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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. On the Gauss-Kronrod quadrature formula for a modified weight function of Chebyshev type
- Abstract
In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of the error of these Gauss-Kronrod formulae for analytic functions are obtained, using techniques of contour integration that were introduced by Gautschi and Varga (cf. Gautschi and Varga SIAM J. Numer. Anal. 20, 1170-1186 1983). Some illustrative numerical examples which show both the accuracy of the Gauss-Kronrod formulas and the sharpness of our estimations are displayed. Though for the sake of brevity we restrict ourselves to the first kind Chebyshev weight, a similar analysis may be carried out for the other three Chebyshev type weights; part of the corresponding computations are included in a final appendix.
- Published
- 2022
4. A note on generalized averaged Gaussian formulas for a class of weight functions
- Abstract
In the recent paper Notaris (Numer. Math., 142:129-147,2019) it has been introduced a new and useful class of nonnegative measures for which the well-known Gauss-Kronrod quadrature formulae coincide with the generalized averaged Gaussian quadrature formulas. In such a case, the given generalized averaged Gaussian quadrature formulas are of the higher degree of precision, and can be numerically constructed by an effective and simple method; see Spalevic (Math. Comp., 76:1483-1492,2007). Moreover, as almost immediate consequence of our results from Spalevic (Math. Comp.,76:1483-1492,2007) and that theory, we prove the main statements in Notaris (Numer. Math.,142:129-147,2019) in a different manner, by means of the Jacobi tridiagonal matrix approach.
- Published
- 2020
5. Complex moment-based methods for differential eigenvalue problems
- Author
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Imakura, Akira, Morikuni, Keiichi, and Takayasu, Akitoshi
- Subjects
Applied Mathematics ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Numerical Analysis (math.NA) - Abstract
This paper considers computing partial eigenpairs of differential eigenvalue problems (DEPs) such that eigenvalues are in a certain region on the complex plane. Recently, based on a "solve-then-discretize" paradigm, an operator analogue of the FEAST method has been proposed for DEPs without discretization of the coefficient operators. Compared to conventional "discretize-then-solve" approaches that discretize the operators and solve the resulting matrix problem, the operator analogue of FEAST exhibits much higher accuracy; however, it involves solving a large number of ordinary differential equations (ODEs). In this paper, to reduce the computational costs, we propose operation analogues of Sakurai-Sugiura-type complex moment-based eigensolvers for DEPs using higher-order complex moments and analyze the error bound of the proposed methods. We show that the number of ODEs to be solved can be reduced by a factor of the degree of complex moments without degrading accuracy, which is verified by numerical results. Numerical results demonstrate that the proposed methods are over five times faster compared with the operator analogue of FEAST for several DEPs while maintaining almost the same high accuracy. This study is expected to promote the "solve-then-discretize" paradigm for solving DEPs and contribute to faster and more accurate solutions in real-world applications., Comment: 26 pages, 9 figures
- Published
- 2022
6. A class of C2 quasi-interpolating splines free of Gibbs phenomenon
- Author
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Sergio Amat, David Levin, Juan Ruiz-Álvarez, Juan C. Trillo, Dionisio F. Yáñez, Universidad Politécnica de Cartagena, and Universidad de Valencia
- Subjects
Splines ,Computer aided design (modeling of curves) ,12 Matemáticas ,C2 regularity ,Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,Matemática Aplicada ,Adaption to discontinuities ,Quasi-interpolation ,ComputingMethodologies_COMPUTERGRAPHICS - Abstract
In many applications, it is useful to use piecewise polynomials that satisfy certain regularity conditions at the joint points. Cubic spline functions emerge as good candidates having C2 regularity. On the other hand, if the data points present discontinuities, the classical spline approximations produce Gibbs oscillations. In a recent paper, we have introduced a new nonlinear spline approximation avoiding the presence of these oscillations. Unfortunately, this new reconstruction loses the C2 regularity. This paper introduces a new nonlinear spline that preserves the regularity at all the joint points except at the end points of an interval containing a discontinuity, and that avoids the Gibbs oscillations. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was funded by the Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18, by the national research project MMTM2015-64382-P and PID2019-108336GB-I00 (MINECO/FEDER), by grant MTM2017-83942 funded by Spanish MINECO and by grant PID2020-117211GB-I00 funded by MCIN/AEI/10.13039/501100011033.
- Published
- 2022
7. 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
8. 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
9. Tensor extrapolation methods with applications
- Author
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Rachid Sadaka, Khalide Jbilou, Fatemeh Panjeh Ali Beik, and A. El Ichi
- Subjects
Sequence ,Applied Mathematics ,Numerical analysis ,Theory of computation ,Singular value decomposition ,Extrapolation ,Applied mathematics ,Tensor ,Algebra over a field ,Mathematics ,Matrix polynomial - Abstract
In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for tensors corresponding to T-product. Furthermore, we discuss on the solution of least-squares problem associated with a tensor equation using Tensor Singular Value Decomposition (TSVD). Motivated by the effectiveness of some proposed vector extrapolation methods in earlier papers, we describe how an extrapolation technique can be also implemented on the sequence of tensors produced by truncated TSVD (TTSVD) for solving possibly ill-posed tensor equations.
