Classification and Applications of Randomized Functional Numerical Algorithms for the Solution of Second-Kind Fredholm Integral Equations

Systematization of numerical randomized functional algorithms for approximation of solutions to second-kind Fredholm integral equation is performed in this paper. Three types of algorithms are emphasized: projection algorithms, grid algorithms, and projection-grid algorithms. Disadvantages of grid a...

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Vydané v:Journal of mathematical sciences (New York, N.Y.) Ročník 254; číslo 5; s. 589 - 605
Hlavný autor: Voytishek, A. V.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.05.2021
Springer
Springer Nature B.V
Predmet:
Algorithms
Fredholm equations
Integral equations
Kernels
Mathematics
Mathematics and Statistics
Projection
numerical solution
65C40
grid algorithm
computational kernel
45B05
randomized algorithm
65C05
projection algorithm
second-kind Fredholm integral equation
ISSN:1072-3374, 1573-8795
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Abstract Systematization of numerical randomized functional algorithms for approximation of solutions to second-kind Fredholm integral equation is performed in this paper. Three types of algorithms are emphasized: projection algorithms, grid algorithms, and projection-grid algorithms. Disadvantages of grid algorithms that require calculation of values of the kernel of integral equations at fixed points are revealed (practically, the kernels of equation have integrable singularities and this calculation is impossible). Thus, for applied problems related to the solution of second-kind Fredholm integral equations, projection or projection-grid randomized algorithms are more convenient.
AbstractList Systematization of numerical randomized functional algorithms for approximation of solutions to second-kind Fredholm integral equation is performed in this paper. Three types of algorithms are emphasized: projection algorithms, grid algorithms, and projection-grid algorithms. Disadvantages of grid algorithms that require calculation of values of the kernel of integral equations at fixed points are revealed (practically, the kernels of equation have integrable singularities and this calculation is impossible). Thus, for applied problems related to the solution of second-kind Fredholm integral equations, projection or projection-grid randomized algorithms are more convenient. Keywords and phrases: second-kind Fredholm integral equation, numerical solution, randomized algorithm, projection algorithm, grid algorithm, computational kernel. AMS Subject Classification: 65C40, 45B05, 65C05
Systematization of numerical randomized functional algorithms for approximation of solutions to second-kind Fredholm integral equation is performed in this paper. Three types of algorithms are emphasized: projection algorithms, grid algorithms, and projection-grid algorithms. Disadvantages of grid algorithms that require calculation of values of the kernel of integral equations at fixed points are revealed (practically, the kernels of equation have integrable singularities and this calculation is impossible). Thus, for applied problems related to the solution of second-kind Fredholm integral equations, projection or projection-grid randomized algorithms are more convenient.
Audience Academic
Author Voytishek, A. V.
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Keywords numerical solution
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grid algorithm
computational kernel
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randomized algorithm
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projection algorithm
second-kind Fredholm integral equation
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References_xml – reference: S. V. Rogazinsky, “Statistical modeling based on the projection method for the nonlinear Boltzmann equation,” in: Proc. Conf. “Marchuk Readings–2017”, Omega-Print, Novosibirsk (2017), pp. 82.
– reference: BakhvalovNSNumerical Methods [in Russian]1975MoscowNauka
– reference: MarchukGIAgoshkovVIIntroduction to Projection-Grid Methods [in Russian]1981MoscowNauka0642.65037
– reference: BulgakovaTEVoytishekAVConditional optimization of the randomized iterative methodZh. Vychisl. Mat. Mat. Fiz.20094971148115725993861224.65083
– reference: G. A. Mikhailov and A. V. Voytishek, Numerical Statistical Modeling. Monte Carlo Methods [in Russian], Akademia, Moscow (2006).
– reference: MikhailovGAOptimization of Weight Monte Carlo Methods [in Russian]1987MoscowNauka
– reference: KantorovichLVAkilovGPFunctional Analysis [in Russian]1984MoscowNauka0555.46001
– reference: VladimirovVSEquations of Mathematical Physics [in Russian]1981MoscowNauka
– reference: VoytishekAVFunctional Estimates of the Monte Carlo Method [in Russian]2007NovosibirskNovosibirsk State Univ
– reference: A. V. Voytishek, “Randomized Iterative Numerical Models and Algorithms,” LAP LAMBERT Academic Publishing (2017).
– reference: MarchukGIMethods of Computational Mathematics [in Russian]1980MoscowNauka
– reference: ShkarupaEVVoytishekAVConvergence of discrete-stochastic numerical procedures with independent or weakly dependent estimators at grid nodesJ. Stat. Plan. Inform.200085199211175925010.1016/S0378-3758(99)00081-6
– reference: SabelfeldKKMonte Carlo methods in Boundary-Value Problems [in Russian]1989MoscowNauka
– reference: M. D. Ramazanov, D. Ya. Rakhmatullin, L. Z. Valeeva, and E. L. Bannikova, “Solution of integral equations on multiprocessor computing systems,” Zh. Sib. Federal Univ. Tekh. Yekhnol., 2, No. 1, 69–87 (2009).
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– reference: G. A. Mikhailov, Weight Monte Carlo Methods, Novosibirsk (2000).
– reference: A. S. Frolov and N. N. Chentsov, “Application of dependent tests in the Monte Carlo method for obtaining smooth curves,” in: Proc. Conf. on Probability theory and Mathematical Statistics, Vilnius (1962), pp. 425–437.
– reference: KonovalovANIntroduction to Computational Methods of Linear Algebra [in Russian]1993NovosibirskNauka
– reference: IvanovVMKulchitskyOYA method of numerical solution of integral equations on a random gridDiffer. Uravn.19902623333411050399
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SubjectTerms Algorithms
Fredholm equations
Integral equations
Kernels
Mathematics
Mathematics and Statistics
Projection
Title Classification and Applications of Randomized Functional Numerical Algorithms for the Solution of Second-Kind Fredholm Integral Equations
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