386 results
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2. 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
3. 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
4. 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
5. 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
6. 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
7. 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
8. Comparison of probabilistic and deterministic point sets on the sphere
- Author
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Peter J. Grabner and T. A. Stepanyuk
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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
9. 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
10. On the existence of optimal meshes in every convex domain on the plane
- Author
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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
11. Superconvergence of kernel-based interpolation
- Author
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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
12. 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
13. 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
14. 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
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15. Generalized conditioning based approaches to computing confidence intervals for solutions to stochastic variational inequalities
- Author
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Shu Lu and Michael Lamm
- Subjects
021103 operations research ,General Mathematics ,Computation ,Numerical analysis ,0211 other engineering and technologies ,Asymptotic distribution ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Confidence interval ,Nonlinear programming ,Piecewise linear function ,Convergence (routing) ,Variational inequality ,Applied mathematics ,0101 mathematics ,Software ,Mathematics - Abstract
Stochastic variational inequalities (SVI) provide a unified framework for the study of a general class of nonlinear optimization and Nash-type equilibrium problems with uncertain model data. Often the true solution to an SVI cannot be found directly and must be approximated. This paper considers the use of a sample average approximation (SAA), and proposes a new method to compute confidence intervals for individual components of the true SVI solution based on the asymptotic distribution of SAA solutions. We estimate the asymptotic distribution based on one SAA solution instead of generating multiple SAA solutions, and can handle inequality constraints without requiring the strict complementarity condition in the standard nonlinear programming setting. The method in this paper uses the confidence regions to guide the selection of a single piece of a piecewise linear function that governs the asymptotic distribution of SAA solutions, and does not rely on convergence rates of the SAA solutions in probability. It also provides options to control the computation procedure and investigate effects of certain key estimates on the intervals.
- Published
- 2018
16. On the relation of the spectral test to isotropic discrepancy and L-approximation in Sobolev spaces
- Author
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Mathias Sonnleitner and Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Isotropy ,Mathematical analysis ,Convex set ,Boundary (topology) ,010103 numerical & computational mathematics ,01 natural sciences ,Upper and lower bounds ,Spectral test ,Sobolev space ,Dimension (vector space) ,Unit cube ,0101 mathematics ,Mathematics - Abstract
This paper is a follow-up to the recent paper of Pillichshammer and Sonnleitner (2020) [12] . We show that the isotropic discrepancy of a lattice point set is at most d 2 2 ( d + 1 ) times its spectral test, thereby correcting the dependence on the dimension d and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test – and with it the isotropic discrepancy – is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.
- Published
- 2021
17. Comparing Two Numerical Methods for Approximating a New Giving Up Smoking Model Involving Fractional Order Derivatives
- Author
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Baha Alzalg, Vedat Suat Erturk, Anwar Zeb, Gul Zaman, Shaher Momani, and Ondokuz Mayıs Üniversitesi
- Subjects
Caputo fractional derivative ,Numerical solution ,General Mathematics ,Numerical analysis ,010102 general mathematics ,General Physics and Astronomy ,010103 numerical & computational mathematics ,General Chemistry ,Differential transform method ,01 natural sciences ,Smoking dynamics ,Generalized Euler method ,Euler method ,symbols.namesake ,symbols ,General Earth and Planetary Sciences ,Applied mathematics ,Order (group theory) ,0101 mathematics ,General Agricultural and Biological Sciences ,Giving Up Smoking ,Mathematics - Abstract
Alzalg, Baha/0000-0002-1839-8083; Momani, Shaher M./0000-0002-6326-8456; WOS: 000413785300005 In a recent paper (Zeb et al. in Appl Math Model 37(7):5326-5334, 2013), the authors presented a new model of giving up smoking model. In the present paper, the dynamics of this new model involving the Caputo derivative was studied numerically. For this purpose, generalized Euler method and the multistep generalized differential transform method are employed to compute accurate approximate solutions to this new giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between these two methods and the well-known Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.
- Published
- 2017
18. On the entropy numbers of the mixed smoothness function classes
- Author
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Vladimir Temlyakov
- Subjects
Numerical Analysis ,Multivariate statistics ,Nonlinear approximation ,Greedy approximation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Analysis ,Mathematics - Abstract
Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is based on recent developments of nonlinear approximation, in particular, on greedy approximation. This method consists of the following two steps strategy. At the first step we obtain bounds of the best m -term approximations with respect to a dictionary. At the second step we use general inequalities relating the entropy numbers to the best m -term approximations. For the lower bounds we use the volume estimates method, which is a well known powerful method for proving the lower bounds for the entropy numbers. It was used in a number of previous papers.
- Published
- 2017
19. Infinite-Dimensional $$\ell ^1$$ ℓ 1 Minimization and Function Approximation from Pointwise Data
- Author
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Ben Adcock
- Subjects
Pointwise ,Truncation ,General Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Function approximation ,symbols ,A priori and a posteriori ,Jacobi polynomials ,Applied mathematics ,Minification ,0101 mathematics ,Analysis ,Mathematics - Abstract
We consider the problem of approximating a smooth function from finitely many pointwise samples using $$\ell ^1$$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three advantages of this approach are as follows. First, it provides interpolatory approximations in the absence of noise. Second, it does not require a priori bounds on the expansion tail in order to be implemented. In particular, the truncation strategy we introduce as part of this framework is independent of the function being approximated, provided the function has sufficient regularity. Third, it allows one to explain the key role weights play in the minimization, namely, that of regularizing the problem and removing aliasing phenomena. In the second part of this paper, we present a worst-case error analysis for this approach. We provide a general recipe for analyzing this technique for arbitrary deterministic sets of points. Finally, we use this tool to show that weighted $$\ell ^1$$ minimization with Jacobi polynomials leads to an optimal method for approximating smooth, one-dimensional functions from scattered data.
