1,218 results
Search Results
2. Remarks on E. A. Rahmanov's paper 'on the asymptotics of the ratio of orthogonal polynomials'
- Author
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Paul Nevai and Attila Máté
- Subjects
Discrete mathematics ,Mathematics(all) ,Numerical Analysis ,Statement (logic) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Orthogonal polynomials ,0101 mathematics ,Analysis ,Mathematics ,Counterexample - Abstract
It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.
- Published
- 1982
3. Implementable tensor methods in unconstrained convex optimization
- Author
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Yurii Nesterov, UCL - SSH/LIDAM/CORE - Center for operations research and econometrics, and UCL - SSH/IMMAQ/CORE - Center for operations research and econometrics
- Subjects
tensor mehtods ,90C06 ,General Mathematics ,0211 other engineering and technologies ,65K05 ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Upper and lower bounds ,90C25 ,Worst-case complexity bounds ,High-order methods ,Tensor methods ,Tensor (intrinsic definition) ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Mathematics ,021103 operations research ,Full Length Paper ,Regular polygon ,Order (ring theory) ,Function (mathematics) ,Lower complexity bounds ,Convex optimization ,Rate of convergence ,Software - Abstract
In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level $$O\left( {1 \over k^4}\right) $$ O 1 k 4 , where k is the number of iterations. This is very close to the lower bound of the order $$O\left( {1 \over k^5}\right) $$ O 1 k 5 , which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods.
- Published
- 2021
4. The Four-Parameter PSS Method for Solving the Sylvester Equation
- Author
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Yan-Ran Li, Xin-Hui Shao, and Hai-Long Shen
- Subjects
Iterative method ,General Mathematics ,lcsh:Mathematics ,Positive and skew-Hermitian iterative method ,Value (computer science) ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,Paper based ,lcsh:QA1-939 ,01 natural sciences ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Sylvester equation ,FPPSS iterative method ,0202 electrical engineering, electronic engineering, information engineering ,Computer Science (miscellaneous) ,Applied mathematics ,Order (group theory) ,Computer Science::Programming Languages ,0101 mathematics ,Coefficient matrix ,Engineering (miscellaneous) ,Mathematics - Abstract
In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter&rsquo, s value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime.
- Published
- 2019
- Full Text
- View/download PDF
5. Global optimization in Hilbert space
- Author
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Benoît Chachuat, Boris Houska, Engineering & Physical Science Research Council (EPSRC), and Commission of the European Communities
- Subjects
Technology ,Optimization problem ,Mathematics, Applied ,0211 other engineering and technologies ,CONVEX COMPUTATION ,010103 numerical & computational mathematics ,02 engineering and technology ,ELLIPSOIDS ,01 natural sciences ,90C26 ,93B40 ,Convergence analysis ,0102 Applied Mathematics ,Branch-and-lift ,CUT ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,Operations Research & Management Science ,0103 Numerical and Computational Mathematics ,Bounded function ,Physical Sciences ,symbols ,49M30 ,Calculus of variations ,INTEGRATION ,SET ,Complexity analysis ,Complete search ,Operations Research ,General Mathematics ,APPROXIMATIONS ,Set (abstract data type) ,symbols.namesake ,Applied mathematics ,ALGORITHM ,0101 mathematics ,INTERSECTION ,Global optimization ,0802 Computation Theory and Mathematics ,Science & Technology ,Infinite-dimensional optimization ,Hilbert space ,Computer Science, Software Engineering ,Constraint (information theory) ,Computer Science ,Software - Abstract
We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.
- Published
- 2017
6. Improved structural methods for nonlinear differential-algebraic equations via combinatorial relaxation
- Author
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Taihei Oki
- Subjects
Computer Science - Symbolic Computation ,FOS: Computer and information sciences ,Dynamical systems theory ,General Mathematics ,Mathematics::Optimization and Control ,010103 numerical & computational mathematics ,0102 computer and information sciences ,Symbolic Computation (cs.SC) ,01 natural sciences ,Computer Science::Systems and Control ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Applied mathematics ,Computer Science::Symbolic Computation ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Numerical analysis ,Applied Mathematics ,Relaxation (iterative method) ,Numerical Analysis (math.NA) ,Solver ,Numerical integration ,Nonlinear system ,Computational Mathematics ,Optimization and Control (math.OC) ,010201 computation theory & mathematics ,Differential algebraic equation ,Equation solving - Abstract
Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, the structural methods fail if the DAE has numerical or symbolic cancellations. For such DAEs, methods have been proposed to modify them to other DAEs to which the structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for a class of DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. The augmentation method has advantages that the equation solving is not needed and the sparsity of DAEs is retained. It is shown in numerical experiments that both methods, especially the augmentation method, successfully modify high-index DAEs that the DAE solver in MATLAB cannot handle., Comment: A preliminary version of this paper is to appear in Proceedings of the 44th International Symposium on Symbolic and Algebraic Computation (ISSAC 2019), Beijing, China, July 2019
- Published
- 2021
7. Convergence Analysis of Schwarz Waveform Relaxation for Nonlocal Diffusion Problems
- Author
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Ke Li, Yunxiang Zhao, and Dali Guo
- Subjects
Article Subject ,Discretization ,Differential equation ,General Mathematics ,General Engineering ,Relaxation (iterative method) ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Convolution ,Fractional calculus ,Quadrature (mathematics) ,010101 applied mathematics ,Rate of convergence ,QA1-939 ,Applied mathematics ,TA1-2040 ,0101 mathematics ,Temporal discretization ,Mathematics - Abstract
Diffusion equations with Riemann–Liouville fractional derivatives are Volterra integro-partial differential equations with weakly singular kernels and present fundamental challenges for numerical computation. In this paper, we make a convergence analysis of the Schwarz waveform relaxation (SWR) algorithms with Robin transmission conditions (TCs) for these problems. We focus on deriving good choice of the parameter involved in the Robin TCs, at the continuous and fully discretized level. Particularly, at the space-time continuous level, we show that the derived Robin parameter is much better than the one predicted by the well-understood equioscillation principle. At the fully discretized level, the problem of determining a good Robin parameter is studied in the convolution quadrature framework, which permits us to precisely capture the effects of different temporal discretization methods on the convergence rate of the SWR algorithms. The results obtained in this paper will be preliminary preparations for our further study of the SWR algorithms for integro-partial differential equations.