- Published
- 2020
10. Newton’s method with fractional derivatives and various iteration processes via visual analysis
- Author
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Krzysztof Gdawiec, Agnieszka Lisowska, and Wiesław Kotarski
- Subjects
Polynomial ,Applied Mathematics ,Numerical analysis ,Stability (learning theory) ,Fractional derivative ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,Newton method ,Fixed-point iteration ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Convergence (routing) ,symbols ,Polynomiography ,Applied mathematics ,Iterations ,0101 mathematics ,010301 acoustics ,Complex plane ,Newton's method ,Mathematics - Abstract
The aim of this paper is to visually investigate the dynamics and stability of the process in which the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives in the standard Newton root-finding method. Additionally, instead of the standard Picard iteration, the Mann, Khan, Ishikawa and S iterations are used. This process when applied to polynomials on complex plane produces images showing basins of attractions for polynomial zeros or images representing the number of iterations required to achieve any polynomial root. The images are called polynomiographs. In this paper, we use the colouring according to the number of iterations which reveals the speed of convergence and dynamic properties of processes visualised by polynomiographs. Moreover, to investigate the stability of the methods, we use basins of attraction. To compare numerically the modified root-finding methods among them, we demonstrate their action for polynomialz3− 1 on a complex plane.
- Published
- 2020
11. 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
12. 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
13. Almost sure stability of the Euler–Maruyama method with random variable stepsize for stochastic differential equations
- Author
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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
14. Numerical integration in log-Korobov and log-cosine spaces
- Author
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Gunther Leobacher, Peter Kritzer, Friedrich Pillichshammer, and Josef Dick
- Subjects
Unit sphere ,Discrete mathematics ,Function space ,Applied Mathematics ,Hilbert space ,Numerical Analysis (math.NA) ,Numerical integration ,Combinatorics ,symbols.namesake ,Square-integrable function ,Unit cube ,FOS: Mathematics ,65D30, 65D32 ,symbols ,Mathematics - Numerical Analysis ,Reproducing kernel Hilbert space ,Mathematics ,Real number - Abstract
Quasi-Monte Carlo (QMC) rules are equal weight quadrature rules for approximating integrals over the s-dimensional unit cube [0, 1]s. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on [0, 1]s with square integrable partial mixed derivatives of order ?. Using Parseval's identity, this smoothness can be defined for all real numbers ? > 1/2. In this setting, the condition ? > 1/2 is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with ? arbitrarily close to 1/2. To obtain such estimates we introduce a log-scale for function spaces with smoothness close to 1/2, which we call log-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order ??(N?1/2(logN)?μ(1??)/2)$\mathcal {O}(N^{-1/2} (\log N)^{-\mu (1-\lambda )/2})$ for any 1/μ 1 is a certain power in the log-scale. A second result is concerned with tractability of numerical integration for weighted Korobov spaces with product weights (?j)j??$(\gamma _{j})_{j \in \mathbb {N}}$. Previous results have shown that if ?j=1??j?${\sum }_{j=1}^{\infty } \gamma _{j}^{\tau } < \infty $ for some 1/(2?) ≤ 1 one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where ? is close to 1/(2?), namely we show dimension independent error bounds under the condition that ?j=1??jmax{1,log?j?1}μ(1??)${\sum }_{j=1}^{\infty } \gamma _{j} \max \{1, \log \gamma _{j}^{-1}\}^{\mu (1-\lambda )} < \infty $ for some 1/μ ≤ 1. The essential tool in our analysis is a log-scale Jensen's inequality.The results described above also apply to integration in log-cosine spaces using tent-transformed lattice rules.
- Published
- 2015
15. Estimating the trace of matrix functions with application to complex networks
- Author
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Rafael Díaz Fuentes, Marco Donatelli, Caterina Fenu, and Giorgio Mantica
- Subjects
Applied Mathematics - Abstract
The approximation of trace(f(Ω)), where f is a function of a symmetric matrix Ω, can be challenging when Ω is exceedingly large. In such a case even the partial Lanczos decomposition of Ω is computationally demanding and the stochastic method investigated by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996) is preferred. Moreover, in the last years, a partial global Lanczos method has been shown to reduce CPU time with respect to partial Lanczos decomposition. In this paper we review these techniques, treating them under the unifying theory of measure theory and Gaussian integration. This allows generalizing the stochastic approach, proposing a block version that collects a set of random vectors in a rectangular matrix, in a similar fashion to the partial global Lanczos method. We show that the results of this technique converge quickly to the same approximation provided by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996), while the block approach can leverage the same computational advantages as the partial global Lanczos. Numerical results for the computation of the Von Neumann entropy of complex networks prove the robustness and efficiency of the proposed block stochastic method.
- Published
- 2022
16. Drift-implicit Euler scheme for sandwiched processes driven by Hölder noises
- Author
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Giulia Di Nunno, Yuliya Mishura, and Anton Yurchenko-Tytarenko
- Subjects
60H10, 60H35, 60G22, 91G30 ,Applied Mathematics ,Probability (math.PR) ,FOS: Mathematics - Abstract
In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary $λ$-Hölder continuous process, $λ\in(0,1)$. We prove that, under some mild moment assumptions on the Hölder constant of the noise, the $L^r(Ω;L^\infty([0,T]))$-rate of convergence is equal to $λ$. To exemplify, we consider numerical schemes for the generalized Cox--Ingersoll-Ross and Tsallis--Stariolo--Borland models. The results are illustrated by simulations., 24 pages, 3 figures
- Published
- 2022
17. Solution of ill-posed problems with Chebfun
- Author
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A. Alqahtani, T. Mach, and L. Reichel
- Subjects
Applied Mathematics - Abstract
The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from finite-dimensional linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve ill-posed problems with a regularize-first approach numerically. This allows a user to work with functions instead of vectors and with integral operators instead of matrices. The solution process therefore is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods. Furthermore, the difficult process of explicitly choosing a suitable discretization is not required.