- Published
- 2017
20. Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness
- Author
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Van Kien Nguyen
- Subjects
Statistics and Probability ,Mathematics::Functional Analysis ,Numerical Analysis ,Pure mathematics ,Control and Optimization ,Algebra and Number Theory ,Smoothness (probability theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Sobolev space ,Continuation ,FOS: Mathematics ,0101 mathematics ,Mathematics::Representation Theory ,Lp space ,Mathematics - Abstract
This paper is a continuation of the papers [21] and [22]. Here we shall investigate the asymptotic behaviour of Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness into Lebesgue spaces., 33 pages
- Published
- 2016
21. On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations
- Author
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Sanda Micula
- Subjects
fixed-point theory ,Physics and Astronomy (miscellaneous) ,General Mathematics ,Numerical analysis ,lcsh:Mathematics ,Fixed-point theorem ,010103 numerical & computational mathematics ,Fixed point ,lcsh:QA1-939 ,01 natural sciences ,Integral equation ,010101 applied mathematics ,mixed Volterra–Fredholm integral equations ,cubature formulas ,Chemistry (miscellaneous) ,Fixed-point iteration ,Convergence (routing) ,Picard iteration ,Computer Science (miscellaneous) ,Applied mathematics ,Uniqueness ,0101 mathematics ,Trapezoidal rule ,numerical approximation ,Mathematics - Abstract
In this paper, we propose a class of simple numerical methods for approximating solutions of one-dimensional mixed Volterra&ndash, Fredholm integral equations of the second kind. These methods are based on fixed point results for the existence and uniqueness of the solution (results which also provide successive iterations of the solution) and suitable cubature formulas for the numerical approximations. We discuss in detail a method using Picard iteration and the two-dimensional composite trapezoidal rule, giving convergence conditions and error estimates. The paper concludes with numerical experiments and a discussion of the methods proposed.
- Published
- 2019
22. Equilibrium problems in weakly admissible external fields created by pointwise charges
- Author
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J.F. Sánchez Lara, Ramón Orive, Franck Wielonsky, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Universidad de La Laguna [Tenerife - SP] (ULL), and Universidad de Granada = University of Granada (UGR)
- Subjects
Pointwise ,Numerical Analysis ,Pure mathematics ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,010103 numerical & computational mathematics ,Infinity ,01 natural sciences ,Measure (mathematics) ,Potential theory ,Compact space ,Simple (abstract algebra) ,FOS: Mathematics ,Uniqueness ,Complex Variables (math.CV) ,0101 mathematics ,[MATH]Mathematics [math] ,Complex plane ,Analysis ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,media_common - Abstract
The main subject of this paper is equilibrium problems on an unbounded conductor $\Sigma$ of the complex plane in the presence of a weakly admissible external field. An admissible external field $Q$ on $\Sigma$ satisfies, along with other mild conditions, the following growth property at infinity: $$\lim_{|x| \rightarrow \infty}(Q(x) - \log |x|) = +\infty.$$ This condition guarantees the existence and uniqueness of the equilibrium measure in the presence of $Q$, and the compactness of its support. In the last 10-15 years, several papers have dealt with weakly admissible external fields, in the sense that $Q$ satisfies a weaker condition at infinity, namely, $$\exists M\in(-\infty,\infty],\quad\liminf_{|x| \rightarrow \infty}(Q(x) - \log |x|) = M.$$ Under this last assumption, there still exists a unique equilibrium measure in the external field $Q$, but the support need not be a compact subset of $\Sigma$ anymore. In most examples considered in the literature the support is indeed unbounded. Our main goal in this paper is to illustrate this topic by means of a simple class of external fields on the real axis created by a pair of attractive and repellent charges in the complex plane, and to study the dynamics of the associated equilibrium measures as the strength of the charges evolves. As one of our findings, we exhibit configurations where the support of the equilibrium measure in a weakly admissible external field is a compact subset of the real axis. To achieve our goal, we extend some results from potential theory, known for admissible external fields, to the weakly admissible case. These new results may be of independent interest. Finally, the so--called signed equilibrium measure is an important tool in our analysis. Its relationship with the (positive) equilibrium measure is also explored., Comment: To appear in Journal of Approximation Theory
- Published
- 2019
23. A higher order Faber spline basis for sampling discretization of functions
- Author
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Tino Ullrich and Nadiia Derevianko
- Subjects
Numerical Analysis ,Mathematics::Functional Analysis ,Discretization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Basis function ,42C40, 46E35, 41A15, 65T60 ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,Lipschitz continuity ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Piecewise linear function ,Spline (mathematics) ,Compact space ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Remainder ,Real line ,Analysis ,Mathematics - Abstract
This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber–Schauder basis (Faber, 1908) except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebel’s monograph (Triebel, 2012) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylor’s remainder formula to the dual Chui–Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel–Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber–Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber–Schauder situation.
