterative scheme is resilient to bounded perturbations, which is important from the superiorization methodology point of view.
- Published
- 2015
46. Analysis of Schwarz waveform relaxation for the coupled Ekman boundary layer problem with continuously variable coefficients
- Author
-
Florian Lemarié, Sophie Thery, Eric Blayo, Charles Pelletier, Mathematics and computing applied to oceanic and atmospheric flows (AIRSEA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Grenoble Alpes (UGA)-Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), ANR-16-CE01-0007,COCOA,Méthodes mathématiquement et physiquement consistantes pour le couplage océan-atmosphère(2016), Earth and Life Institute [Louvain-La-Neuve] (ELI), and Université Catholique de Louvain = Catholic University of Louvain (UCL)
- Subjects
coupled Ekman problem ,010504 meteorology & atmospheric sciences ,multiphysics coupling ,Context (language use) ,010103 numerical & computational mathematics ,01 natural sciences ,numerical climate modeling ,Convergence (routing) ,MESH: Schwarz waveform relaxation ,Variable coefficients ,Multiphysics coupling ,Coupled Ekman problem ,Numerical climate modeling ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Waveform ,Applied mathematics ,0101 mathematics ,Physics::Atmospheric and Oceanic Physics ,0105 earth and related environmental sciences ,Mathematics ,[SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere ,Applied Mathematics ,Numerical analysis ,continuously variable coefficients ,Schwarz waveform relaxation ,Boundary layer ,Variable (computer science) ,13. Climate action ,Theory of computation ,Relaxation (approximation) ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
International audience; In this paper we present a global-in-time non-overlapping Schwarz method applied to the Ekman boundary layer problem. Such a coupled problem is representative of large-scale atmospheric and oceanic flows in the vicinity of the air-sea interface. Schwarz waveform relaxation (SWR) algorithms provide attractive methods for ensuring a "tight coupling" between the ocean and the atmosphere. However the convergence study of such algorithms in this context raises a number of challenges. Numerous convergence studies of Schwarz methods have been carried out in idealized settings, but the underlying assumptions to make these studies tractable may prohibit them to be directly extended to the complexity of climate models. We illustrate this aspect on the coupled Ekman problem, which includes several essential features inherent to climate modeling while being simple enough for analytical results to be derived. We investigate its well-posedness and derive an appropriate SWR algorithm. Sufficient conditions for ensuring its convergence for different viscosity profiles are then established. Finally, we illustrate the relevance of our theoretical analysis with numerical results and suggest ways to improve the computational cost of the coupling. Our study emphasizes the fact that the convergence properties can be highly sensitive to some model characteristics such as the geometry of the problem and the use of continuously variable viscosity coefficients.
- Published
- 2021
47. An efficient Gauss–Newton algorithm for solving regularized total least squares problems
- Author
-
Hossein Zare and Masoud Hajarian
- Subjects
Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,System of linear equations ,01 natural sciences ,Regularization (mathematics) ,Gauss–Newton algorithm ,010101 applied mathematics ,Overdetermined system ,Non-linear least squares ,Applied mathematics ,0101 mathematics ,Total least squares ,Coefficient matrix ,Mathematics - Abstract
The total least squares (TLS) method is a well-known technique for solving an overdetermined linear system of equations Ax ≈ b, that is appropriate when both the coefficient matrix A and the right-hand side vector b are contaminated by some noise. For ill-posed TLS poblems, regularization techniques are necessary to stabilize the computed solution; otherwise, TLS produces a noise-dominant output. In this paper, we show that the regularized total least squares (RTLS) problem can be reformulated as a nonlinear least squares problem and can be solved by the Gauss–Newton method. Due to the nature of the RTLS problem, we present an appropriate method to choose a good regularization parameter and also a good initial guess. Finally, the efficiency of the proposed method is examined by some test problems.