- Published
- 2022
18. A Riemannian inexact Newton dogleg method for constructing a symmetric nonnegative matrix with prescribed spectrum
- Author
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Zhi Zhao, Teng-Teng Yao, Zheng-Jian Bai, and Xiao-Qing Jin
- Subjects
Applied Mathematics ,FOS: Mathematics ,Numerical Analysis (math.NA) ,Mathematics::Differential Geometry ,Mathematics - Numerical Analysis ,15A18, 65F08, 65F18, 65F15 - Abstract
This paper is concerned with the inverse problem of constructing a symmetric nonnegative matrix from realizable spectrum. We reformulate the inverse problem as an underdetermined nonlinear matrix equation over a Riemannian product manifold. To solve it, we develop a Riemannian underdetermined inexact Newton dogleg method for solving a general underdetermined nonlinear equation defined between Riemannian manifolds and Euclidean spaces. The global and quadratic convergence of the proposed method is established under some mild assumptions. Then we solve the inverse problem by applying the proposed method to its equivalent nonlinear matrix equation and a preconditioner for the perturbed normal Riemannian Newton equation is also constructed. Numerical tests show the efficiency of the proposed method for solving the inverse problem., 32 pages, 6 figures
- Published
- 2022
19. On the Gauss-Kronrod quadrature formula for a modified weight function of Chebyshev type
- Author
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Ramón Orive, Aleksandar V. Pejčev, Miodrag M. Spalević, and Ljubica Mihić
- Subjects
Applied Mathematics - Abstract
In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of the error of these Gauss–Kronrod formulae for analytic functions are obtained, using techniques of contour integration that were introduced by Gautschi and Varga (cf. Gautschi and Varga SIAM J. Numer. Anal. 20, 1170–1186 1983). Some illustrative numerical examples which show both the accuracy of the Gauss–Kronrod formulas and the sharpness of our estimations are displayed. Though for the sake of brevity we restrict ourselves to the first kind Chebyshev weight, a similar analysis may be carried out for the other three Chebyshev type weights; part of the corresponding computations are included in a final appendix.
- Published
- 2022
20. An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data
- Author
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Bi-Yun Zhu, Xue-Yang Li, and Ai-Guo Xiao
- Subjects
Applied Mathematics - Abstract
In this paper, we focus on developing a high efficient algorithm for solving d-dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity have a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis for spatial discretization, and all the sine coefficients can be divided into several levels. The sine coefficients with different levels are discretized by temporal basis with different scales, which can lead to the STSG method. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order O(2-Jd) with O(2JJ) degrees of freedom (DOF) for d=1 and O(2Jd) DOF for d>1, where J denotes the maximal level of sine coefficients. However, the standard STSG is not suitable to simulate the singularity of TFDE at the initial time. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Then, the modified STSG is used to construct the fully discrete scheme for solving TFDE. The great advantage of STSG method can be shown in the comparative numerical experiment.Mathematics Subject Classification (2010) 35R11 · 65M70 · 65T40 · 68Q25
- Published
- 2023
21. The behavior of the Gauss-Radau upper bound of the error norm in CG
- Author
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Gérard Meurant and Petr Tichý
- Subjects
Applied Mathematics - Abstract
Consider the problem of solving systems of linear algebraic equations $$\varvec{A}\varvec{x}=\varvec{b}$$ A x = b with a real symmetric positive definite matrix $$\varvec{A}$$ A using the conjugate gradient (CG) method. To stop the algorithm at the appropriate moment, it is important to monitor the quality of the approximate solution. One of the most relevant quantities for measuring the quality of the approximate solution is the $$\varvec{A}$$ A -norm of the error. This quantity cannot be easily computed; however, it can be estimated. In this paper we discuss and analyze the behavior of the Gauss-Radau upper bound on the $$\varvec{A}$$ A -norm of the error, based on viewing CG as a procedure for approximating a certain Riemann-Stieltjes integral. This upper bound depends on a prescribed underestimate $$\varvec{\mu }$$ μ to the smallest eigenvalue of $$\varvec{A}$$ A . We concentrate on explaining a phenomenon observed during computations showing that, in later CG iterations, the upper bound loses its accuracy, and is almost independent of $$\varvec{\mu }$$ μ . We construct a model problem that is used to demonstrate and study the behavior of the upper bound in dependence of $$\varvec{\mu }$$ μ , and developed formulas that are helpful in understanding this behavior. We show that the above-mentioned phenomenon is closely related to the convergence of the smallest Ritz value to the smallest eigenvalue of $$\varvec{A}$$ A . It occurs when the smallest Ritz value is a better approximation to the smallest eigenvalue than the prescribed underestimate $$\varvec{\mu }$$ μ . We also suggest an adaptive strategy for improving the accuracy of the upper bounds in the previous iterations.