- Published
- 2019
- Full Text
- View/download PDF
24. Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method
- Author
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Weam Alharbi and Sergei Petrovskii
- Subjects
Power series ,Mittag-Leffler function ,lcsh:Mathematics ,General Mathematics ,Numerical analysis ,Homotopy ,fractional derivative ,010103 numerical & computational mathematics ,lcsh:QA1-939 ,01 natural sciences ,Ambartsumian equation ,Domain (mathematical analysis) ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,Convergence (routing) ,Computer Science (miscellaneous) ,symbols ,Applied mathematics ,0101 mathematics ,Homotopy perturbation method ,homotopy perturbation method ,Engineering (miscellaneous) ,Mathematics - Abstract
The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.
- Published
- 2020
25. Stability of Fredholm properties on interpolation Banach spaces
- Author
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Mieczysław Mastyło, Natan Kruglyak, and Irina Asekritova
- Subjects
Mathematics::Functional Analysis ,Numerical Analysis ,Pure mathematics ,Functor ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Surjective function ,Operator (computer programming) ,Interpolation space ,0101 mathematics ,Equivalence (measure theory) ,Analysis ,Interpolation ,Mathematics - Abstract
The main aim of this paper is to prove novel results on stability of the semi-Fredholm property of operators on interpolation spaces generated by interpolation functors. The methods are based on some general ideas we develop in the paper. This allows us to extend some previous work in literature to the abstract setting. We show an application to interpolation methods introduced by Cwikel–Kalton–Milman–Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also some other well known methods of interpolation. A by-product of these results get the stability of isomorphisms on Calderon products of Banach function lattices. We also study the important characteristics in operator Banach space theory, the so-called modules of injection and surjection, and we prove interpolation estimates of these modules of operators on scales of the Calderon complex interpolation spaces.
- Published
- 2020
26. Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals
- Author
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Peter Kritzer, Henryk Woźniakowski, and Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Polynomial ,Control and Optimization ,Algebra and Number Theory ,Logarithm ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,010103 numerical & computational mathematics ,01 natural sciences ,Exponential polynomial ,Exponential function ,Singular value ,symbols.namesake ,Tensor product ,Bounded function ,symbols ,0101 mathematics ,Mathematics - Abstract
We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an e -approximation for the d -variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on e − 1 . The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on e − 1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential ( s , t ) -weak tractability with max ( s , t ) ≥ 1 . For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential ( s , t ) -weak tractability with max ( s , t ) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).
- Published
- 2020
27. A note on tractability of multivariate analytic problems
- Author
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Yongping Liu and Guiqiao Xu
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Multivariate statistics ,Matching (statistics) ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Random variate ,Linear problem ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we study d -variate approximation for weighted Korobov spaces in the worst-case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. We give matching necessary and sufficient conditions for some notions of tractability in terms of two weight parameters. Our result is an affirmative answer to a problem which is left open in a recent paper of Kritzer, Pillichshammer and Wo?niakowski.
- Published
- 2016
28. On approximation properties of generalized Kantorovich-type sampling operators
- Author
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Olga Orlova and Gert Tamberg
- Subjects
Numerical Analysis ,Generalization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Microlocal analysis ,Sampling (statistics) ,010103 numerical & computational mathematics ,Spectral theorem ,Singular integral ,Operator theory ,01 natural sciences ,Electronic mail ,Fourier integral operator ,Convolution ,Kernel (statistics) ,0101 mathematics ,Operator norm ,Analysis ,Mathematics - Abstract
In this paper, we generalize the notion of Kantorovich-type sampling operators using the Fejer-type singular integral. By means of these operators we are able to reconstruct signals (functions) which are not necessarily continuous. Moreover, our generalization allows us to take the measurement error into account. The goal of this paper is to estimate the rate of approximation by the above operators via high-order modulus of smoothness.
- Published
- 2016
29. Solving a system of split variational inequality problems
- Author
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K. R. Kazmi
- Subjects
Sequence ,Mathematical optimization ,021103 operations research ,Iterative method ,General Mathematics ,Numerical analysis ,0211 other engineering and technologies ,Hilbert space ,010103 numerical & computational mathematics ,02 engineering and technology ,Algebraic geometry ,01 natural sciences ,symbols.namesake ,Variational inequality ,Projection method ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we introduce a system of split variational inequality problems in real Hilbert spaces. Using a projection method, we propose an iterative algorithm for solving this system of split variational inequality problems. Further, we prove that the sequence generated by the iterative algorithm converges strongly to a solution of the system of split variational inequality problems. Furthermore, we discuss some consequences of the main result. The iterative algorithms and results presented in this paper generalize, unify and improve the previously known results of this area.
- Published
- 2015
30. Calculating the spectral factorization and outer functions by sampling-based approximations—Fundamental limitations
- Author
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Volker Pohl and Holger Boche
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Sampling (statistics) ,Spectral density ,010103 numerical & computational mathematics ,Function (mathematics) ,Dirichlet's energy ,Spectral theorem ,Hardy space ,Singular integral ,01 natural sciences ,symbols.namesake ,symbols ,0101 mathematics ,Closed-form expression ,Analysis ,Mathematics - Abstract
This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because of a singular integral in this expression. Therefore approximation methods are usually applied to calculate the spectral factor. It is shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor for all densities in this set. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. The paper will also show that the same results hold for sampling-based algorithms for the calculation of the outer function in the theory of Hardy spaces.
