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Journal articles on the topic 'Linear equations; Schwarz methods'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 February 2022

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1

Antonietti, Paola F., Blanca Ayuso de Dios, Susanne C. Brenner, and Li-yeng Sung. "Schwarz Methods for a Preconditioned WOPSIP Method for Elliptic Problems." Computational Methods in Applied Mathematics 12, no. 3 (2012): 241–72. http://dx.doi.org/10.2478/cmam-2012-0021.

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Abstract We propose and analyze several two-level non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) discretization of a second order boundary value problem. We show that the preconditioners are scalable and that the condition number of the resulting preconditioned linear systems of equations is independent of the penalty parameter and is of order H/h, where H and h represent the mesh sizes of the coarse and fine partitions, respectively. Numerical experiments that illustrate the performance of the proposed two-level Schwarz methods a

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2

Nagid, Nabila, and Hassan Belhadj. "New approach for accelerating the nonlinear Schwarz iterations." Boletim da Sociedade Paranaense de Matemática 38, no. 4 (2019): 51–69. http://dx.doi.org/10.5269/bspm.v38i4.37018.

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The vector Epsilon algorithm is an effective extrapolation method used for accelerating the convergence of vector sequences. In this paper, this method is used to accelerate the convergence of Schwarz iterative methods for stationary linear and nonlinear partial differential equations (PDEs). The vector Epsilon algorithm is applied to the vector sequences produced by additive Schwarz (AS) or restricted additive Schwarz (RAS) methods after discretization. Some convergence analysis is presented, and several test-cases of analytical problems are performed in order to illustrate the interest of su

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3

Antoine, Xavier, Fengji Hou, and Emmanuel Lorin. "Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 4 (2018): 1569–96. http://dx.doi.org/10.1051/m2an/2017048.

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This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for linear equations with variable coefficients and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in t

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4

Wu, Shu-Lin. "Schwarz Waveform Relaxation for Heat Equations with Nonlinear Dynamical Boundary Conditions." Abstract and Applied Analysis 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/474608.

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We are interested in solving heat equations with nonlinear dynamical boundary conditions by using domain decomposition methods. In the classical framework, one first discretizes the time direction and then solves a sequence of state steady problems by the domain decomposition method. In this paper, we consider the heat equations at spacetime continuous level and study a Schwarz waveform relaxation algorithm for parallel computation purpose. We prove the linear convergence of the algorithm on long time intervals and show how the convergence rate depends on the size of overlap and the nonlineari

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5

Antoine, X., and E. Lorin. "An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations." Numerische Mathematik 137, no. 4 (2017): 923–58. http://dx.doi.org/10.1007/s00211-017-0897-3.

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6

Lapenta, Giovanni, and Wei Jiang. "Implicit Temporal Discretization and Exact Energy Conservation for Particle Methods Applied to the Poisson–Boltzmann Equation." Plasma 1, no. 2 (2018): 242–58. http://dx.doi.org/10.3390/plasma1020021.

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We report on a new multiscale method approach for the study of systems with wide separation of short-range forces acting on short time scales and long-range forces acting on much slower scales. We consider the case of the Poisson–Boltzmann equation that describes the long-range forces using the Boltzmann formula (i.e., we assume the medium to be in quasi local thermal equilibrium). We develop a new approach where fields and particle information (mediated by the equations for their moments) are solved self-consistently. The new approach is implicit and numerically stable, providing exact energy

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7

Гурьева, Я. Л., and В. П. Ильин. "On acceleration technologies of parallel decomposition methods." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 1 (April 2, 2015): 146–54. http://dx.doi.org/10.26089/nummet.v16r115.

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Одним из главных препятствий масштабированному распараллеливанию алгебраических методов декомпозиции для решения сверхбольших разреженных систем линейных алгебраических уравнений (СЛАУ) является замедление скорости сходимости аддитивного итерационного алгоритма Шварца в подпространствах Крылова при увеличении количества подобластей. Целью настоящей статьи является сравнительный экспериментальный анализ различных приeмов ускорения итераций: параметризованное пересечение подобластей, использование специальных интерфейсных условий на границах смежных подобластей, а также применение грубосеточной

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8

Zhou, H., and H. A. A. Tchelepi. "Two-Stage Algebraic Multiscale Linear Solver for Highly Heterogeneous Reservoir Models." SPE Journal 17, no. 02 (2012): 523–39. http://dx.doi.org/10.2118/141473-pa.

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Summary An efficient Two-Stage Algebraic Multiscale Solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU), or the Additive Schwarz (AS) method is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative s

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9

Kong, Fande, and Xiao-Chuan Cai. "Scalability study of an implicit solver for coupled fluid-structure interaction problems on unstructured meshes in 3D." International Journal of High Performance Computing Applications 32, no. 2 (2016): 207–19. http://dx.doi.org/10.1177/1094342016646437.