- Published
- 2021
8. Analysis of backward Euler projection FEM for the Landau–Lifshitz equation
- Author
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Weiwei Sun and Rong An
- Subjects
010101 applied mathematics ,Computational Mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,0101 mathematics ,Projection (set theory) ,01 natural sciences ,Backward Euler method ,Landau–Lifshitz–Gilbert equation ,Finite element method ,Mathematics - Abstract
The paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau–Lifshitz equation with the point-wise constraint $|{\textbf{m}}|=1$. The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been used in many areas in the past several decades. However, error analysis for the commonly used method has not been done since the classical energy approach cannot be applied directly. In this paper we present an optimal $\textbf{L}^2$ error analysis of the backward Euler sphere projection method by using quadratic or higher order finite elements under a time step condition $\tau =O(\epsilon _0 h)$ with some small $\epsilon _0>0$. The analysis is based on more precise estimates of the extra error caused by the sphere projection in both $\textbf{L}^2$ and $\textbf{H}^1$ norms, and the classical estimate of dual norm. Numerical experiment is provided to confirm our theoretical analysis.
- Published
- 2021
9. Sampling Discretization of Integral Norms
- Author
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Alexei Shadrin, Feng Dai, Andriy Prymak, Sergey Tikhonov, and Vladimir Temlyakov
- Subjects
Discretization ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Probabilistic logic ,010103 numerical & computational mathematics ,Extension (predicate logic) ,01 natural sciences ,Computational Mathematics ,Uniform norm ,Entropy (information theory) ,Applied mathematics ,0101 mathematics ,Trigonometry ,Analysis ,Subspace topology ,Mathematics - Abstract
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun only recently. In this paper we obtain a conditional theorem for all integral norms $$L_q$$ , $$1\le q
- Published
- 2021
10. On a new class of functional equations satisfied by polynomial functions
- Author
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Chisom Prince Okeke, Timothy Nadhomi, Maciej Sablik, and Tomasz Szostok
- Subjects
Polynomial functions ,Polynomial ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Fr'echet operator ,Functional equations ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Continuity of monomial functions ,Monomial functions ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Linear combination ,Linear equation ,Mathematics - Abstract
The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation$$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$F(x+y)-F(x)-F(y)=yf(x)+xf(y)considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.
- Published
- 2021
11. Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon
- Author
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Gaurav Dhar, Loïc Bourdin, Mathématiques & Sécurité de l'information (XLIM-MATHIS), XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), and Dhar, Gaurav
- Subjects
function of bounded variations ,General Mathematics ,Stieltjes integral ,0211 other engineering and technologies ,[MATH] Mathematics [math] ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,digital control ,[SPI.AUTO]Engineering Sciences [physics]/Automatic ,optimal control ,Variational principle ,Pontryagin maximum principle ,Applied mathematics ,[MATH]Mathematics [math] ,0101 mathematics ,Mathematics ,021103 operations research ,state constraints ,Numerical analysis ,[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC] ,Riemann–Stieltjes integral ,Ekeland variational principle ,shooting method ,Optimal control ,indirect method ,Nonlinear system ,[SPI.AUTO] Engineering Sciences [physics]/Automatic ,Sampled-data control ,Control system ,Bounded function ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Riemann-Stieltjes integral ,Software ,Hamiltonian (control theory) - Abstract
International audience; Sampled-data control systems have steadily been gaining interest for their applications in automatic engineering where they are implemented as digital controllers and recently results have been obtained in optimal control theory for nonlinear sampled-data control systems and certain generalizations. In this paper we derive a Pontryagin maximum principle for general nonlinear finite-dimensional optimal sampled-data control problems with running inequality state constraints. In particular, we obtain a nonpositive averaged Hamiltonian gradient condition with the adjoint vector being a function of bounded variations. Our proof is based on the Ekeland variational principle. In general, optimal control problems with running inequality state constraints are difficult to solve using numerical methods due to the discontinuities (the jumps and the singular part) of the adjoint vector. However in our case we find that under certain general hypotheses the adjoint vector only experiences jumps at most at the sampling times and moreover the trajectory only contacts the running inequality state constraints at most at the sampling times. We call this behavior a bouncing trajectory phenomenon and it constitutes the second major focus of this paper. Finally taking advantage of the bouncing trajectory phenomenon we numerically solve three examples with different kinds of constraints and in several dimensions.
- Published
- 2020
12. Minimization arguments in analysis of variational-hemivariational inequalities
- Author
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Weimin Han and Mircea Sofonea
- Subjects
Applied Mathematics ,General Mathematics ,Hilbert space ,Structure (category theory) ,General Physics and Astronomy ,Contrast (statistics) ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Contact mechanics ,Compact space ,symbols ,Applied mathematics ,Minification ,0101 mathematics ,Mathematics - Abstract
In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.
- Published
- 2022
- Full Text
- View/download PDF
13. Optimal error estimates and recovery technique of a mixed finite element method for nonlinear thermistor equations
- Author
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Huadong Gao, Chengda Wu, and Weiwei Sun
- Subjects
010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Applied Mathematics ,General Mathematics ,Thermistor ,Applied mathematics ,010103 numerical & computational mathematics ,Mixed finite element method ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with optimal error estimates and recovery technique of a classical mixed finite element method for the thermistor problem, which is governed by a parabolic/elliptic system with strong nonlinearity and coupling. The method is based on a popular combination of the lowest-order Raviart–Thomas mixed approximation for the electric potential/field $(\phi , \boldsymbol{\theta })$ and the linear Lagrange approximation for the temperature $u$. A common question is how the first-order approximation influences the accuracy of the second-order approximation to the temperature in such a strongly coupled system, while previous work only showed the first-order accuracy $O(h)$ for all three components in a traditional way. In this paper, we prove that the method produces the optimal second-order accuracy $O(h^2)$ for $u$ in the spatial direction, although the accuracy for the potential/field is in the order of $O(h)$. And more importantly, we propose a simple one-step recovery technique to obtain a new numerical electric potential/field of second-order accuracy. The analysis presented in this paper relies on an $H^{-1}$-norm estimate of the mixed finite element methods and analysis on a nonclassical elliptic map. We provide numerical experiments in both two- and three-dimensional spaces to confirm our theoretical analyses.