- Published
- 2020
31. Multivariate bounded variation functions of Jordan–Wiener type
- Author
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Alexander Brudnyi and Yu. Brudnyi
- Subjects
Pointwise ,Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Predual ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,Linear subspace ,Separable space ,Bounded function ,Bounded variation ,Differentiable function ,0101 mathematics ,Analysis ,Mathematics - Abstract
We introduce and study spaces of multivariate functions of bounded variation generalizing the classical Jordan and Wiener spaces. Multivariate generalizations of the Jordan space were given by several prominent researchers. However, each of the proposed concepts preserves only few properties of Jordan variation which are designed to a selected application. In contrast, the multivariate generalization of the Jordan space presented in this paper preserves all known and reveals some previously unknown properties of the space. These, in turn, are special cases of the basic properties of the introduced spaces proved in the paper. Specifically, the first part of the paper describes structure properties of functions of bounded ( k , p ) -variation ( V p k functions). It includes assertions on discontinuity sets and pointwise differentiability of V p k functions and their Luzin type and C ∞ approximations. The second part presents results on Banach structure of V p k spaces, namely, atomic decomposition and constructive characterization of their predual spaces. As a result, we obtain the so-called two-stars theorems describing V p k spaces as second duals of their separable subspaces consisting of functions of “vanishing variation”.
- Published
- 2020
32. Sampling discretization error of integral norms for function classes
- Author
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Vladimir N. Temlyakov
- Subjects
Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Discretization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Supervised learning ,010103 numerical & computational mathematics ,Discretization error ,01 natural sciences ,Numerical integration ,Approximation error ,Norm (mathematics) ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
The new ingredient of this paper is that we consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research – supervised learning theory and numerical integration – can be used in sampling discretization of the square norm on different function classes.
- Published
- 2019
33. Adaptive regularization with cubics on manifolds
- Author
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Coralia Cartis, Nicolas Boumal, Naman Agarwal, and Brian Bullins
- Subjects
Hessian matrix ,021103 operations research ,Generalization ,General Mathematics ,Numerical analysis ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Lipschitz continuity ,01 natural sciences ,Arc (geometry) ,symbols.namesake ,Optimization and Control (math.OC) ,symbols ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Exponential map (Riemannian geometry) ,Newton's method ,Mathematics - Optimization and Control ,Software ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Adaptive regularization with cubics (ARC) is an algorithm for unconstrained, non-convex optimization. Akin to the popular trust-region method, its iterations can be thought of as approximate, safe-guarded Newton steps. For cost functions with Lipschitz continuous Hessian, ARC has optimal iteration complexity, in the sense that it produces an iterate with gradient smaller than $\varepsilon$ in $O(1/\varepsilon^{1.5})$ iterations. For the same price, it can also guarantee a Hessian with smallest eigenvalue larger than $-\varepsilon^{1/2}$. In this paper, we study a generalization of ARC to optimization on Riemannian manifolds. In particular, we generalize the iteration complexity results to this richer framework. Our central contribution lies in the identification of appropriate manifold-specific assumptions that allow us to secure these complexity guarantees both when using the exponential map and when using a general retraction. A substantial part of the paper is devoted to studying these assumptions---relevant beyond ARC---and providing user-friendly sufficient conditions for them. Numerical experiments are encouraging., Comment: 48 pages, 3 figures
- Published
- 2018
- Full Text
- View/download PDF
34. Approximation theorems for a family of multivariate neural network operators in Orlicz-type spaces
- Author
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Gianluca Vinti and Danilo Costarelli
- Subjects
Clustering high-dimensional data ,Pure mathematics ,Artificial neural network ,Sigmoidal functions ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Sigmoidal functions, Neural networks operators, Modular convergence,Orlicz spaces ,Neural networks operators ,010103 numerical & computational mathematics ,Sigmoid function ,Type (model theory) ,Orlicz spaces ,01 natural sciences ,Constructive ,Modular convergence ,Convergence (routing) ,0101 mathematics ,Interpolation ,Mathematics - Abstract
In this paper, we study the theory of a Kantorovich version of the multivariate neural network operators. Such operators, are activated by suitable kernels generated by sigmoidal functions. In particular, the main result here proved is a modular convergence theorem in Orlicz spaces. As special cases, convergence theorem in $$L^p$$ -spaces, interpolation spaces, and exponential-type spaces can be deduced. In general, multivariate approximations by constructive neural network algorithms are useful for applications to neurocomputing processes involving high dimensional data. At the end of the paper, several examples of activation functions of sigmoidal-type for which the above theory holds have been described.