- Published
- 2023
22. An algebraically stabilized method for convection–diffusion–reaction problems with optimal experimental convergence rates on general meshes
- Author
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Petr Knobloch
- Subjects
Applied Mathematics - Abstract
Algebraically stabilized finite element discretizations of scalar steady-state convection–diffusion–reaction equations often provide accurate approximate solutions satisfying the discrete maximum principle (DMP). However, it was observed that a deterioration of the accuracy and convergence rates may occur for some problems if meshes without local symmetries are used. The paper investigates these phenomena both numerically and analytically and the findings are used to design a new algebraic stabilization called Symmetrized Monotone Upwind-type Algebraically Stabilized (SMUAS) method. It is proved that the SMUAS method is linearity preserving and satisfies the DMP on arbitrary simplicial meshes. Moreover, numerical results indicate that the SMUAS method leads to optimal convergence rates on general simplicial meshes.
- Published
- 2023
23. Bivariate general Appell interpolation problem
- Author
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F. A. Costabile, M. I. Gualtieri, and A. Napoli
- Subjects
Applied Mathematics - Abstract
In this paper, the solution to a bivariate Appell interpolation problem proposed in a previous work is given. Bounds of the truncation error are considered. Ten new interpolants for real, regular, bivariate functions are constructed. Numerical examples and comparisons with bivariate Bernstein polynomials are considered.
- Published
- 2022
24. Substructured two-grid and multi-grid domain decomposition methods
- Author
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G. Ciaramella and T. Vanzan
- Subjects
Coarse correction ,Applied Mathematics ,Schwarz methods ,Elliptic equations ,Substructured methods ,Multigrid methods ,Domain decomposition methods ,Two-level methods - Abstract
Two-level Schwarz domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature, most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of Schwarz domain decomposition methods to define, for general elliptic problems, a new class of substructured two-level methods, for which both Schwarz smoothers and coarse correction steps are defined on the interfaces (except for the application of the smoother that requires volumetric subdomain solves). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require, like classical multi-grid methods, the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.
- Published
- 2022
25. (Spectral) Chebyshev collocation methods for solving differential equations
- Author
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Pierluigi Amodio, Luigi Brugnano, and Felice Iavernaro
- Subjects
65L06, 65L05 ,Applied Mathematics ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Numerical Analysis (math.NA) ,Mathematics::Numerical Analysis - Abstract
Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli [33]. In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods., Comment: 25 pages, 2 figures, 2 tables
- Published
- 2023
26. A class of spectral conjugate gradient methods for Riemannian optimization
- Author
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Chunming Tang, Wancheng Tan, Shajie Xing, and Haiyan Zheng
- Subjects
Applied Mathematics - Abstract
Spectral conjugate gradient (SCG) methods are combinations of spectral gradient method and conjugate gradient (CG) methods, which have been well studied in Euclidean space. In this paper, we aim to extend this class of methods to solve optimization problems on Riemannian manifolds. Firstly, we present a Riemannian version of the spectral parameter, which guarantees that the search direction always satisfies the sufficient descent property without the help of any line search strategy. Secondly, we introduce a generic algorithmic framework for the Riemannian SCG methods, in which the selection of the CG parameter is very flexible. Under the Riemannian Wolfe conditions, the global convergence of the proposed algorithmic framework is established whenever the absolute value of the CG parameter is no more than the Riemannian Fletcher-Reeves CG parameter. Finally, some preliminary numerical results are reported and compared with several classical Riemannian CG methods, which show that our new methods are efficient.MSC Classification: 65K05 , 90C30
- Published
- 2023
27. Generalized conformable fractional Newton-type method for solving nonlinear systems
- Author
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Giro Candelario, Alicia Cordero, Juan R. Torregrosa, and María P. Vassileva
- Subjects
Applied Mathematics - Abstract
In a recent paper, a conformable fractional Newton-type method was proposed for solving nonlinear equations. This method involves a lower computational cost compared to others fractional iterative methods. Indeed, the theoretical order of convergence is held in practice, and it presents a better numerical behaviour than fractional Newton-type methods formerly proposed, even compared to classical Newton-Raphson method. In this work, we design a generalization of this method for solving nonlinear systems by using a new conformable fractional Jacobian matrix, and a suitable conformable Taylor power series; and it is compared with classical Newton’s scheme. The necessary concepts and results are stated in order to design this method. Convergence analysis is made and a quadratic order of convergence is obtained, as in classical Newton's method. Numerical tests are made, and the Approximated Computational Order ofo Convergence (ACOC) supports the theory. Also, the proposed scheme shows good stability properties observed by means of convergence planes.
- Published
- 2023
28. Dynamics of Newton-like root finding methods
- Author
-
B. Campos, J. Canela, and P. Vindel
- Subjects
Applied Mathematics - Abstract
When exploring the literature, it can be observed that the operator obtained when applying Newton-like root finding algorithms to the quadratic polynomials z2 − c has the same form regardless of which algorithm has been used. In this paper, we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree d polynomials p(z) = zd − c. Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algorithms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods.