- Published
- 2020
14. On QZ steps with perfect shifts and computing the index of a differential-algebraic equation
- Author
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Paul Van Dooren and Nicola Mastronardi
- Subjects
Index (economics) ,Applied Mathematics ,General Mathematics ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Mathematics::Numerical Analysis ,Computational Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,Differential algebraic equation ,QZ algorithm ,eigenvalues ,perfect shifts ,index ,Mathematics - Abstract
In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.
- Published
- 2020
15. A New Class of Difference Methods with Intrinsic Parallelism for Burgers–Fisher Equation
- Author
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Yueyue Pan, Lifei Wu, and Xiaozhong Yang
- Subjects
Article Subject ,General Mathematics ,General Engineering ,Fisher equation ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,Simple (abstract algebra) ,Scheme (mathematics) ,Convergence (routing) ,QA1-939 ,Parallelism (grammar) ,Applied mathematics ,Uniqueness ,TA1-2040 ,0101 mathematics ,Absolute stability ,Mathematics - Abstract
This paper proposes a new class of difference methods with intrinsic parallelism for solving the Burgers–Fisher equation. A new class of parallel difference schemes of pure alternating segment explicit-implicit (PASE-I) and pure alternating segment implicit-explicit (PASI-E) are constructed by taking simple classical explicit and implicit schemes, combined with the alternating segment technique. The existence, uniqueness, linear absolute stability, and convergence for the solutions of PASE-I and PASI-E schemes are well illustrated. Both theoretical analysis and numerical experiments show that PASE-I and PASI-E schemes are linearly absolute stable, with 2-order time accuracy and 2-order spatial accuracy. Compared with the implicit scheme and the Crank–Nicolson (C-N) scheme, the computational efficiency of the PASE-I (PASI-E) scheme is greatly improved. The PASE-I and PASI-E schemes have obvious parallel computing properties, which show that the difference methods with intrinsic parallelism in this paper are feasible to solve the Burgers–Fisher equation.
- Published
- 2020
16. Multivariate approximation of functions on irregular domains by weighted least-squares methods
- Author
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Giovanni Migliorati, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
- Subjects
Christoffel symbols ,Computational complexity theory ,Basis (linear algebra) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Estimator ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Domain (mathematical analysis) ,Computational Mathematics ,Bounded function ,FOS: Mathematics ,Applied mathematics ,Orthonormal basis ,Mathematics - Numerical Analysis ,[MATH]Mathematics [math] ,0101 mathematics ,Mathematics - Abstract
We propose and analyse numerical algorithms based on weighted least squares for the approximation of a real-valued function on a general bounded domain $\Omega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(\Omega)$, the analysis in [6] shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations $m$ of the order $n \log n$. When an $L^2(\Omega)$-orthonormal basis of $V_n$ is available in analytic form, such estimators can be constructed using the algorithms described in [6,Section 5]. If the basis also has product form, then these algorithms have computational complexity linear in $d$ and $m$. In this paper we show that, when $\Omega$ is an irregular domain such that the analytic form of an $L^2(\Omega)$-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from $V_n$, again with $m$ of the order $n \log n$, but using a suitable surrogate basis of $V_n$ orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of $\Omega$ and $V_n$. Numerical results validating our analysis are presented., Comment: Version of the paper accepted for publication
- Published
- 2020
17. Optimal-rate finite-element solution of Dirichlet problems in curved domains with straight-edged tetrahedra
- Author
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Vitoriano Ruas
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Finite element solution ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Tetrahedron ,symbols ,0101 mathematics ,Mathematics - Abstract
In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.
- Published
- 2020
18. A priori analysis of a higher-order nonlinear elasticity model for an atomistic chain with periodic boundary condition
- Author
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Lei Zhang, Hao Wang, and Yangshuai Wang
- Subjects
Applied Mathematics ,General Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Chain (algebraic topology) ,Periodic boundary conditions ,Order (group theory) ,A priori and a posteriori ,Applied mathematics ,0101 mathematics ,Nonlinear elasticity ,Mathematics - Abstract
Nonlinear elastic models are widely used to describe the elastic response of crystalline solids, for example, the well-known Cauchy–Born model. While the Cauchy–Born model only depends on the strain, effects of higher-order strain gradients are significant and higher-order continuum models are preferred in various applications such as defect dynamics and modeling of carbon nanotubes. In this paper we rigorously derive a higher-order nonlinear elasticity model for crystals from its atomistic description in one dimension. We show that, compared to the second-order accuracy of the Cauchy–Born model, the higher-order continuum model in this paper is of fourth-order accuracy with respect to the interatomic spacing in the thermal dynamic limit. In addition we discuss the key issues for the derivation of higher-order continuum models in more general cases. The theoretical convergence results are demonstrated by numerical experiments.
- Published
- 2020
19. Trace finite element methods for surface vector-Laplace equations
- Author
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Thomas Jankuhn and Arnold Reusken
- Subjects
Partial differential equation ,Discretization ,Applied Mathematics ,General Mathematics ,Tangent ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Lagrange multiplier ,Norm (mathematics) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,symbols ,65N30, 65N12, 65N15 ,Applied mathematics ,Vector field ,Penalty method ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Abstract
In this paper we analyze a class of trace finite element methods for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vector field must be tangential to the surface (‘tangent condition’). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. The main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of the Lagrange multiplier.
- Published
- 2020
20. Packing colorings of subcubic outerplanar graphs
- Author
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Nicolas Gastineau, Olivier Togni, Boštjan Brešar, Faculty of Natural Sciences and Mathematics [Maribor], University of Maribor, Laboratoire d'Informatique de Bourgogne [Dijon] (LIB), Université de Bourgogne (UB), and Togni, Olivier
- Subjects
05C15, 05C12, 05C70 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Graph ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Combinatorics ,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] ,Integer ,Outerplanar graph ,Bounded function ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Bipartite graph ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Invariant (mathematics) ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a $(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an $S$-packing coloring for $S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the maximum degree of vertices), and this property does not hold if one of the integers $3$ is replaced by $4$ in the sequence $S$., Comment: 24 pages
- Published
- 2020
21. On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation
- Author
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Qiuyan Xu and Zhiyong Liu
- Subjects
Collocation ,Article Subject ,General Mathematics ,Direct method ,General Engineering ,Boundary (topology) ,Monge–Ampère equation ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Discrete system ,symbols.namesake ,Nonlinear system ,QA1-939 ,symbols ,Applied mathematics ,Radial basis function ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.