- Published
- 2018
35. Best finite constrained approximations of one-dimensional probabilities
- Author
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Arno Berger and Chuang Xu
- Subjects
Numerical Analysis ,Approximations of π ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Probability (math.PR) ,010102 general mathematics ,010103 numerical & computational mathematics ,Infinity ,01 natural sciences ,Monotone polygon ,Rate of convergence ,Voronoi partition ,Step function ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Analysis ,Mathematics - Probability ,Probability measure ,Mathematics ,media_common - Abstract
This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the $L^r$-Kantorovich (or transport) distance, where either the locations or the weights of the approximations' atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. In view of emerging mathematical and statistical applications, special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The approach developed in this paper is elementary; it is based on best approximations of (monotone) $L^r$-functions by step functions, and thus different from, yet naturally complementary to, the classical Voronoi partition approach., To appear in J. Approx. Theory
- Published
- 2017
36. Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials
- Author
-
Abey López-García and Guillermo López Lagomasino
- Subjects
Algebraic properties ,Numerical Analysis ,Sequence ,Recurrence relation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,42C05, 30E10 (Primary) 47B39 (Secondary) ,010103 numerical & computational mathematics ,Star (graph theory) ,01 natural sciences ,Combinatorics ,Mathematics - Spectral Theory ,Mathematics - Classical Analysis and ODEs ,Orthogonal polynomials ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Limit (mathematics) ,0101 mathematics ,Algebraic number ,Spectral Theory (math.SP) ,Analysis ,Complement (set theory) ,Mathematics - Abstract
We investigate the ratio asymptotic behavior of the sequence $(Q_{n})_{n=0}^{\infty}$ of multiple orthogonal polynomials associated with a Nikishin system of $p\geq 1$ measures that are compactly supported on the star-like set of $p+1$ rays $S_{+}=\{z\in\mathbb{C}: z^{p+1}\geq 0\}$. The main algebraic property of these polynomials is that they satisfy a three-term recurrence relation of the form $zQ_{n}(z)=Q_{n+1}(z)+a_{n} Q_{n-p}(z)$ with $a_{n}>0$ for all $n\geq p$. Under a Rakhmanov-type condition on the measures generating the Nikishin system, we prove that the sequence of ratios $Q_{n+1}(z)/Q_{n}(z)$ and the sequence $a_{n}$ of recurrence coefficients are limit periodic with period $p(p+1)$. Our results complement some results obtained by the first author and Mi\~{n}a-D\'{i}az in a recent paper in which algebraic properties and weak asymptotics of these polynomials were investigated. Our results also extend some results obtained by the first author in the case $p=2$., Comment: Minor corrections were done for this version. A continuation of this work is in arXiv: 1907.03002. This paper has 37 pages, 1 table, no figures
- Published
- 2016
37. Error Analysis of Lagrange Interpolation on Tetrahedrons
- Author
-
Takuya Tsuchiya and Kenta Kobayashi
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Lagrange polynomial ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,01 natural sciences ,Angle condition ,65D05, 65N30 ,symbols.namesake ,Error analysis ,Bounded function ,symbols ,Tetrahedron ,FOS: Mathematics ,Applied mathematics ,Mathematics::Metric Geometry ,Mathematics - Numerical Analysis ,0101 mathematics ,Circumscribed circle ,Analysis ,Mathematics - Abstract
This paper describes the analysis of Lagrange interpolation errors on tetrahedrons. In many textbooks, the error analysis of Lagrange interpolation is conducted under geometric assumptions such as shape regularity or the (generalized) maximum angle condition. In this paper, we present a new estimation in which the error is bounded in terms of the diameter and projected circumradius of the tetrahedron. Because we do not impose any geometric restrictions on the tetrahedron itself, our error estimation may be applied to any tetrahedralizations of domains including very thin tetrahedrons., Comment: To appear in Journal of Approximation Theory
- Published
- 2016
- Full Text
- View/download PDF
38. On strata of degenerate polyhedral cones, II: Relations between condition measures
- Author
-
Felipe Cucker, Javier Peña, and Dennis Cheung
- Subjects
Statistics and Probability ,Numerical Analysis ,021103 operations research ,Control and Optimization ,Algebra and Number Theory ,Condition numbers ,Applied Mathematics ,General Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Combinatorics ,Complementarity problems ,Linear programming ,0101 mathematics ,Mathematics - Abstract
In a paper Cheung, Cucker and Peña (in press) [5] that can be seen as the first part of this one, we extended the well-known condition numbers for polyhedral conic systems C(A) Renegar (1994, 1995) [7–9] and C(A) Cheung and Cucker (2001) [3] to versions C¯(A) and C¯(A) that are finite for all input matrices A∈Rn×m. In this paper we compare C¯(A) and C¯(A) with other condition measures for the same problem that are also always finite.
- Published
- 2010
39. Generalized tractability for multivariate problems Part I
- Author
-
Henryk Woźniakowski and Michael Gnewuch
- Subjects
Discrete mathematics ,Statistics and Probability ,Polynomial ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,Information-based complexity ,General Mathematics ,media_common.quotation_subject ,Applied Mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,Space (mathematics) ,Infinity ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Set (abstract data type) ,Tensor product ,Bounded function ,0101 mathematics ,Mathematics ,media_common - Abstract
Many papers study polynomial tractability for multivariate problems. Let n(@?,d) be the minimal number of information evaluations needed to reduce the initial error by a factor of @? for a multivariate problem defined on a space of d-variate functions. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that n(@?,d) is bounded by a polynomial in @?^-^1 and d and this holds for all (@?^-^1,d)@?[1,~)xN. In this paper we study generalized tractability by verifying when n(@?,d) can be bounded by a power of T(@?^-^1,d) for all (@?^-^1,d)@[email protected], where @W can be a proper subset of [1,~)xN. Here T is a tractability function, which is non-decreasing in both variables and grows slower than exponentially to infinity. In this article we consider the set @W=[1,~)x{1,2,...,d^*}@?[1,@?"0^-^1)xN for some d^*>=1 and @?"[email protected]?(0,1). We study linear tensor product problems for which we can compute arbitrary linear functionals as information evaluations. We present necessary and sufficient conditions on T such that generalized tractability holds for linear tensor product problems. We show a number of examples for which polynomial tractability does not hold but generalized tractability does.