- Published
- 2022
29. A null-space approach for large-scale symmetric saddle point systems with a small and non zero (2, 2) block
- Author
-
Jennifer Scott and Miroslav Tůma
- Subjects
Applied Mathematics - Abstract
Null-space methods have long been used to solve large sparse n × n symmetric saddle point systems of equations in which the (2, 2) block is zero. This paper focuses on the case where the (1, 1) block is ill conditioned or rank deficient and the k × k (2, 2) block is non zero and small (k ≪ n). Additionally, the (2, 1) block may be rank deficient. Such systems arise in a range of practical applications. A novel null-space approach is proposed that transforms the system matrix into a nicer symmetric saddle point matrix of order n that has a non zero (2, 2) block of order at most 2k and, importantly, the (1, 1) block is symmetric positive definite. Success of any null-space approach depends on constructing a suitable null-space basis. We propose methods for wide matrices having far fewer rows than columns with the aim of balancing stability of the transformed saddle point matrix with preserving sparsity in the (1, 1) block. Linear least squares problems that contain a small number of dense rows are an important motivation and are used to illustrate our ideas and to explore their potential for solving large-scale systems.
- Published
- 2022
30. Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs
- Author
-
Yonghui Bo, Yushun Wang, and Wenjun Cai
- Subjects
Applied Mathematics ,FOS: Mathematics ,Numerical Analysis (math.NA) ,Mathematics - Numerical Analysis - Abstract
In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such novel strategy is based on the newly developed exponential scalar variable (ESAV) approach that can remove the bounded-from-blew restriction of nonlinear terms in the Hamiltonian functional and provides a totally explicit discretization of the auxiliary variable without computing extra inner products, which make it more effective and applicable than the traditional scalar auxiliary variable (SAV) approach. To achieve arbitrary high-order accuracy and energy preservation, we utilize the symplectic Runge-Kutta method for both solution variables and the auxiliary variable, where the values of internal stages in nonlinear terms are explicitly derived via an extrapolation from numerical solutions already obtained in the preceding calculation. A prediction-correction strategy is proposed to further improve the accuracy. Fourier pseudo-spectral method is then employed to obtain fully discrete schemes. Compared with the SAV schemes, the solution variables and the auxiliary variable in these ESAV schemes are now decoupled. Moreover, when the linear terms are of constant coefficients, the solution variables can be explicitly solved by using the fast Fourier transform. Numerical experiments are carried out for three Hamiltonian PDEs to demonstrate the efficiency and conservation of the ESAV schemes.
- Published
- 2022
31. Erratum to: 'Comments on another hybrid conjugate gradient algorithm for unconstrained optimization by Andrei'
- Author
-
Zhifeng Dai and Fenghua Wen
- Subjects
Combinatorics ,Applied Mathematics ,Numerical analysis ,Conjugate gradient method ,Theory of computation ,Unconstrained optimization ,Algebra over a field ,Mathematics - Abstract
We would like to correct some details of the above paper. According to References [1] and formula (2.8) in the above paper, we give some corrections about the above paper. More precisely, • The formula (2.9) in Theorem 2.1 should be g k+1dk+1 ≤ −δ‖gk+1‖. • Formula (3.1) should be g k sk ≤ −δ‖gk‖. • Formula (3.2) should be g k dk ≤ −δ‖gk‖. • Formula (3.3) should be g k sk ≤ −αkδ‖gk‖. • Formula (3.9) should be g k sk ≤ −αkδ‖gk‖.
- Published
- 2015
32. A new step size rule for the superiorization method and its application in computerized tomography
- Author
-
Touraj Nikazad, L. Afzalipour, Tommy Elfving, and M. Abbasi
- Subjects
Beräkningsmatematik ,Applied Mathematics ,Numerical analysis ,Linear system ,Residual ,Regularization (mathematics) ,Computational Mathematics ,Operator (computer programming) ,Conjugate gradient method ,Convergence (routing) ,Subgradient method ,Algorithm ,Convex feasibility problem ,Strictly quasi-nonexpansive operator ,Convex optimization ,Perturbation resilient iterative method ,Superiorization ,Computed tomography ,Sequential block method ,Mathematics - Abstract
In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the value of the penalty term and the other step reduces the residual of the underlying linear system (using an algorithmic operator T). For the new step size rule, we present a convergence analysis for the case when T belongs to a large subclass of strictly quasi-nonexpansive operators. To examine our algorithm numerically, we consider box constraints and use the total variation (TV) functional as a regularization term. The specific test cases are chosen from computed tomography using both noisy and noiseless data. We compare our algorithm with previously used parameters in superiorization. The T operator is based on sequential block iteration (for which our convergence analysis is valid), but we also use the conjugate gradient method (without theoretical support). Finally, we compare with the well-known "fast iterative shrinkage-thresholding algorithm" (FISTA). The numerical results demonstrate that our new step size rule improves previous step size rules for the superiorization methodology and is competitive with, and in several instances behaves better than, the other methods.
- Published
- 2021
33. On semilocal convergence analysis for two-step Newton method under generalized Lipschitz conditions in Banach spaces
- Author
-
Juan Liang, Yonghui Ling, and Weihua Lin
- Subjects
Nonlinear system ,symbols.namesake ,Operator (computer programming) ,Applied Mathematics ,Numerical analysis ,Convergence (routing) ,symbols ,Banach space ,Applied mathematics ,Lipschitz continuity ,Newton's method ,Mathematics ,Algebraic Riccati equation - Abstract
In the present paper, we consider the semilocal convergence issue of two-step Newton method for solving nonlinear operator equation in Banach spaces. Under the assumption that the first derivative of the operator satisfies a generalized Lipschitz condition, a new semilocal convergence analysis for the two-step Newton method is presented. The Q-cubic convergence is obtained by an additional condition. This analysis also allows us to obtain three important spacial cases about the convergence results based on the premises of Kantorovich, Smale and Nesterov-Nemirovskii types. As applications of our convergence results, a nonsymmetric algebraic Riccati equation arising from transport theory and a two-dimensional nonlinear convection-diffusion equation are provided.