- Published
- 2020
22. GLOBAL RELAXED MODULUS-BASED SYNCHRONOUS BLOCK MULTISPLITTING MULTI-PARAMETERS METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS
- Author
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Xianyu Zuo and Litao Zhang
- Subjects
Numerical linear algebra ,General Mathematics ,Modulus ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,computer.software_genre ,01 natural sciences ,Complementarity (physics) ,Linear complementarity problem ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Matrix analysis ,0101 mathematics ,System matrix ,computer ,Mathematics - Abstract
Recently, Bai and Zhang [Numerical Linear Algebra with Applications, 2013, 20, 425Ƀ439] constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them; Li et al. [Journal of Nanchang University (Natural Science), 2013, 37, 307Ƀ312] studied synchronous block multisplitting iteration methods; Zhang and Li [Computers and Mathematics with Application, 2014, 67, 1954Ƀ1959] analyzed and obtained the weaker convergence results for linear complementarity problems. In this paper, we generalize their algorithms and further study global relaxed modulus-based synchronous block multisplitting multi-parameters methods for linear complementarity problems. Furthermore, we give the weaker convergence results of our new method in this paper when the system matrix is a block $ H_{+}- $matrix. Therefore, new results provide a guarantee for the optimal relaxation parameters, please refer to [A. Hadjidimos, M. Lapidakis and M. Tzoumas, SIAM Journal on Matrix Analysis and Applications, 2012, 33, 97Ƀ110, (dx.doi.org/10.1137/100811222)], where optimal parameters are determined.
- Published
- 2020
23. Extensions of linear operators from hyperplanes and strong uniqueness of best approximation in L(X,W)
- Author
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Paweł Wójcik
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,010103 numerical & computational mathematics ,Codimension ,Extension (predicate logic) ,01 natural sciences ,Projection (linear algebra) ,Operator (computer programming) ,Hyperplane ,Uniqueness ,0101 mathematics ,Analysis ,Subspace topology ,Mathematics - Abstract
The aim of this paper is to present some results concerning the problem of minimal projections and extensions. Let X be a reflexive Banach space and let Y be a closed subspace of X of codimension one. Let W be a finite-dimensional Banach space. We present a new sufficient condition under which any minimal extension of an operator A ∈ L ( Y , W ) is strongly unique. In this paper we show (in some circumstances) that if 1 λ ( Y , X ) , then a minimal projection from X onto Y is a strongly unique minimal projection. Moreover, we introduce and study a new geometric property of normed spaces. In this paper we also present a result concerning the strong unicity of best approximation.
- Published
- 2019
24. On the Convergence of a New Family of Multi-Point Ehrlich-Type Iterative Methods for Polynomial Zeros
- Author
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Petko D. Proinov and Milena Petkova
- Subjects
Polynomial ,iteration functions ,Iterative method ,General Mathematics ,010103 numerical & computational mathematics ,Construct (python library) ,multi-point iterative methods ,Type (model theory) ,01 natural sciences ,Local convergence ,010101 applied mathematics ,error estimates ,Convergence (routing) ,semilocal convergence ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,local convergence ,0101 mathematics ,polynomial zeros ,Engineering (miscellaneous) ,Multi point ,Mathematics - Abstract
In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.
- Published
- 2021
25. Error Estimations for Total Variation Type Regularization
- Author
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Chun Huang, Ziyang Yuan, and Kuan Li
- Subjects
Series (mathematics) ,General Mathematics ,Stability (learning theory) ,010103 numerical & computational mathematics ,Inverse problem ,Type (model theory) ,01 natural sciences ,Regularization (mathematics) ,010101 applied mathematics ,regularization ,total variation ,Rate of convergence ,Consistency (statistics) ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,A priori and a posteriori ,inverse problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
This paper provides several error estimations for total variation (TV) type regularization, which arises in a series of areas, for instance, signal and imaging processing, machine learning, etc. In this paper, some basic properties of the minimizer for the TV regularization problem such as stability, consistency and convergence rate are fully investigated. Both a priori and a posteriori rules are considered in this paper. Furthermore, an improved convergence rate is given based on the sparsity assumption. The problem under the condition of non-sparsity, which is common in practice, is also discussed, the results of the corresponding convergence rate are also presented under certain mild conditions.
- Published
- 2021
26. Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost?
- Author
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B. Cano and Nuria Reguera
- Subjects
Order reduction ,General Mathematics ,Krylov methods ,010103 numerical & computational mathematics ,Exponential integrator ,01 natural sciences ,Exponential function ,010101 applied mathematics ,efficiency ,QA1-939 ,Computer Science (miscellaneous) ,Spite ,Applied mathematics ,avoiding order reduction ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order reduction. The aim of the present paper is to explain the surprising result that, many times, in spite of having to calculate more terms at each step, the computational cost of doing it through Krylov methods decreases instead of increases. This is very interesting since, in that way, the methods improve not only in terms of accuracy, but also in terms of computational cost.
- Published
- 2021
- Full Text
- View/download PDF
27. On the Generalized Laplace Transform
- Author
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Paul Bosch, Héctor José Carmenate García, José M. Rodríguez, José M. Sigarreta, Comunidad de Madrid, and Ministerio de Ciencia, Innovación y Universidades (España)
- Subjects
Work (thermodynamics) ,Physics and Astronomy (miscellaneous) ,Matemáticas ,General Mathematics ,Inverse ,010103 numerical & computational mathematics ,01 natural sciences ,Convolution ,Computer Science (miscellaneous) ,Applied mathematics ,convolution ,0101 mathematics ,Harmonic oscillator ,Mathematics ,Laplace transform ,lcsh:Mathematics ,010102 general mathematics ,Order (ring theory) ,fractional derivative ,Fractional derivative ,lcsh:QA1-939 ,Generalized Laplace transform ,Fractional calculus ,generalized Laplace transform ,Chemistry (miscellaneous) ,Fractional differential - Abstract
This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications. In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations. We would like to thank the referees for their comments, which have improved the paper. The research of José M. Rodríguez and José M. Sigarreta was supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/10.13039/501100011033), Spain. The research of José M. Rodríguez is supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).
