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Быстрые Фурье-солверы для МКЭ высокого порядка с тензорными произведениями для уравнения типа Пуассона
Журнал вычислительной математики и математической физики. 2020. Т. 60. № 2. С. 234-252.
Злотник А.А., Злотник И.А.
Research target:
Mathematics
Language:
Russian
Keywords: Fast Fourier TransformFast direct algorithmhigh order finite element methodPoisson equationбыстрый прямой алгоритмметод конечных элементов высокого порядкауравнение ПуассонаБДПФ
Publication based on the results of:
А.А. Злотник, И.А. Злотник, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2017 Т. 473 № 2 С. 131-137
Представлен новый быстрый прямой алгоритм реализации метода конечных элементов (МКЭ) порядка n>=2 на прямоугольниках для решения различных краевых задач для уравнений типа Пуассона. Он обобщает хорошо известный алгоритм для случая разностных схем или билинейных КЭ (n=1), использующий версии быстрого дискретного преобразования Фурье (БДПФ). Его ядром являются быстрые прямой и обратный алгоритмы разложения по собственным векторам ...
Added: October 25, 2016
Alexander Zlotnik, Ilya Zlotnik, / Cornell University. Series "Working papers by Cornell University". 2016. No. 1609.07758.
We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor product high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson kind. They are based on the well-known Fourier approaches. The key new points are the fast direct and inverse FFT-based algorithms for expansion in eigenvectors ...
Added: September 29, 2016
Alexander Zlotnik, Ilya Zlotnik, / Cornell University. Series "Working papers by Cornell University". 2017. No. 1701.03967.
We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor product high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson
kind. They are based on the well-known Fourier approaches. The key new points are the fast direct and inverse FFT-based algorithms for expansion in eigenvectors of the 1D ...
Added: January 17, 2017
Zlotnik A., Zlotnik I. A., Journal of Computational and Applied Mathematics 2018
We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor product high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson kind. They are based on the well-known Fourier approaches. The key new points are the fast direct and inverse FFT-based algorithms for expansion ...
Added: April 8, 2017
Zlotnik A.A., Zlotnik I.A., Computational Mathematics and Mathematical Physics 2020 Vol. 60 No. 2 P. 240-257
We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor products finite element method (FEM) based on the tensor products of the 1D high-order FEM spaces on multi-dimensional rectangular parallelepipeds for solving the $N$-dimensional Poisson type equation $-\Delta u+\alpha u=f$ ($N\geq 2$) with the Dirichlet boundary conditions. They are based ...
Added: May 19, 2020
Zlotnik A.A., Zlotnik I.A., Doklady Mathematics 2017 Vol. 95 No. 2 P. 129-135
A new fast direct algorithm for implementing a finite element method (FEM) of order on rectangles as applied to boundary value problems for Poisson-type equations is described that extends a well-known algorithm for the case of difference schemes or bilinear finite elements (n = 1). Its core consists of fast direct and inverse algorithms for ...
Added: February 28, 2017
Veretennikov A., Теорiя Ймовiрностей та Математична Статистика 2016 Vol. 95 P. 178-188
Poisson equation in the whole space was studied earlier for so called ergodic generators L corresponding to homogeneous Markov diffusions. Solving this equation is one of the main tools for diffusion approximation in the theory of stochastic averaging and homogenisation. Here a similar equation with a potential is considered, firstly because it is natural for ...
Added: October 17, 2017
Malovichko M., Yavich N., Razorenova A. et al., / Cornell University Library. Series math.RT "arXiv:1808.06395 [math.RT]". 2022.
This paper introduces a novel numerical method for the inverse problem of electroencephalography(EEG). We pose the inverse EEG problem as an optimal control (OC) problem for Poisson’s equation. The optimality conditions lead to a variational system of differential equations. It is discretized directly in finite-element spaces leading to a system of linear equations with a ...
Added: October 31, 2022
Popkov Y., Popkov A., Dubnov Y. A., Автоматика и телемеханика 2020 № 7 С. 148-172
A randomized forecasting method based on the generation of ensembles of entropy-optimal forecasting trajectories is developed. The latter are generated by randomized dynamic regression models containing random parameters, measurement noises, and a random input. The probability density functions of random parameters and measurement noises are estimated using real data within the randomized machine learning procedure. ...
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Beklemishev L. D., Оноприенко А. А., Математический сборник 2015 Т. 206 № 9 С. 3-20
We formulate some term rewriting systems in which the number of computation steps is finite for each output, but this number cannot be bounded by a provably total computable function in Peano arithmetic PA. Thus, the termination of such systems is unprovable in PA. These systems are derived from an independent combinatorial result known as the Worm ...