- Published
- 2021
34. Generation of test matrices with specified eigenvalues using floating-point arithmetic
- Author
-
Takeshi Ogita and Katsuhisa Ozaki
- Subjects
Pure mathematics ,Numerical linear algebra ,Matrix (mathematics) ,Floating point ,Applied Mathematics ,Product (mathematics) ,Jordan normal form ,Block matrix ,Round-off error ,computer.software_genre ,computer ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper concerns test matrices for numerical linear algebra using an error-free transformation of floating-point arithmetic. For specified eigenvalues given by a user, we propose methods of generating a matrix whose eigenvalues are exactly known based on, for example, Schur or Jordan normal form and a block diagonal form. It is also possible to produce a real matrix with specified complex eigenvalues. Such test matrices with exactly known eigenvalues are useful for numerical algorithms in checking the accuracy of computed results. In particular, exact errors of eigenvalues can be monitored. To generate test matrices, we first propose an error-free transformation for the product of three matrices YSX. We approximate S by ${S^{\prime }}$ S ′ to compute ${YS^{\prime }X}$ Y S ′ X without a rounding error. Next, the error-free transformation is applied to the generation of test matrices with exactly known eigenvalues. Note that the exactly known eigenvalues of the constructed matrix may differ from the anticipated given eigenvalues. Finally, numerical examples are introduced in checking the accuracy of numerical computations for symmetric and unsymmetric eigenvalue problems.
- Published
- 2021
35. A Hessenberg-type algorithm for computing PageRank Problems
- Author
-
Xian-Ming Gu, Chun Wen, Zhao-Li Shen, Siu-Long Lei, Ke Zhang, Bruno Carpentieri, and Computational and Numerical Mathematics
- Subjects
Hessenberg process ,Applied Mathematics ,Numerical analysis ,Computation ,Arnoldi process ,Ritz values ,Numerical Analysis (math.NA) ,PageRank vector ,Solver ,Krylov subspace method ,law.invention ,65F15, 65F10, 65Y20 ,Dimension (vector space) ,PageRank ,law ,Theory of computation ,Convergence (routing) ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Algorithm ,Subspace topology ,Mathematics - Abstract
PageRank is a widespread model for analysing the relative relevance of nodes within large graphs arising in several applications. In the current paper, we present a cost-effective Hessenberg-type method built upon the Hessenberg process for the solution of difficult PageRank problems. The new method is very competitive with other popular algorithms in this field, such as Arnoldi-type methods, especially when the damping factor is close to $1$ and the dimension of the search subspace is large. The convergence and the complexity of the proposed algorithm are investigated. Numerical experiments are reported to show the efficiency of the new solver for practical PageRank computations., 4 Figures, 6 Tables. 19 pages, the current version has been improved further and accepted by {\em Numerical Algorithms}
- Published
- 2021
36. 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
37. 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
38. The FMM accelerated PIES with the modified binary tree in solving potential problems for the domains with curvilinear boundaries
- Author
-
Andrzej Kużelewski and Eugeniusz Zieniuk
- Subjects
Curvilinear coordinates ,Binary tree ,Applied Mathematics ,Fast multipole method ,Numerical analysis ,Boundary (topology) ,010103 numerical & computational mathematics ,01 natural sciences ,Integral equation ,010101 applied mathematics ,Personal computer ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
The paper presents an accelerating of solving potential boundary value problems (BVPs) with curvilinear boundaries by modified parametric integral equations system (PIES). The fast multipole method (FMM) known from the literature was included into modified PIES. To consider complex curvilinear shapes of a boundary, the modification of a binary tree used by the FMM is proposed. The FMM combined with the PIES, called the fast PIES, also allows a significant reduction of random access memory (RAM) utilization. Therefore, it is possible to solve complex engineering problems on a standard personal computer (PC). The proposed algorithm is based on the modified PIES and allows for obtaining accurate solutions of complex BVPs described by the curvilinear boundary at a reasonable time on the PC.
- Published
- 2021
39. Generalized cross validation for ℓp-ℓq minimization
- Author
-
Alessandro Buccini and Lothar Reichel
- Subjects
Applied Mathematics ,010103 numerical & computational mathematics ,Inverse problem ,01 natural sciences ,Regularization (mathematics) ,Cross-validation ,Term (time) ,010101 applied mathematics ,Noise ,Theory of computation ,Applied mathematics ,Minification ,Sensitivity (control systems) ,0101 mathematics ,Mathematics - Abstract
Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of ap-norm, and a regularization term, that is defined in terms of aq-norm. We allow 0
- Published
- 2021
40. Newton method for ℓ0-regularized optimization
- Author
-
Naihua Xiu, Lili Pan, and Shenglong Zhou
- Subjects
Quadratic growth ,Sequence ,math.OC ,0103 Numerical and Computational Mathematics ,Applied Mathematics ,Numerical analysis ,Numerical & Computational Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Stationary point ,Regularization (mathematics) ,010101 applied mathematics ,symbols.namesake ,Optimization and Control (math.OC) ,0102 Applied Mathematics ,Norm (mathematics) ,Bounded function ,FOS: Mathematics ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Newton's method ,0802 Computation Theory and Mathematics ,Mathematics - Abstract
As a tractable approach, regularization is frequently adopted in sparse optimization. This gives rise to regularized optimization, which aims to minimize the ℓ0 norm or its continuous surrogates that characterize the sparsity. From the continuity of surrogates to the discreteness of the ℓ0 norm, the most challenging model is the ℓ0-regularized optimization. There is an impressive body of work on the development of numerical algorithms to overcome this challenge. However, most of the developed methods only ensure that either the (sub)sequence converges to a stationary point from the deterministic optimization perspective or that the distance between each iteration and any given sparse reference point is bounded by an error bound in the sense of probability. In this paper, we develop a Newton-type method for the ℓ0-regularized optimization and prove that the generated sequence converges to a stationary point globally and quadratically under the standard assumptions, theoretically explaining that our method can perform surprisingly well.