- Published
- 2021
28. Primal-dual optimization algorithms over Riemannian manifolds: an iteration complexity analysis
- Author
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Shiqian Ma, Junyu Zhang, and Shuzhong Zhang
- Subjects
Curvilinear coordinates ,021103 operations research ,Optimization problem ,General Mathematics ,Numerical analysis ,0211 other engineering and technologies ,Duality (optimization) ,010103 numerical & computational mathematics ,02 engineering and technology ,Riemannian manifold ,01 natural sciences ,Tensor (intrinsic definition) ,Euclidean geometry ,Embedding ,Applied mathematics ,0101 mathematics ,Software ,Mathematics - Abstract
In this paper we study nonconvex and nonsmooth multi-block optimization over Euclidean embedded (smooth) Riemannian submanifolds with coupled linear constraints. Such optimization problems naturally arise from machine learning, statistical learning, compressive sensing, image processing, and tensor PCA, among others. By utilizing the embedding structure, we develop an ADMM-like primal-dual approach based on decoupled solvable subroutines such as linearized proximal mappings, where the duality is with respect to the embedded Euclidean spaces. First, we introduce the optimality conditions for the afore-mentioned optimization models. Then, the notion of $$\epsilon $$ -stationary solutions is introduced as a result. The main part of the paper is to show that the proposed algorithms possess an iteration complexity of $$O(1/\epsilon ^2)$$ to reach an $$\epsilon $$ -stationary solution. For prohibitively large-size tensor or machine learning models, we present a sampling-based stochastic algorithm with the same iteration complexity bound in expectation. In case the subproblems are not analytically solvable, a feasible curvilinear line-search variant of the algorithm based on retraction operators is proposed. Finally, we show specifically how the algorithms can be implemented to solve a variety of practical problems such as the NP-hard maximum bisection problem, the $$\ell _q$$ regularized sparse tensor principal component analysis and the community detection problem. Our preliminary numerical results show great potentials of the proposed methods.
- Published
- 2019
29. Reproducing kernel orthogonal polynomials on the multinomial distribution
- Author
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Robert C. Griffiths and Persi Diaconis
- Subjects
Numerical Analysis ,Stationary distribution ,Markov chain ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Poisson kernel ,010103 numerical & computational mathematics ,Kravchuk polynomials ,01 natural sciences ,Combinatorics ,symbols.namesake ,Kernel (statistics) ,Orthogonal polynomials ,symbols ,Test statistic ,Multinomial distribution ,0101 mathematics ,Analysis ,Mathematics - Abstract
Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
- Published
- 2019
30. An extension of the Hermite–Hadamard inequality for convex and s-convex functions
- Author
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Péter Kórus
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,010103 numerical & computational mathematics ,Extension (predicate logic) ,01 natural sciences ,Iterated integrals ,Hermite–Hadamard inequality ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Convex function ,Mathematics - Abstract
The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.
- Published
- 2019
31. Comparison of probabilistic and deterministic point sets on the sphere
- Author
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Peter J. Grabner and T. A. Stepanyuk
- Subjects
Unit sphere ,Numerical Analysis ,Sequence ,Applied Mathematics ,General Mathematics ,Existential quantification ,010102 general mathematics ,Probabilistic logic ,Sampling (statistics) ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Point (geometry) ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .
- Published
- 2019
32. Approximation by modified Jain–Baskakov operators
- Author
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Marius Mihai Birou, Vishnu Narayan Mishra, and Preeti Sharma
- Subjects
Baskakov operator ,General Mathematics ,010102 general mathematics ,Applied mathematics ,Basis function ,Asymptotic formula ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Modulus of continuity ,Mathematics - Abstract
In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions e 0 {e_{0}} and e 1 {e_{1}} . It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.
- Published
- 2019
33. A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing
- Author
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Can Huang, Kim Ngan Le, and Martin Stynes
- Subjects
Forcing (recursion theory) ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,Gronwall's inequality ,Applied mathematics ,Fokker–Planck equation ,0101 mathematics ,Mathematics - Abstract
First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker–Planck equation with general forcing. SIAM J. Numer. Anal.,54 1763–1784) to solve the time-fractional Fokker–Planck equation on a domain $\varOmega \times [0,T]$ with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the $L^2(\varOmega )$ norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative $\alpha $ approaches the classical value of $1$. Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker–Planck problem, we present a new $L^2(\varOmega )$ convergence proof that avoids a flaw in the analysis of Le et al.’s paper for the semidiscrete (backward Euler scheme in time) method.
- Published
- 2019
34. On the existence of optimal meshes in every convex domain on the plane
- Author
-
András Kroó
- Subjects
Numerical Analysis ,Polynomial ,Conjecture ,Degree (graph theory) ,Plane (geometry) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Polytope ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Cardinality ,Polygon mesh ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we study the so called optimal polynomial meshes for domains in K ⊂ R d , d ≥ 2 . These meshes are discrete point sets Y n of cardinality c n d which have the property that ‖ p ‖ K ≤ A ‖ p ‖ Y n for every polynomial p of degree at most n with a constant A > 1 independent of n . It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C 2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d = 2 .
- Published
- 2019
35. Optimal proximal augmented Lagrangian method and its application to full Jacobian splitting for multi-block separable convex minimization problems
- Author
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Bingsheng He, Feng Ma, and Xiaoming Yuan
- Subjects
021103 operations research ,Augmented Lagrangian method ,Applied Mathematics ,General Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Computer Science::Numerical Analysis ,01 natural sciences ,Separable space ,Computational Mathematics ,symbols.namesake ,Block (telecommunications) ,Jacobian matrix and determinant ,Convex optimization ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
The augmented Lagrangian method (ALM) is fundamental in solving convex programming problems with linear constraints. The proximal version of ALM, which regularizes ALM’s subproblem over the primal variable at each iteration by an additional positive-definite quadratic proximal term, has been well studied in the literature. In this paper we show that it is not necessary to employ a positive-definite quadratic proximal term for the proximal ALM and the convergence can be still ensured if the positive definiteness is relaxed to indefiniteness by reducing the proximal parameter. An indefinite proximal version of the ALM is thus proposed for the generic setting of convex programming problems with linear constraints. We show that our relaxation is optimal in the sense that the proximal parameter cannot be further reduced. The consideration of indefinite proximal regularization is particularly meaningful for generating larger step sizes in solving ALM’s primal subproblems. When the model under discussion is separable in the sense that its objective function consists of finitely many additive function components without coupled variables, it is desired to decompose each ALM’s subproblem over the primal variable in Jacobian manner, replacing the original one by a sequence of easier and smaller decomposed subproblems, so that parallel computation can be applied. This full Jacobian splitting version of the ALM is known to be not necessarily convergent, and it has been studied in the literature that its convergence can be ensured if all the decomposed subproblems are further regularized by sufficiently large proximal terms. But how small the proximal parameter could be is still open. The other purpose of this paper is to show the smallest proximal parameter for the full Jacobian splitting version of ALM for solving multi-block separable convex minimization models.