Added: March 13, 2016
D. V. Gribanov, D.S. Malyshev, P. M. Pardalos et al., Journal of Combinatorial Optimization 2018 Vol. 35 No. 4 P. 1128-1146
In this paper, we present fixed-parameter tractable algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems’ formulations are near square. The parameter is the maximum absolute value of the rank minors in the corresponding matrices. Additionally, we present fixed-parameter tractable algorithms ...
Added: February 19, 2018
Malyshev D., Вестник Нижегородского университета им. Н.И. Лобачевского 2008 № 6 С. 141-146
Рассматривается понятие граничного класса, которое является полезным инструментом для анализа вычислительной сложности задач на графах. Исследуются два конкретных класса графов, и приводятся задачи, для которых эти классы являются граничными. ...
Added: August 31, 2012
Belomestny D., Iosipoi L., Mathematics and Computers in Simulation 2021 No. 181 P. 351-363
Markov Chain Monte Carlo methods become increasingly popular in applied mathematics as a tool for numerical integration with respect to complex and high-dimensional distributions. However, application of MCMC methods to heavy-tailed distributions and distributions with analytically intractable densities turns out to be rather problematic. In this paper, we propose a novel approach towards the use ...
Added: October 31, 2020
Revenko A., Kuznetsov S., Fundamenta Informaticae 2012 Vol. 4 No. 115 P. 377-394
An approach for studying relations between properties of functions on sets is proposed. The approach is based on Attribute Exploration. 16 properties of functions are considered, among them monotonicity, idempotency, path independence, exchange properties, convexity, etc. Example functions are partially computer generated on the powersets of sets with 2, 3 and 4 elements. Attribute Exploration ...
Added: December 31, 2012
Kryuchkov M., Rusakov S. V., Вестник Ижевского государственного технического университета 2015 № 2(66) С. 110-112
This paper describes the results of testing the neuronal technical trend indicator according to the exchange rate of Brent oil in 2014. Testing of the model was carried out on three time series, which characterized by their features. ...
Added: August 31, 2015
Malyshev D., Дискретный анализ и исследование операций 2012 Т. 19 № 4 С. 66-72
Рассматривается конструктивный подход к формированию новых случаев эффективной разрешимости задачи о независимом множестве в семействе наследственных частей множества графов Free({P5,C5}). Именно, доказывается, что если эта задача полиномиально разрешима в классе Free({P5,C5,G}), то для любого графа H, который может быть индуктивно получен из G применением к текущему графу сложения с K1 или умножения на K1, эта ...
Added: August 31, 2012
Lanham : University Press of America, 2012
The history of logic and analytic philosophy in Central and Eastern Europe is still known to very few people. As an exception to the rule, only two scientific schools became internationally popular: the Vienna Circle and the Lvov-Warsaw School. Nevertheless, the countries included in this region have not only joint history, but also joint cultural ...
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Barcelona : IEEE, 2017
International Conference on Control, Decision and Information Technologies. ...
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МГУ, 2014
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Sirotkin D., Malyshev D., Дискретная математика 2017 Т. 29 № 3 С. 114-125
Задача о независимом множестве для заданного обыкновенного графа состоит в вычислении размера наибольшего множества его попарно несмежных вершин. Предлагается новый способ редукции графов. С его помощью получено новое доказательство NP-полноты задачи о независимом множестве в классе планарных графов и доказана NP-полнота данной задачи в классе плоских графов, имеющих только треугольные внутренние грани, с максимальной степенью ...
Added: September 7, 2017
Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 3 P. 573-607
Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal ...
Added: October 9, 2012
Babash A. V., М. : ИНФРА-М, РИОР, 2013
Пособие предназначено для студентов высших учебных заведений, обучающихся по специальности «Прикладная информатика (в экономике)». Оно также содержит методический материал для ряда инновационных курсов лекций по профилю «Информационная безопасность» и может быть использовано и для блока дисциплин этого профиля. Ряд представленных результатов полезен специалистам и аспирантам, специализирующихся в указанной области. ...
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Vyalyi M., Дискретная математика 1991 Т. 3 № 3 С. 35-45
Added: October 17, 2014
Akopov A. S., Beklaryan L. A., Saghatelyan A. K., Environmental Modelling and Software 2019 Vol. 116 P. 7-25
Urban greenery such as trees can effectively reduce air pollution in a natural and eco-friendly way. However, how to spatially locate and arrange greenery in an optimal way remains as a challenging task. We developed an agent-based model of air pollution dynamics to support the optimal allocation and configuration of tree clusters in a city. The Pareto ...
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