- Published
- 2021
41. Extrapolated sequential constraint method for variational inequality over the intersection of fixed-point sets
- Author
-
Nimit Nimana and Mootta Prangprakhon
- Subjects
Applied Mathematics ,Numerical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Fixed point ,Constraint (information theory) ,Intersection ,Optimization and Control (math.OC) ,Conjugate gradient method ,Convergence (routing) ,Variational inequality ,FOS: Mathematics ,Method of steepest descent ,Applied mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
This paper deals with the solving of variational inequality problem where the constrained set is given as the intersection of a number of fixed-point sets. To this end, we present an extrapolated sequential constraint method. At each iteration, the proposed method is updated based on the ideas of a hybrid conjugate gradient method used to accelerate the well-known hybrid steepest descent method, and an extrapolated cyclic cutter method for solving a common fixed point problem. We prove strong convergence of the method under some suitable assumptions of step-size sequences. We finally show the numerical efficiency of the proposed method compared to some existing methods.
- Published
- 2021
42. On the performance of curvature stabilization time stepping methods for double-diffusive natural convection flows in the presence of magnetic field
- Author
-
Songul Kaya, Fatma G. Eroglu, and Aytekin Cibik
- Subjects
Natural convection ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,010103 numerical & computational mathematics ,Curvature ,01 natural sciences ,Magnetic field ,010101 applied mathematics ,Time stepping ,Flow (mathematics) ,Theory of computation ,Numerical tests ,0101 mathematics ,Mathematics - Abstract
This paper considers analytically and numerically the curvature-based stabilization method for double-diffusive natural convection flows in the presence of magnetic field. Unconditionally stable and optimally accurate second-order approximations are obtained. The impact of curvature stabilization on different flow fields are presented for several numerical test problems.
- Published
- 2021
43. Barycentric prolate interpolation and pseudospectral differentiation
- Author
-
Yan Tian
- Subjects
Preconditioner ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Barycentric coordinate system ,Collocation (remote sensing) ,01 natural sciences ,010101 applied mathematics ,Theory of computation ,Convergence (routing) ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics ,Interpolation - Abstract
In this paper, we provide further illustrations of prolate interpolation and pseudospectral differentiation based on the barycentric perspectives. The convergence rates of the barycentric prolate interpolation and pseudospectral differentiation are derived. Furthermore, we propose the new preconditioner, which leads to the well-conditioned prolate collocation scheme. Numerical examples are included to show the high accuracy of the new method. We apply this approach to solve the second-order boundary value problem and Helmholtz problem.
- Published
- 2021
44. Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure
- Author
-
Antoine Tambue and Aurelien Junior Noupelah
- Subjects
poisson random measure ,Fractional Brownian motion ,errors estimate ,Discretization ,Applied Mathematics ,Numerical analysis ,finite element method ,fractional Brownian motion ,Poisson random measure ,010103 numerical & computational mathematics ,Type (model theory) ,Exponential integrator ,VDP::Matematikk og Naturvitenskap: 400::Matematikk: 410 ,01 natural sciences ,Measure (mathematics) ,stochastic parabolic partial differential equations ,010101 applied mathematics ,Stochastic partial differential equation ,Mathematics::Probability ,timestepping methods ,finite element methods ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter $H>\frac {1}{2}$ H > 1 2 and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains the optimal convergence rates of order $\mathcal {O}(h^{2}+\varDelta t)$ O ( h 2 + Δ t ) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise.
- Published
- 2020
45. New subspace minimization conjugate gradient methods based on regularization model for unconstrained optimization
- Author
-
Hongwei Liu, Zexian Liu, and Ting Zhao
- Subjects
Line search ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Regularization (mathematics) ,010101 applied mathematics ,Conjugate gradient method ,Norm (mathematics) ,Theory of computation ,Applied mathematics ,Minification ,0101 mathematics ,Subspace topology ,Mathematics - Abstract
In this paper, two new subspace minimization conjugate gradient methods based on p-regularization models are proposed, where a special scaled norm in p-regularization model is analyzed. Different choices of special scaled norm lead to different solutions to the p-regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. R-linear convergence of the proposed methods is also analyzed. Numerical results show that, for the CUTEr library, the proposed methods are superior to four conjugate gradient methods, which were proposed by Hager and Zhang (SIAM J. Optim. 16(1):170–192, 2005), Dai and Kou (SIAM J. Optim. 23(1):296–320, 2013), Liu and Liu (J. Optim. Theory. Appl. 180(3):879–906, 2019) and Li et al. (Comput. Appl. Math. 38(1):2019), respectively.
