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About 10,000 papers for “Numerical Methods and Algorithms”

Numerical Methods, Algorithms and Results

Olena O. Bulatsyk, B. Z. Kat︠s︡enelenbaum, Yury P. Topolyuk et al. · 2010

This chapter analyzes some numerical methods that applies to the formulated problems. It describes some theoretical aspects of the iterative methods for direct minimization of the functionals and for the solution of nonlinear integral equations and their systems. The chapter also describes the modification of the Newton method that is applied to the concrete problems. As a rule, this method is most effective when used together with the above iterative ones. A specific numerical technique known as the ‘opposite directions’ method is utilized for the optimization of linearly extended systems. Almost all the methods are demonstrated on two-dimensional problems described by one-dimensional equations. The chapter discusses the method of generalized separation of variables using examples of the optimization problem for two dimensional antenna arrays. The problems concerning eigen functions of certain homogeneous integral equations which describe the eigenwaves and eigenoscillations of physical systems are considered. Controlled Vocabulary Terms antenna arrays; eigenvalues and eigenfunctions; integral equations; Newton method; numerical analysis; optimisation

Cited by 0

Numerical methods: principles, analyses, and algorithms

Choice Reviews Online · 2010

1. Background to Numerical Methods 2. Scope of Mathematical and Numerical Methods 3. Errors and their Propagation 4. Programming Tools and Techniques 5. Numerical Solution of One-Dimensional Non-Linear Equations 6. Numerical Solution of Linear Equations: Direct Methods 7. Numerical Solutions for Matrix Inversion 8. Numerical SOlution of Linear Systems of Equations: Iterative Methods 9. Eigenvalues and Eigenvalues 10. Numerical Solution of Systems of Nonlinear Equations 11. Linear Least Squares Problem 12. Interpolation and Extrapolation 13. Numerical Differentiation 14. Numeical Integration 15. Numerical Solution of Ordinary Differential Equations: Initial Value Problem 16. Numerical Solution of Ordinary Differential Equations: Boundary value Problem 17. Partial Differential Equations 18. Numerical Solution of Parabolic Partial Differential Equations 19. Numerical Solution of Hyperbolic Partial Differential Equations 20. Numerical Solution of Elliptic Partial Differential Equations 22. Advances in Numerical Methods Using Neurocomputing Paradigm 23. Numerical Solution of Differential Equations

Cited by 10

Composite Numerical Methods and Scalable Tile Algorithms

Dimitar Slavchev · Biomath Communications · 2022

Numerical modeling is an important tool when studying various natural processes and phenomena. Fractional diffusion can be used for modeling many processes in biology, for example in silico experiments in molecular biology and medicine design, protein diffusion within cells, complex media geometry, etc. The problem is usually reduced to a system of linear algebraic equations and in many cases this system has a dense coefficient matrix. Numerically solving such problems with the traditional LU factorization is a computationally expensive endeavour - $O(n^3)$. In this paper we explore the use of a hierarchical compression method based on Hierarchical Semi-Separable compression and ULV-like factorization from the STRUctured Matrices PACKage (STRUMPACK) software library for a flow around airfoils problem discretized with Boundary Element Method and fractional diffusion problem discretized with the Finite Element Method. The HSS based method promises better overall computational complexity of $O(r^2n)$ for problems with suitable structure - low rank off-diagonal blocks. Here $r$ is the maximum rank of the off-diagonal blocks. We present analysis of the performance and accuracy of the HSS based method and compare it with the state of the art direct LU factorization solvers. This paper is based on the PhD thesis Composite Numerical Methods and Scalable Tile Algorithms of the author defended on 17.05.2022 in the Institute of Information and Communication Technologies at the Bulgarian Academy of Sciences.

Cited by 0Open access

Numerical methods: design, analysis, and computer implementation of algorithms

Greenbaum, Anne, Chartier, Timothy P · CERN Document Server (European Organization for Nuclear Research) · 2012

Numerical Methods provides a clear and concise exploration of standard numerical analysis topics, as well as nontraditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals. Filled with appealing examples that will motivate students, the textbook considers modern application areas, such as information retrieval and animation, and classical topics from physics and engineering. Exercises use MATLAB and promote understanding of computational results. The book gives instructors the flexibility to emphasize different aspects--design, analysis, or computer implementation--of numerical algorithms, depending on the background and interests of students. Designed for upper-division undergraduates in mathematics or computer science classes, the textbook assumes that students have prior knowledge of linear algebra and calculus, although these topics are reviewed in the text. Short discussions of the history of numerical methods are interspersed throughout the chapters. The book also includes polynomial interpolation at Chebyshev points, use of the MATLAB package Chebfun, and a section on the fast Fourier transform. Supplementary materials are available online. Clear and concise exposition of standard numerical analysis topics Explores nontraditional topics, such as mathematical modeling and Monte Carlo methods Covers modern applications, including information retrieval and animation, and classical applications from physics and engineering Promotes understanding of computational results through MATLAB exercises Provides flexibility so instructors can emphasize mathematical or applied/computational aspects of numerical methods or a combination Includes recent results on polynomial interpolation at Chebyshev points and use of the MATLAB package Chebfun Short discussions of the history of numerical methods interspersed throughout Supplementary materials available online.