- Published
- 2021
48. A new branch-and-cut algorithm for non-convex quadratic programming via alternative direction method and semidefinite relaxation
- Author
-
Huixian Wu, Sikai Chen, and Hezhi Luo
- Subjects
Hessian matrix ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Quadratic equation ,Theory of computation ,Convergence (routing) ,symbols ,Quadratic programming ,Relaxation (approximation) ,0101 mathematics ,Algorithm ,Branch and cut ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We consider a non-convex quadratic program (QP) with linear and convex quadratic constraints that arises from a broad range of applications and is known to be NP-hard. In this paper, we first prove that the alternative direction method converges to a local solution of the underlying QP problem. We then propose a new branch-and-cut algorithm that finds a globally optimal solution to the underlying QP problem within a pre-specified 𝜖-tolerance by integrating the alternative direction method with semidefinite relaxation and disjunctive cut techniques. We establish the global convergence of the algorithm and estimate its complexity. Preliminary numerical results demonstrate that the proposed algorithm can effectively find a globally optimal solution to medium-scale QP instances in which the number of negative eigenvalues of the Hessian matrix in the objective function is less than or equals 20.
- Published
- 2021
49. On the unitary block-decomposability of 1-parameter matrix flows and static matrices
- Author
-
Frank Uhlig
- Subjects
Applied Mathematics ,Diagonal ,010103 numerical & computational mathematics ,01 natural sciences ,Matrix similarity ,Square (algebra) ,010101 applied mathematics ,Combinatorics ,Matrix (mathematics) ,Flow (mathematics) ,Differentiable function ,0101 mathematics ,Invariant (mathematics) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
For general complex or real 1-parameter matrix flows A(t)n, n and for static matrices $A \in \mathbb {C}_{n,n}$ alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity Cn, n as A(t) = C− 1 ⋅ diag(A1(t),...,Al(t)) ⋅ C or A = C− 1 ⋅diag(A1,...,Al) ⋅ C with each diagonal block Ak(t) or Ak square and their number l exceeding 1 if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the MATLAB eig computed ‘eigenvectors’ of one associated flow matrix B(ta) to find the coarsest simultaneous block structure for all flow matrices B(tb). The method works efficiently for all time-varying matrix flows A(t), be they real or complex, normal, with Jordan structures or repeated eigenvalues, differentiable, continuous, or discontinuous in t, and likewise for all fixed entry matrices A. Our intended aim is to discover unitarily diagonal-block decomposable flows as they originate in real-time from sensor given data for time-varying matrix problems that are unitarily invariant. Then, the complexities of their numerical treatments decrease by adopting ‘divide and conquer’ methods for their diagonal blocks. In the process, we discover and study k-normal fixed entry matrix classes that can be decomposed under unitary similarities into various k-dimensional block-diagonal forms.
- Published
- 2021
50. An infeasible interior-point arc-search algorithm for nonlinear constrained optimization
- Author
-
Makoto Yamashita, Yaguang Yang, and Einosuke Iida
- Subjects
Line search ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Nonlinear programming ,90C51, 90C30 ,010101 applied mathematics ,Optimization and Control (math.OC) ,Search algorithm ,Theory of computation ,Path (graph theory) ,Convergence (routing) ,FOS: Mathematics ,Benchmark (computing) ,0101 mathematics ,Mathematics - Optimization and Control ,Algorithm ,Interior point method ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
In this paper, we propose an infeasible arc-search interior-point algorithm for solving nonlinear programming problems. Most algorithms based on interior-point methods are categorized as line search since they compute a next iterate on a straight line determined by a search direction which approximates the central path. The proposed arc-search interior-point algorithm uses an arc for the approximation. We discuss convergence properties of the proposed algorithm. We also conduct numerical experiments on the CUTEst benchmark problems and compare the performance of the proposed arc-search algorithm with that of a line-search algorithm. Numerical results indicate that the proposed arc-search algorithm reaches the optimal solution using fewer iterations but longer times than a line-search algorithm. A modification that leads to a faster arc-search algorithm is also discussed.
- Published
- 2021
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