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Fluid-structure interaction (FSI) problems are computationally very challenging. In this paper we consider the monolithic approach for solving the fully coupled FSI problem. Most existing techniques, such as multigrid methods, do not work well for the coupled system since the system consists of elliptic, parabolic and hyperbolic components all together. Other approaches based on direct solvers do not scale to large numbers of processors. In this paper, we introduce a multilevel unstructured mesh Schwarz preconditioned Newton–Krylov method for the implicitly discretized, fully coupled system of

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10

Phillips, Peter C. B., and Werner Ploberger. "Posterior Odds Testing for a Unit Root with Data-Based Model Selection." Econometric Theory 10, no. 3-4 (1994): 774–808. http://dx.doi.org/10.1017/s026646660000877x.

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The Kalman filter is used to derive updating equations for the Bayesian data density in discrete time linear regression models with stochastic regressors. The implied “Bayes model” has time varying parameters and conditionally heterogeneous error variances. A σ-finite Bayes model measure is given and used to produce a new-model-selection criterion (PIC) and objective posterior odds tests for sharp null hypotheses like the presence of a unit root. This extends earlier work by Phillips and Ploberger [18]. Autoregressive-moving average (ARMA) models are considered, and a general test of trend-sta

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11

Colli Franzone, Piero, Luca F. Pavarino, and Simone Scacchi. "Bioelectrical effects of mechanical feedbacks in a strongly coupled cardiac electro-mechanical model." Mathematical Models and Methods in Applied Sciences 26, no. 01 (2015): 27–57. http://dx.doi.org/10.1142/s0218202516500020.

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The aim of this work is to investigate by means of numerical simulations the effects of myocardial deformation due to muscle contraction on the bioelectrical activity of the cardiac tissue. The three-dimensional electro-mechanical model considered consists of the following four components: the quasi-static orthotropic finite elasticity equations for the deformation of the cardiac tissue; the active tension model for the intracellular calcium dynamics and cross-bridge binding; the orthotropic Bidomain model for the electrical current flow through the tissue; the membrane model of the cardiac my

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12

JAKOBSEN, ESPEN ROBSTAD. "ON THE RATE OF CONVERGENCE OF APPROXIMATION SCHEMES FOR BELLMAN EQUATIONS ASSOCIATED WITH OPTIMAL STOPPING TIME PROBLEMS." Mathematical Models and Methods in Applied Sciences 13, no. 05 (2003): 613–44. http://dx.doi.org/10.1142/s0218202503002660.

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We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. A

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13

Stankevich, I. V., and P. S. Aronov. "Mathematical Modeling of the Contact Interaction of Two Elastic Bodies Using the Mortar Method." Mathematics and Mathematical Modeling, no. 3 (August 3, 2018): 26–44. http://dx.doi.org/10.24108/mathm.0318.0000112.

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The article discusses the development of an algorithm for solving contact problems of elasticity theory. Solving such problems is often associated with necessity of using mismatched grids. Their joining can be carried out both with the help of iterative procedures that form the so-called Schwarz alternating methods, and with the help of the Lagrange multipliers method or the penalty method. The algorithm constructed in the article uses the mortar method for matching the finite elements on the contact line. All these methods of joining the grids make it possible to ensure continuity of displace

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14

Gnatyuk, M. A., and V. M. Morozov. "An integral equation technique for the analysis of phased array antenna with matching step discontinuities." Journal of Physics and Electronics 26, no. 2 (2018): 101–6. http://dx.doi.org/10.15421/331833.

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Applying the integral equation method of overlapping partial domains and the Schwartz alternating method to solving an electromagnetic wave diffraction problem is considered in this paper. The infinite rectangular waveguide phased array antenna scanning in H plane which waveguides have step matching discontinuities is represented. The whole field definition domain is sliced into three overlapping partial domains. The system of integral representations for unknown Ey components of the electrical field vector in each domain is set up using Greenʼs functions. Unknown functions in each domain are

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15

Dolean, V., M. J. Gander, and L. Gerardo-Giorda. "Optimized Schwarz Methods for Maxwell's Equations." SIAM Journal on Scientific Computing 31, no. 3 (2009): 2193–213. http://dx.doi.org/10.1137/080728536.

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16

Sanabria Malagón, Camilo. "Schwarz maps of algebraic linear ordinary differential equations." Journal of Differential Equations 263, no. 11 (2017): 7123–40. http://dx.doi.org/10.1016/j.jde.2017.08.002.

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17

Migliorati, Giovanni, and Alfio Quarteroni. "Multilevel Schwarz methods for elliptic partial differential equations." Computer Methods in Applied Mechanics and Engineering 200, no. 25-28 (2011): 2282–96. http://dx.doi.org/10.1016/j.cma.2011.03.017.