- Published
- 2020
46. Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets
- Author
-
Delfim F. M. Torres, Pedro M. Lima, and Somayeh Nemati
- Subjects
Applied Mathematics ,Numerical analysis ,Third-kind Volterra integral equations ,Jacobi wavelets ,Collocation points ,34D05, 45E10, 65T60 ,Gauss–Jacobi quadrature ,Basis function ,Numerical Analysis (math.NA) ,Volterra integral equation ,Quadrature (mathematics) ,symbols.namesake ,Algebraic equation ,Wavelet ,FOS: Mathematics ,symbols ,Gaussian quadrature ,Applied mathematics ,Mathematics - Numerical Analysis ,Mathematics - Abstract
We propose a spectral collocation method, based on the generalized Jacobi wavelets along with the Gauss-Jacobi quadrature formula, for solving a class of third-kind Volterra integral equations. To do this, the interval of integration is first transformed into the interval [-1,1], by considering a suitable change of variable. Then, by introducing special Jacobi parameters, the integral part is approximated using the Gauss-Jacobi quadrature rule. An approximation of the unknown function is considered in terms of Jacobi wavelets functions with unknown coefficients, which must be determined. By substituting this approximation into the equation, and collocating the resulting equation at a set of collocation points, a system of linear algebraic equations is obtained. Then, we suggest a method to determine the number of basis functions necessary to attain a certain precision. Finally, some examples are included to illustrate the applicability, efficiency, and accuracy of the new scheme., Comment: This is a preprint of a paper whose final and definite form is with 'Numer. Algorithms', Print ISSN 1017-1398, Electronic ISSN 1572-9265, available at [https://www.springer.com/journal/11075]. Submitted 12-Oct-2019; Revised 17-Dec-2019; Accepted 11-Feb-2020
- Published
- 2020
47. Linearized Krylov subspace Bregman iteration with nonnegativity constraint
- Author
-
Lothar Reichel, Mirjeta Pasha, and Alessandro Buccini
- Subjects
Iterative method ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Krylov subspace ,01 natural sciences ,Projection (linear algebra) ,010101 applied mathematics ,Constraint (information theory) ,Theory of computation ,Bregman iteration ,Applied mathematics ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
Bregman-type iterative methods have received considerable attention in recent years due to their ease of implementation and the high quality of the computed solutions they deliver. However, these iterative methods may require a large number of iterations and this reduces their usefulness. This paper develops a computationally attractive linearized Bregman algorithm by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. The projection reduces the computational effort required for each iteration. A variant of this solution method, in which nonnegativity of each computed iterate is imposed, also is described. Extensive numerical examples illustrate the performance of the proposed methods.
- Published
- 2020
48. Numerical treatment of the generalized Love integral equation
- Author
-
Giada Serafini, Luisa Fermo, and Maria Grazia Russo
- Subjects
symbols.namesake ,Applied Mathematics ,Numerical analysis ,Theory of computation ,symbols ,Nyström method ,Gaussian quadrature ,Applied mathematics ,Numerical tests ,Integral equation ,Quadrature (mathematics) ,Mathematics - Abstract
In this paper, the generalized Love integral equation has been considered. In order to approximate the solution, a Nyström method based on a mixed quadrature rule has been proposed. Such a rule is a combination of a product and a “dilation” quadrature formula. The stability and convergence of the described numerical procedure have been discussed in suitable weighted spaces and the efficiency of the method is shown by some numerical tests.
- Published
- 2020
49. Local error estimation and step size control in adaptive linear multistep methods
- Author
-
Gustaf Söderlind, Imre Fekete, Yiannis Hadjimichael, and Carmen Arévalo
- Subjects
Applied Mathematics ,Numerical analysis ,Estimator ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Variable (computer science) ,Control theory ,Initial value problem ,0101 mathematics ,Constant (mathematics) ,Algorithm ,Mathematics ,Linear multistep method - Abstract
In a k-step adaptive linear multistep methods the coefficients depend on the k − 1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, chp+ 1y(p+ 1)(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic regime. In this paper we derive dynamic asymptotic models of the local error and its estimator, and show how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enable the use of controllers with enhanced stability, producing more regular step size sequences. Numerical examples illustrate the impact of dynamically compensated control, and that the proper choice of error estimator affects efficiency.
- Published
- 2020
50. Extension of the LP-Newton method to conic programming problems via semi-infinite representation
- Author
-
Mirai Tanaka and Takayuki Okuno
- Subjects
symbols.namesake ,Sequence ,Linear programming ,Semi-infinite ,Applied Mathematics ,Numerical analysis ,Theory of computation ,symbols ,Applied mathematics ,Polytope ,Newton's method ,Projection (linear algebra) ,Mathematics - Abstract
The LP-Newton method solves linear programming (LP) problems by repeatedly projecting a current point onto a certain relevant polytope. In this paper, we extend the algorithmic framework of the LP-Newton method to conic programming (CP) problems via a linear semi-infinite programming (LSIP) reformulation. In this extension, we produce a sequence by projection onto polyhedral cones constructed from LP problems obtained by finitely relaxing the LSIP problem equivalent to the CP problem. We show global convergence of the proposed algorithm under mild assumptions. To investigate its efficiency, we apply our proposed algorithm and a primal-dual interior-point method to second-order cone programming problems and compare the obtained results.
- Published
- 2020
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