Cited by 0

Extremal polynomials and methods of optimization of numerical algorithms

V.I. Lebedev · Sbornik Mathematics · 2004

Chebyshev-Markov-Bernstein-Szego polynomials Cn(x )e xtremal on (−1, 1) with weight functions w(x )=( 1 +x) α (1 − x) β / Sl(x ), where α, β =0 , 1 and Sl(x )= m=1 (1 − ckTlk (x)) > 0, are considered. A universal formula for their representation in trigonometric form is presented. Optimal distributions of the nodes of the weighted interpolation and explicit quadrature formulae of Gauss, Markov, Lobatto, and Rado types are obtained for integrals with weight p(x )= w 2 (x)(1 − x 2 ) −1/2 . The parameters of optimal Chebyshev iterative methods reduc- ing the error optimally by comparison with the initial error defined in another norm are determined. For each stage of the Fedorenko-Bakhvalov method itera- tion parameters are determined which take account of the results of the previous calculations. Chebyshev filters with weight are constructed. Iterative methods of the solution of equations containing compact operators are studied. Bibliography: 45 titles.

Cited by 6

Analytical methods and numerical algorithms for pricing path dependent derivatives

Pingping Zeng · 2014

b1334487 HKUST Electronic Theses Analytical methods and numerical algorithms for pricing path dependent derivatives by Zeng Pingping thesis 2014 xviii, 191 pages : illustrations ; 30 cm Developing computational and approximation methods for discrete path dependent options is one of…Read more ›

Cited by 0

Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics

Justin Solomon · Medical Entomology and Zoology · 2015

Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics presents a new approach to numerical analysis for modern computer scientists. Using examples from a broad base of computational tasks, including data processing, computational photography, and animation, the textbook introduces numerical modeling and algorithmic design from a practical standpoint and provides insight into the theoretical tools needed to support these skills. The book covers a wide range of topicsfrom numerical linear algebra to optimization and differential equationsfocusing on real-world motivation and unifying themes. It incorporates cases from computer science research and practice, accompanied by highlights from in-depth literature on each subtopic. Comprehensive end-of-chapter exercises encourage critical thinking and build students intuition while introducing extensions of the basic material. The text is designed for advanced undergraduate and beginning graduate students in computer science and related fields with experience in calculus and linear algebra. For students with a background in discrete mathematics, the book includes some reminders of relevant continuous mathematical background.

Cited by 30Open access

Algorithms and Numerical Methods for High Dimensional Financial Market Models

Revista De, Clasificación Jel G · 2010

A major challenge in Computational Finance is the pricing of options that depend on a large number of risk factors. Prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding parabolic equation. A number of problems in high-dimensional spaces have been addressed by the usual technique of separation of variables. In order to use the separated representation for numerical analysis applications, many algorithms and operations need to be translated into this framework. The aim this paper is present and review some of these techniques in the context of High Dimensional Financial Markets Models.

Cited by 0

Bioethanol production optimization by direct numerical methods and evolutionary algorithms

Cecilia Fernández, Nadia Pantano, María Carla Groff et al. · IEEE Latin America Transactions · 2024

This paper develops a dynamic optimization methodology based on direct numerical methods, for the bioethanol fed-batch production from glucose and fructose as a substrate. The mathematical model that governs the process consists of six differential equations and is highly nonlinear. The proposed strategy uses the Fourier trigonometric basis and normalized orthogonal polynomials for substrate feeding rate parameterization. Then, evolutionary algorithms and gradient methods are combined to search parameters that generate the best control action. This parameterization methodology requires a minimum number of parameters to optimize. Also, the continuous and differentiable nature of the optimal profile enables its direct implementation in the physical process, eliminating the necessity for filtering or smoothing it. In addition, they are ideal for bioprocesses, in which it is preferable to avoid abrupt changes in the operating modes of the process to promote cell growth. As a result, using only 3 parameters, a 3.5% increase in ethanol production was achieved, while the reference uses at least 10 parameters and provides a stepped feed profile. The simulations have yielded promising results, making this proposal an alternative with excellent potential for process optimization.

Cited by 2Open access