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18

Aksoy, Ü., and A. O. Çelebi. "Schwarz problem for higher order linear equations in a polydisc." Complex Variables and Elliptic Equations 62, no. 10 (2017): 1558–69. http://dx.doi.org/10.1080/17476933.2016.1254627.

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19

Boulbrachene, Messaoud. "Finite Element Convergence Analysis of a Schwarz Alternating Method for Nonlinear Elliptic PDEs." Sultan Qaboos University Journal for Science [SQUJS] 24, no. 2 (2020): 109. http://dx.doi.org/10.24200/squjs.vol24iss2pp109-121.

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In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for nonlinear elliptic partial differential equations in the context of linear subdomain problems and nonmatching grids. The method stands on the combination of the convergence of linear Schwarz sequences with standard finite element L-error estimate for linear problems.

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20

Miyajima, Keiichi, Artur Korniłowicz, and Yasunari Shidama. "Contracting Mapping on Normed Linear Space." Formalized Mathematics 20, no. 4 (2012): 291–301. http://dx.doi.org/10.2478/v10037-012-0035-8.

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Summary In this article, we described the contracting mapping on normed linear space. Furthermore, we applied that mapping to ordinary differential equations on real normed space. Our method is based on the one presented by Schwarz [29].

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21

Li, Shishun, Xinping Shao, and Xiao-Chuan Cai. "Multilevel Space-Time Additive Schwarz Methods for Parabolic Equations." SIAM Journal on Scientific Computing 40, no. 5 (2018): A3012—A3037. http://dx.doi.org/10.1137/17m113808x.

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22

Blayo, Eric, David Cherel, and Antoine Rousseau. "Towards Optimized Schwarz Methods for the Navier–Stokes Equations." Journal of Scientific Computing 66, no. 1 (2015): 275–95. http://dx.doi.org/10.1007/s10915-015-0020-9.

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23

Toselli, Andrea. "Overlapping Schwarz methods for Maxwell's equations in three dimensions." Numerische Mathematik 86, no. 4 (2000): 733–52. http://dx.doi.org/10.1007/pl00005417.

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24

Guo, Guangbao. "Schwarz Methods for Quasi-Likelihood in Generalized Linear Models." Communications in Statistics - Simulation and Computation 37, no. 10 (2008): 2027–36. http://dx.doi.org/10.1080/03610910802311700.

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25

Dmitriev, М. О. "Determination of individual teleroentgenographic characteristics of the face profile in ukrainian young men and girls with orthognathic bite." Biomedical and Biosocial Anthropology, no. 32 (September 20, 2018): 28–34. http://dx.doi.org/10.31393/bba32-2018-04.

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Modern dentistry requires the definition of individualized values of teleroentgenographic indicators. To solve such problems, methods of regression and correlation analysis are increasingly used, which help to establish not only the existence of various relationships between the anatomical structures of the head and the parameters of the dento-jaw system, but also allow more accurately predict the change in the contour of soft facial tissue in response to orthodontic treatment. The purpose of the study is to develop mathematical models for the determination of individual teleroentgenographic c

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26

Martin, Véronique. "Schwarz Waveform Relaxation Algorithms for the Linear Viscous Equatorial Shallow Water Equations." SIAM Journal on Scientific Computing 31, no. 5 (2009): 3595–625. http://dx.doi.org/10.1137/070691450.

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27

Magoulès, Frédéric, Pascal Laurent-Gengoux, and Florent Pruvost. "Preconditioners for Schwarz relaxation methods applied to differential algebraic equations." International Journal of Computer Mathematics 91, no. 8 (2014): 1775–89. http://dx.doi.org/10.1080/00207160.2013.862524.

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28

Tran, Minh-Binh. "Overlapping optimized Schwarz methods for parabolic equations in $n$ dimensions." Proceedings of the American Mathematical Society 141, no. 5 (2012): 1627–40. http://dx.doi.org/10.1090/s0002-9939-2012-11522-9.

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29

Lui, S. H. "On Schwarz Alternating Methods for the Incompressible Navier--Stokes Equations." SIAM Journal on Scientific Computing 22, no. 6 (2001): 1974–86. http://dx.doi.org/10.1137/s1064827598347411.

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30

Klawonn, Axel, and Luca F. Pavarino. "Overlapping Schwarz methods for mixed linear elasticity and Stokes problems." Computer Methods in Applied Mechanics and Engineering 165, no. 1-4 (1998): 233–45. http://dx.doi.org/10.1016/s0045-7825(98)00059-0.

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31

Pauler, D. "The Schwarz criterion and related methods for normal linear models." Biometrika 85, no. 1 (1998): 13–27. http://dx.doi.org/10.1093/biomet/85.1.13.