- Published
- 2007
- Full Text
- View/download PDF
40. Stochastic Optimization Using a Trust-Region Method and Random Models
- Author
-
Matt Menickelly, Ruobing Chen, and Katya Scheinberg
- Subjects
Trust region ,021103 operations research ,General Mathematics ,Numerical analysis ,0211 other engineering and technologies ,Sample (statistics) ,010103 numerical & computational mathematics ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,Stationary point ,Optimization and Control (math.OC) ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Stochastic optimization ,Noise (video) ,0101 mathematics ,Mathematics - Optimization and Control ,Software ,Mathematics - Abstract
In this paper, we propose and analyze a trust-region model-based algorithm for solving unconstrained stochastic optimization problems. Our framework utilizes random models of an objective function $f(x)$, obtained from stochastic observations of the function or its gradient. Our method also utilizes estimates of function values to gauge progress that is being made. The convergence analysis relies on requirements that these models and these estimates are sufficiently accurate with sufficiently high, but fixed, probability. Beyond these conditions, no assumptions are made on how these models and estimates are generated. Under these general conditions we show an almost sure global convergence of the method to a first order stationary point. In the second part of the paper, we present examples of generating sufficiently accurate random models under biased or unbiased noise assumptions. Lastly, we present some computational results showing the benefits of the proposed method compared to existing approaches that are based on sample averaging or stochastic gradients., Comment: Revised version posted September 23, 2016. Originally posted April 17, 2015
- Published
- 2015
- Full Text
- View/download PDF
41. On the Robustness of Regularized Pairwise Learning Methods Based on Kernels
- Author
-
Andreas Christmann and Ding-Xuan Zhou
- Subjects
Statistics and Probability ,FOS: Computer and information sciences ,Control and Optimization ,General Mathematics ,Mathematics - Statistics Theory ,Machine Learning (stat.ML) ,010103 numerical & computational mathematics ,Statistics Theory (math.ST) ,01 natural sciences ,010104 statistics & probability ,Pairwise learning ,Statistics - Machine Learning ,FOS: Mathematics ,Applied mathematics ,Entropy (information theory) ,Empirical risk minimization ,0101 mathematics ,Mathematics ,Numerical Analysis ,Algebra and Number Theory ,Applied Mathematics ,Regular polygon ,Lipschitz continuity ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Support vector machine ,Bounded function ,62G05, 62G08, 62G35, 62G20, 62F40 ,Minification - Abstract
Regularized empirical risk minimization including support vector machines plays an important role in machine learning theory. In this paper regularized pairwise learning (RPL) methods based on kernels will be investigated. One example is regularized minimization of the error entropy loss which has recently attracted quite some interest from the viewpoint of consistency and learning rates. This paper shows that such RPL methods have additionally good statistical robustness properties, if the loss function and the kernel are chosen appropriately. We treat two cases of particular interest: (i) a bounded and non-convex loss function and (ii) an unbounded convex loss function satisfying a certain Lipschitz type condition., Comment: 36 pages, 1 figure
- Published
- 2015
- Full Text
- View/download PDF
42. Approximation with scaled shift-invariant spaces by means of quasi-projection operators
- Author
-
Rong-Qing Jia
- Subjects
Mathematics(all) ,General Mathematics ,Wavelet Analysis ,Cascade algorithms ,Moduli of smoothness ,010103 numerical & computational mathematics ,Spectral theorem ,Quasi-interpolation ,01 natural sciences ,0101 mathematics ,Mathematics ,Approximation theory ,Numerical Analysis ,Lipschitz spaces ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Operator theory ,Lipschitz continuity ,Compact operator on Hilbert space ,Approximation order ,Sobolev space ,Spline (mathematics) ,Shift-invariant spaces ,Sobolev spaces ,De Boor's algorithm ,Quasi-projection ,Analysis - Abstract
The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis.