- Published
- 2019
36. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- Author
-
Juan Carlos Cortés, Marc Jornet, and Julia Calatayud
- Subjects
Non-autonomous and random dynamical systems ,Adaptive generalized Polynomial Chaos ,General Mathematics ,Comparative case ,Monte Carlo method ,random Fröbenius method ,Random Frobenius method ,010103 numerical & computational mathematics ,01 natural sciences ,93e03 ,non-autonomous and random dynamical systems ,computational uncertainty quantification ,Stochastic Galerkin projection technique ,Linear differential equation ,34f05 ,QA1-939 ,Applied mathematics ,Order (group theory) ,60h35 ,0101 mathematics ,Uncertainty quantification ,Mathematics ,Final version ,Computational uncertainty quantification ,random fröbenius method ,93E03 ,adaptive generalized Polynomial Chaos ,stochastic Galerkin projection technique ,010101 applied mathematics ,Frobenius method ,34F05 ,60H35 ,MATEMATICA APLICADA - Abstract
[EN] This paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Frobenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Frobenius method, in which the random input coefficients may be expressed via a Karhunen-Loeve expansion., This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the valuable comments raised by the reviewer, which have improved the final version of the paper.
- Published
- 2018
37. Geometric properties of F-normed Orlicz spaces
- Author
-
Yunan Cui, Paweł Kolwicz, Radosław Kaczmarek, and Henryk Hudzik
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Function space ,Applied Mathematics ,General Mathematics ,Uniform convergence ,010102 general mathematics ,Monotonic function ,010103 numerical & computational mathematics ,01 natural sciences ,Linear subspace ,Monotone polygon ,Norm (mathematics) ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Mathematics - Abstract
The paper deals with F-normed functions and sequence spaces. First, some general results on such spaces are presented. But most of the results in this paper concern various monotonicity properties and various Kadec–Klee properties of F-normed Orlicz functions and sequence spaces and their subspaces of elements with order continuous norm, when they are generated by monotone Orlicz functions on $${\mathbb {R}}_{+}$$ and equipped with the classical Mazur–Orlicz F-norm. Strict monotonicity, lower (and upper) local uniform monotonicity and uniform monotonicity in the classical sense as well as their orthogonal counterparts are considered. It follows from the criteria that are presented for these properties that all the above classical monotonicity properties except for uniform monotonicity differ from their orthogonal counterparts [in contrast to Kothe spaces (see Hudzik et al. in Rocky Mt J Math 30(3):933–950, 2000)]. The Kadec–Klee properties that are considered in this paper correspond to various kinds of convergence: convergence locally in measure and convergence globally in measure for function spaces, uniform convergence and coordinatewise convergence in the case of sequence spaces.
- Published
- 2018
38. Superconvergence of kernel-based interpolation
- Author
-
Robert Schaback
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,Open problem ,Hilbert space ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Positive-definite matrix ,Superconvergence ,Eigenfunction ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Spline (mathematics) ,FOS: Mathematics ,symbols ,Applied mathematics ,Mathematics - Numerical Analysis ,Boundary value problem ,0101 mathematics ,Spline interpolation ,Analysis ,Mathematics - Abstract
From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization, and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings.
- Published
- 2018
39. Stream function formulation of surface Stokes equations
- Author
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Arnold Reusken
- Subjects
010101 applied mathematics ,Surface (mathematics) ,Computational Mathematics ,Applied Mathematics ,General Mathematics ,Stream function ,010103 numerical & computational mathematics ,Mechanics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we present a derivation of the surface Helmholtz decomposition, discuss its relation to the surface Hodge decomposition and derive a well-posed stream function formulation of a class of surface Stokes problems. We consider a $C^2$ connected (not necessarily simply connected) oriented hypersurface $\varGamma \subset \mathbb{R}^3$ without boundary. The surface gradient, divergence, curl and Laplace operators are defined in terms of the standard differential operators of the ambient Euclidean space $\mathbb{R}^3$. These representations are very convenient for the implementation of numerical methods for surface partial differential equations. We introduce surface $\mathbf H({\mathop{\rm div}}_{\varGamma})$ and $\mathbf H({\mathop{\rm curl}}_{\varGamma})$ spaces and derive useful properties of these spaces. A main result of the paper is the derivation of the Helmholtz decomposition, in terms of these surface differential operators, based on elementary differential calculus. As a corollary of this decomposition we obtain that for a simply connected surface to every tangential divergence-free velocity field there corresponds a unique scalar stream function. Using this result the variational form of the surface Stokes equation can be reformulated as a well-posed variational formulation of a fourth-order equation for the stream function. The latter can be rewritten as two coupled second-order equations, which form the basis for a finite element discretization. A particular finite element method is explained and the results of a numerical experiment with this method are presented.