- Published
- 2004
- Full Text
- View/download PDF
43. Cubature formulas, discrepancy, and nonlinear approximation
- Author
-
Vladimir Temlyakov
- Subjects
Statistics and Probability ,Class (set theory) ,Approximation theory ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,Generalization ,General Mathematics ,Applied Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,Mathematical proof ,01 natural sciences ,Numerical integration ,Algebra ,Bounded function ,Applied mathematics ,0101 mathematics ,Discrepancy theory ,Mathematics - Abstract
The main goal of this paper is to demonstrate connections between the following three big areas of research: the theory of cubature formulas (numerical integration), the discrepancy theory, and nonlinear approximation. In Section 1, we discuss a relation between results on cubature formulas and on discrepancy. In particular, we show how standard in the theory of cubature formulas settings can be translated into the discrepancy problem and into a natural generalization of the discrepancy problem. This leads to a concept of the r-discrepancy. In Section 2, we present results on a relation between construction of an optimal cubature formula with m knots for a given function class and best nonlinear m-term approximation of a special function determined by the function class. The nonlinear m-term approximation is taken with regard to a redundant dictionary also determined by the function class. Sections 3 and 4 contain some known results on the lower and the upper estimates of errors of optimal cubature formulas for the class of functions with bounded mixed derivative. One of the important messages (well known in approximation theory) of this paper is that the theory of discrepancy is closely connected with the theory of cubature formulas for the classes of functions with bounded mixed derivative. We have included in the paper both new results (Section 2) and known results. We included some known results with their proofs for the following two reasons. First of all we want to make the paper self-contained (within reasonable limits). Secondly, we selected the proofs which demonstrate different methods and are not very much technically involved. Section 5 contains historical notes on discrepancy and cubature formulas, some further comments and remarks. Historical remarks on nonlinear approximation are included in Section 2. We want to point out that this paper is not a complete survey in any of the above-mentioned areas. We did not even try to provide a complete list of results in those areas. We rather wanted to highlight the most typical results in cubature formulas (Sections 3 and 4) and show their relation to the discrepancy theory.
- Published
- 2003
- Full Text
- View/download PDF
44. Asymptotic behavior of splitting schemes involving time-subcycling techniques
- Author
-
Guillaume Dujardin, Pauline Lafitte, Quantitative methods for stochastic models in physics (MEPHYSTO), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université libre de Bruxelles (ULB), Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Mathématiques Appliquées aux Systèmes - EA 4037 (MAS), Ecole Centrale Paris, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, and Université de Lille-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Mathematical optimization ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Ode ,Stability (learning theory) ,Context (language use) ,010103 numerical & computational mathematics ,01 natural sciences ,Numerical integration ,010101 applied mathematics ,Computational Mathematics ,Mathematics - Analysis of PDEs ,Rate of convergence ,Linear differential equation ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Limit (mathematics) ,0101 mathematics ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
This paper deals with the numerical integration of well-posed multiscale systems of ODEs or evolutionary PDEs. As these systems appear naturally in engineering problems, time-subcycling techniques are widely used every day to improve computational efficiency. These methods rely on a decomposition of the vector field in a fast part and a slow part and take advantage of that decomposition. This way, if an unconditionnally stable (semi-)implicit scheme cannot be easily implemented, one can integrate the fast equations with a much smaller time step than that of the slow equations, instead of having to integrate the whole system with a very small time-step to ensure stability. Then, one can build a numerical integrator using a standard composition method, such as a Lie or a Strang formula for example. Such methods are primarily designed to be convergent in short-time to the solution of the original problems. However, their longtime behavior rises interesting questions, the answers to which are not very well known. In particular, when the solutions of the problems converge in time to an asymptotic equilibrium state, the question of the asymptotic accuracy of the numerical longtime limit of the schemes as well as that of the rate of convergence is certainly of interest. In this context, the asymptotic error is defined as the difference between the exact and numerical asymptotic states. The goal of this paper is to apply that kind of numerical methods based on splitting schemes with subcycling to some simple examples of evolutionary ODEs and PDEs that have attractive equilibrium states, to address the aforementioned questions of asymptotic accuracy, to perform a rigorous analysis, and to compare them with their counterparts without subcycling. Our analysis is developed on simple linear ODE and PDE toy-models and is illustrated with several numerical experiments on these toy-models as well as on more complex systems. Lie and, Comment: IMA Journal of Numerical Analysis, Oxford University Press (OUP): Policy A - Oxford Open Option A, 2015
- Published
- 2014
- Full Text
- View/download PDF
45. Complexity of Weighted Approximation over Rd
- Author
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Henryk Woźniakowski and G. W. Wasilkowski
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Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Weight function ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,Approximation algorithm ,integration ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Continuation ,worst case complexity ,Approximation error ,Norm (mathematics) ,weighted multivariate approximation ,0202 electrical engineering, electronic engineering, information engineering ,Worst-case complexity ,Uniform boundedness ,Partial derivative ,0101 mathematics ,Mathematics - Abstract
We study approximation of multivariate functions defined over Rd. We assume that all rth order partial derivatives of the functions considered are continuous and uniformly bounded. Approximation algorithms U(f) only use the values of f or its partial derivatives up to order r. We want to recover the function f with small error measured in a weighted Lq norm with a weight function ρ. We study the worst case (information) complexity which is equal to the minimal number of function and derivative evaluations needed to obtain error ε. We provide necessary and sufficient conditions in terms of the weight ρ and the parameters q and r for the weighted approximation problem to have finite complexity. We also provide conditions guaranteeing that the complexity is of the same order as the complexity of the classical approximation problem over a finite domain. Since the complexity of the weighted integration problem is equivalent to the complexity of the weighted approximation problem with q=1, the results of this paper also hold for weighted integration. This paper is a continuation of [7], where weighted approximation over R was studied.