- Published
- 2018
40. The weak core inverse
- Author
-
Néstor Thome, D. E. Ferreyra, Fabián Eduardo Levis, and A. N. Priori
- Subjects
Pure mathematics ,Multilinear algebra ,Class (set theory) ,Generalized inverse ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Inverse ,Generalized inverses ,010103 numerical & computational mathematics ,Weak group inverse ,01 natural sciences ,Square matrix ,Core EP decomposition ,Core (graph theory) ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Core inverse ,Mathematics - Abstract
[EN] In this paper, we introduce a new generalized inverse, called weak core inverse (or, in short, WC inverse) of a complex square matrix. This new inverse extends the notion of the core inverse defined by Baksalary and Trenkler (Linear Multilinear Algebra 58(6):681-697, 2010). We investigate characterizations, representations, and properties for this generalized inverse. In addition, we introduce weak core matrices (or, in short, WC matrices) and we show that these matrices form a more general class than that given by the known weak group matrices, recently investigated by H. Wang and X. Liu., In what follows, we detail the acknowledgements. D.E. Ferreyra, F.E. Levis, A.N. Priori - Partially supported by Universidad Nacional de Rio Cuarto (Grant PPI 18/C559) and CONICET (Grant PIP 112-201501-00433CO). D.E. Ferreyra F.E. Levis - Partially supported by ANPCyT (Grant PICT 201803492). D.E. Ferreyra, N. Thome -Partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (Grant Resol. Nro. 135/19). N. Thome -Partially supported by Ministerio de Economia, Industria y Competitividad of Spain (Grant Red de Excelencia MTM2017-90682-REDT) and by Universidad Nacional del Sur of Argentina (Grant 24/L108). We would like to thank the Referees for their valuable comments and suggestions which helped us to considerably improve the presentation of the paper
- Published
- 2021
41. A Nonconstant Shape Parameter-Dependent Competing Risks’ Model in Accelerate Life Test Based on Adaptive Type-II Progressive Hybrid Censoring
- Author
-
Yan Wang, Yimin Shi, and Min Wu
- Subjects
Article Subject ,Scale (ratio) ,General Mathematics ,General Engineering ,010103 numerical & computational mathematics ,Bivariate analysis ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Censoring (statistics) ,Gompertz distribution ,Shape parameter ,Confidence interval ,010104 statistics & probability ,Bayes' theorem ,QA1-939 ,Applied mathematics ,Statistics::Methodology ,Uniqueness ,0101 mathematics ,TA1-2040 ,Mathematics - Abstract
In this paper, the dependent competing risks’ model is considered in the constant-stress accelerated life test under the adaptive type-II progressive hybrid-censored scheme. The dependency between failure causes is modeled by Marshall–Olkin bivariate Gompertz distribution. The scale and shape parameters in the model both change with the stress levels, and the failure causes of some test units are unknown. Then, the maximum likelihood estimations and approximation confidence intervals of the unknown parameters are considered. And, the necessary and sufficient condition is established for the existence and uniqueness of the maximum likelihood estimations for unknown parameters. The Bayes approach is also employed to estimate the unknown parameters under suitable prior distributions. The Bayes estimations and highest posterior credible intervals of the unknown parameters are obtained. Finally, a simulation experiment has been performed to illustrate the methods proposed in this paper.
- Published
- 2021
42. An Integrated Genetic Algorithm and Homotopy Analysis Method to Solve Nonlinear Equation Systems
- Author
-
Hala A. Omar
- Subjects
Article Subject ,Differential equation ,General Mathematics ,Homotopy ,General Engineering ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Square (algebra) ,010101 applied mathematics ,Linear map ,Nonlinear system ,Genetic algorithm ,Convergence (routing) ,QA1-939 ,Applied mathematics ,0101 mathematics ,TA1-2040 ,Homotopy analysis method ,Mathematics - Abstract
Solving nonlinear equation systems for engineering applications is one of the broadest and most essential numerical studies. Several methods and combinations were developed to solve such problems by either finding their roots mathematically or formalizing such problems as an optimization task to obtain the optimal solution using a predetermined objective function. This paper proposes a new algorithm for solving square and nonsquare nonlinear systems combining the genetic algorithm (GA) and the homotopy analysis method (HAM). First, the GA is applied to find out the solution. If it is realized, the algorithm is terminated at this stage as the target solution is determined. Otherwise, the HAM is initiated based on the GA stage’s computed initial guess and linear operator. Moreover, the GA is utilized to calculate the optimum value of the convergence control parameter (h) algebraically without plotting the h-curves or identifying the valid region. Four test functions are examined in this paper to verify the proposed algorithm’s accuracy and efficiency. The results are compared to the Newton HAM (NHAM) and Newton homotopy differential equation (NHDE). The results corroborated the superiority of the proposed algorithm in solving nonlinear equation systems efficiently.
- Published
- 2021
43. Neural Network Method for Solving Time-Fractional Telegraph Equation
- Author
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Lelisa Kebena Bijiga and Wubshet Ibrahim
- Subjects
Optimization problem ,Artificial neural network ,Article Subject ,Differential equation ,General Mathematics ,General Engineering ,010103 numerical & computational mathematics ,Function (mathematics) ,Telegrapher's equations ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,QA1-939 ,Applied mathematics ,Development (differential geometry) ,Boundary value problem ,0101 mathematics ,Fractional differential ,TA1-2040 ,Mathematics - Abstract
Recently, the development of neural network method for solving differential equations has made a remarkable progress for solving fractional differential equations. In this paper, a neural network method is employed to solve time-fractional telegraph equation. The loss function containing initial/boundary conditions with adjustable parameters (weights and biases) is constructed. Also, in this paper, a time-fractional telegraph equation was formulated as an optimization problem. Numerical examples with known analytic solutions including numerical results, their graphs, weights, and biases were also discussed to confirm the accuracy of the method used. Also, the graphical and tabular results were analyzed thoroughly. The mean square errors for different choices of neurons and epochs have been presented in tables along with graphical presentations.
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- 2021
44. A Numerical Method for Compressible Model of Contamination from Nuclear Waste in Porous Media
- Author
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Zhifeng Wang
- Subjects
Article Subject ,General Mathematics ,Numerical analysis ,Linear system ,General Engineering ,010103 numerical & computational mathematics ,Mixed finite element method ,Grid ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Asymptotically optimal algorithm ,Compressibility ,QA1-939 ,Applied mathematics ,0101 mathematics ,TA1-2040 ,Porous medium ,Mathematics - Abstract
This paper studies and analyzes a model describing the flow of contaminated brines through the porous media under severe thermal conditions caused by the radioactive contaminants. The problem is approximated based on combining the mixed finite element method with the modified method of characteristics. In order to solve the resulting algebraic nonlinear equations efficiently, a two-grid method is presented and discussed in this paper. This approach includes a small nonlinear system on a coarse grid with size H and a linear system on a fine grid with size h . It follows from error estimates that asymptotically optimal accuracy can be obtained as long as the mesh sizes satisfy H = O h 1 / 3 .