- Published
- 2001
46. Approximating Fixed Points of Weakly Contracting Mappings
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K. Sikorski, Leonid Khachiyan, and Z. Huang
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Discrete mathematics ,Statistics and Probability ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,Iterative method ,General Mathematics ,Applied Mathematics ,Dimension (graph theory) ,010103 numerical & computational mathematics ,Function (mathematics) ,Fixed point ,01 natural sciences ,Ellipsoid ,Upper and lower bounds ,010101 applied mathematics ,Combinatorics ,Approximation error ,Simple (abstract algebra) ,0101 mathematics ,Mathematics - Abstract
We consider the problem of approximating fixed points of contractive functions whose contraction factor is close to 1. In a previous paper (1993, K. Sikorski et al. , J. Complexity 9 , 181–200), we proved that for the absolute error criterion, the upper bound on the number of function evaluations to compute e -approximations is O( n 3 (ln(1/ e )+ln(1/(1− q ))+ln n )) in the worst case, where 0 q n is the dimension of the problem. This upper bound is achieved by the circumscribed ellipsoid (CE) algorithm combined with a dimensional deflation process. In this paper we present an inscribed ellipsoid (IE) algorithm that enjoys O( n (ln(1/ e )+ln(1/(1− q )))) bound. For q close to 1, the IE algorithm thus runs in many fewer iterations than the simple iteration method, that requires ⌈ln(1/ e )/ln(1/ q )⌉ function evaluations. Our analysis also implies that: (1) The dimensional deflation procedure in the CE algorithm is not necessary and that the resulting “plain” CE algorithm enjoys O( n 2 (log(1/ e )+log(1/(1− q )))) upper bound on the number of function evaluations. (2) The IE algorithm solves the problem in the residual sense, i.e., computes x such that ‖ f ( x )− x ‖⩽ δ , with O( n ln(1/ δ )) function evaluations for every q ⩽1.
- Published
- 1999
- Full Text
- View/download PDF
47. Numerical Integration and Discrepancy Under Smoothness Assumption and Without It
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Vladimir Temlyakov
- Subjects
Approximation theory ,Work (thermodynamics) ,Smoothness (probability theory) ,General Mathematics ,Numerical analysis ,010102 general mathematics ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Numerical integration ,Computational Mathematics ,Nonlinear approximation ,Applied mathematics ,0101 mathematics ,Analysis ,Discrepancy theory ,Mathematics - Abstract
The goal of this paper is threefold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work in proving lower bounds for recently developed new type of discrepancy—the smooth discrepancy. Third, we consider numerical integration in classes, for which we do not impose any smoothness assumptions. We illustrate how nonlinear approximation, in particular greedy approximation, allows us to guarantee some rate of decay of errors of numerical integration even in such a general setting with no smoothness assumptions.
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- 2021
48. A numerical method for two-dimensional Hammerstein integral equations
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Sanda Micula
- Subjects
Collocation ,General Mathematics ,Numerical analysis ,010102 general mathematics ,010103 numerical & computational mathematics ,Linear interpolation ,Superconvergence ,01 natural sciences ,Integral equation ,Numerical integration ,Collocation method ,Applied mathematics ,0101 mathematics ,Interpolation ,Mathematics - Abstract
"In this paper we investigate a collocation method for the approximate solution of Hammerstein integral equations in two dimensions. As in [8], col- location is applied to a reformulation of the equation in a new unknown, thus reducing the computational cost and simplifying the implementation. We start with a special type of piecewise linear interpolation over triangles for a refor- mulation of the equation. This leads to a numerical integration scheme that can then be extended to any bounded domain in R2, which is used in collocation. We analyze and prove the convergence of the method and give error estimates. As the quadrature formula has a higher degree of precision than expected with linear interpolation, the resulting collocation method is superconvergent, thus requiring fewer iterations for a desired accuracy. We show the applicability of the proposed scheme on numerical examples and discuss future research ideas in this area."
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- 2021
49. Comparative Study on Numerical Methods for Singularly Perturbed Advanced-Delay Differential Equations
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A. Tamilselvan, Praveen Agarwal, Porpattama Hammachukiattikul, R. Vadivel, Elango Sekar, and Nallappan Gunasekaran
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Article Subject ,Differential equation ,General Mathematics ,Numerical analysis ,Finite difference ,010103 numerical & computational mathematics ,Delay differential equation ,01 natural sciences ,010101 applied mathematics ,Mathematics Subject Classification ,Norm (mathematics) ,Convergence (routing) ,QA1-939 ,Piecewise ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).
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- 2021
50. Global analysis of an environmental and death transmission model for Ebola outbreak with perturbation
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Samwel K. Tarus, Abdul A. Kamara, and Xiangjun Wang
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0209 industrial biotechnology ,Computer simulation ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Feasible region ,Perturbation (astronomy) ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Stability (probability) ,020901 industrial engineering & automation ,Exponential stability ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,0101 mathematics ,Basic reproduction number ,Matrix method ,Mathematics - Abstract
In this paper, we inspect the global asymptotic stability (GAS) of Ebola–epidemic using a modified Susceptible-Infected-Deceased-Pathogen (SIDP) model with perturbation. The feasible region is obtained, and we determine the basic reproduction number using the next-generation matrix method. The GAS of the disease-free and endemic equilibria are discuss and provide parameter conditions for which the stability exists. We show theoretically that the contaminated environmental Ebola human–deceased transmission model is GAS with and without probability. Numerical simulation to support our theoretical findings are presented.
- Published
- 2021
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