- Published
- 2021
45. Convergence of a multidimensional Glimm-like scheme for the transport of fronts
- Author
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Olivier Hurisse, Thierry Gallouët, Aix Marseille Université (AMU), Mécanique des Fluides, Energies et Environnement (EDF R&D MFEE), EDF R&D (EDF R&D), and EDF (EDF)-EDF (EDF)
- Subjects
Scheme (programming language) ,Convection ,Class (set theory) ,Advection ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,010103 numerical & computational mathematics ,Glimm’s scheme ,01 natural sciences ,Projection (linear algebra) ,multidimensional problem ,Computational Mathematics ,Convergence (routing) ,Applied mathematics ,random choice ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,front propagation ,Preprint ,0101 mathematics ,computer ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Mathematics ,computer.programming_language - Abstract
International audience; This paper is devoted to the numerical analysis of a numerical scheme dedicated to the simulation of front advection (see https://hal.archives-ouvertes.fr/hal-02940407v1 for a preprint version presenting this scheme and some numerical results). The latter has been recently proposed and it is based on the ideas used for the Glimm's scheme. It relies on a two-step approach: a convection step is followed by a projection step which is based on a random choice. The main advantage of this scheme is that it applies to multi-dimensional problems. In the present paper a convergence result for this scheme is provided for a particular class of multi-dimensional problems. This work has been accepted for publication in IMA Journal of Numerical Analysis: https://doi.org/10.1093/imanum/drab053.
- Published
- 2020
46. On the Cauchy Problem of Vectorial Thermostatted Kinetic Frameworks
- Author
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Marco Menale, Carlo Bianca, Bruno Carbonaro, Bianca, Carlo, Carbonaro, Bruno, and Menale, Marco
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State variable ,Physics and Astronomy (miscellaneous) ,integro-differential equation ,General Mathematics ,Complex system ,010103 numerical & computational mathematics ,complexity ,kinetic theory ,Cauchy problem ,nonlinearity ,Mathematical models, Boltzmann equation, Vlasov equation, Kinetic Theory for Active Particles, well-posedness problems ,01 natural sciences ,Quadratic equation ,Computer Science (miscellaneous) ,Applied mathematics ,Initial value problem ,Uniqueness ,0101 mathematics ,Mathematics ,Variable (mathematics) ,lcsh:Mathematics ,010102 general mathematics ,lcsh:QA1-939 ,Nonlinear system ,Chemistry (miscellaneous) - Abstract
This paper is devoted to the derivation and mathematical analysis of new thermostatted kinetic theory frameworks for the modeling of nonequilibrium complex systems composed by particles whose microscopic state includes a vectorial state variable. The mathematical analysis refers to the global existence and uniqueness of the solution of the related Cauchy problem. Specifically, the paper is divided in two parts. In the first part the thermostatted framework with a continuous vectorial variable is proposed and analyzed. The framework consists of a system of partial integro-differential equations with quadratic type nonlinearities. In the second part the thermostatted framework with a discrete vectorial variable is investigated. Real world applications, such as social systems and crowd dynamics, and future research directions are outlined in the paper.
- Published
- 2020
47. A note on Korobov lattice rules for integration of analytic functions
- Author
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Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,11K45, 65D30 ,01 natural sciences ,Numerical integration ,Periodic function ,Lattice (order) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Fourier series ,SIMPLE algorithm ,Analytic function ,Mathematics - Abstract
We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension d . Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matching necessary and sufficient conditions, so far no explicit algorithm has been known which yields the desired result. In this paper we close this gap and prove that Korobov lattice rules are suitable algorithms in order to achieve exponential-weak tractability for integration in weighted Korobov spaces of analytic periodic functions.
- Published
- 2020
- Full Text
- View/download PDF
48. A Smoothing Newton Method with a Mixed Line Search for Monotone Weighted Complementarity Problems
- Author
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Xiaoqin Jiang and He Huang
- Subjects
021103 operations research ,Line search ,Article Subject ,General Mathematics ,0211 other engineering and technologies ,General Engineering ,010103 numerical & computational mathematics ,02 engineering and technology ,Engineering (General). Civil engineering (General) ,System of linear equations ,01 natural sciences ,Complementarity (physics) ,symbols.namesake ,Monotone polygon ,Complementarity theory ,Robustness (computer science) ,QA1-939 ,symbols ,Applied mathematics ,TA1-2040 ,0101 mathematics ,Newton's method ,Mathematics ,Smoothing - Abstract
In this paper, we present a smoothing Newton method for solving the monotone weighted complementarity problem (WCP). In each iteration of our method, the iterative direction is achieved by solving a system of linear equations and the iterative step length is achieved by adopting a line search. A feature of the line search criteria used in this paper is that monotone and nonmonotone line search are mixed used. The proposed method is new even when the WCP reduces to the standard complementarity problem. Particularly, the proposed method is proved to possess the global convergence under a weak assumption. The preliminary experimental results show the effectiveness and robustness of the proposed method for solving the concerned WCP.
- Published
- 2020
- Full Text
- View/download PDF
49. On a System of k-Difference Equations of Order Three
- Author
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Ibrahim Yalcinkaya, Yong-Min Li, Hijaz Ahmad, and Durhasan Turgut Tollu
- Subjects
Article Subject ,General Mathematics ,General Engineering ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,Order (business) ,QA1-939 ,Applied mathematics ,0101 mathematics ,TA1-2040 ,Mathematics - Abstract
In this paper, we deal with the global behavior of the positive solutions of the system of k -difference equations u n + 1 1 = α 1 u n − 1 1 / β 1 + α 1 u n − 2 2 r 1 , u n + 1 2 = α 2 u n − 1 2 / β 2 + α 2 u n − 2 3 r 2 , … , u n + 1 k = α k u n − 1 k / β k + α k u n − 2 1 r k , n ∈ ℕ 0 , where the initial conditions u − l i l = 0,1,2 are nonnegative real numbers and the parameters α i , β i , γ i , and r i are positive real numbers for i = 1,2 , … , k , by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.
- Published
- 2020
50. Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces
- Author
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Filip Turoboś, Jacek Jachymski, and Katarzyna Chrząszcz
- Subjects
Intersection theorem ,Pure mathematics ,Sequence ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,Metric space ,Iterated function ,Metrization theorem ,Metric (mathematics) ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Contraction principle ,Mathematics - Abstract
Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are now experiencing their well-deserved renaissance. The aim of this paper is to present two improvements of Frink’s metrization theorem along with some fixed point results for single-valued mappings on quasimetric spaces. Moreover, Cantor’s intersection theorem for sequences of sets which are not necessarily closed is established in a quasimetric setting. This enables us to give a new proof of a quasimetric version of the Banach Contraction Principle obtained by Bakhtin. We also provide error estimates for a sequence of iterates of a mapping, which seem to be new even in a metric setting.
- Published
- 2018
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