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32

Sun, Zhe, Jinping Zeng, and Donghui Li. "Semismooth Newton Schwarz iterative methods for the linear complementarity problem." BIT Numerical Mathematics 50, no. 2 (2010): 425–49. http://dx.doi.org/10.1007/s10543-010-0261-9.

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33

Li, Shishun, Rongliang Chen, and Xinping Shao. "Parallel two-level space–time hybrid Schwarz method for solving linear parabolic equations." Applied Numerical Mathematics 139 (May 2019): 120–35. http://dx.doi.org/10.1016/j.apnum.2019.01.016.

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34

Beuchler, Sven, and Martin Purrucker. "Schwarz Type Solvers for -FEM Discretizations of Mixed Problems." Computational Methods in Applied Mathematics 12, no. 4 (2012): 369–90. http://dx.doi.org/10.2478/cmam-2012-0030.

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AbstractThis paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development of an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair which satisfies the discrete inf

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35

Wu, Shu-Lin, and Cheng-Ming Huang. "Quasi-optimized Schwarz methods for reaction diffusion equations with time delay." Journal of Mathematical Analysis and Applications 385, no. 1 (2012): 354–70. http://dx.doi.org/10.1016/j.jmaa.2011.06.052.

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36

Bouajaji, M. El, V. Dolean, M. J. Gander, and S. Lanteri. "Optimized Schwarz Methods for the Time-Harmonic Maxwell Equations with Damping." SIAM Journal on Scientific Computing 34, no. 4 (2012): A2048—A2071. http://dx.doi.org/10.1137/110842995.

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37

Jiang, Yao-Lin, and Hui Zhang. "Schwarz waveform relaxation methods for parabolic equations in space-frequency domain." Computers & Mathematics with Applications 55, no. 12 (2008): 2924–39. http://dx.doi.org/10.1016/j.camwa.2007.11.025.

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38

Lui, S. H. "On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs." Numerische Mathematik 93, no. 1 (2002): 109–29. http://dx.doi.org/10.1007/bf02679439.

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39

Schwandt, Hartmut. "Parallel interval Newton-like Schwarz methods for almost linear parabolic problems." Journal of Computational and Applied Mathematics 199, no. 2 (2007): 437–44. http://dx.doi.org/10.1016/j.cam.2005.07.042.

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40

Simoncini, V. "Computational Methods for Linear Matrix Equations." SIAM Review 58, no. 3 (2016): 377–441. http://dx.doi.org/10.1137/130912839.

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41

Dolean, Victorita, Martin J. Gander, and Erwin Veneros. "Asymptotic analysis of optimized Schwarz methods for maxwell’s equations with discontinuous coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 6 (2018): 2457–77. http://dx.doi.org/10.1051/m2an/2018041.

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Discretized time harmonic Maxwell’s equations are hard to solve by iterative methods, and the best currently available methods are based on domain decomposition and optimized transmission conditions. Optimized Schwarz methods were the first ones to use such transmission conditions, and this approach turned out to be so fundamentally important that it has been rediscovered over the last years under the name sweeping preconditioners, source transfer, single layer potential method and the method of polarized traces. We show here how one can optimize transmission conditions to take benefit from th

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42

Stephan, Ernst P., Matthias Maischak, and Florian Leydecker. "Some schwarz methods for integral equations on surfaces-h and p versions." Computing and Visualization in Science 8, no. 3-4 (2005): 211–16. http://dx.doi.org/10.1007/s00791-005-0011-8.

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43

Liu, Cuiyu, and Chen-liang Li. "A Preconditioned Multisplitting and Schwarz Method for Linear Complementarity Problem." Journal of Applied Mathematics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/519017.

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The preconditioner presented by Hadjidimos et al. (2003) can improve on the convergence rate of the classical iterative methods to solve linear systems. In this paper, we extend this preconditioner to solve linear complementarity problems whose coefficient matrix isM-matrix orH-matrix and present a multisplitting and Schwarz method. The convergence theorems are given. The numerical experiments show that the methods are efficient.

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44

Gander, Martin, Laurence Halpern, Frédéric Magoulès, and François-Xavier Roux. "Analysis of Patch Substructuring Methods." International Journal of Applied Mathematics and Computer Science 17, no. 3 (2007): 395–402. http://dx.doi.org/10.2478/v10006-007-0032-1.

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Analysis of Patch Substructuring MethodsPatch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains, condensated on the interfaces, to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one patch per interface. We show that this method is equivalen

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45

Pavarino, Luca F., and Elena Zampieri. "Overlapping Schwarz and Spectral Element Methods for Linear Elasticity and Elastic Waves." Journal of Scientific Computing 27, no. 1-3 (2006): 51–73. http://dx.doi.org/10.1007/s10915-005-9047-7.

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46

Blayo, Eric, Antoine Rousseau, and Manel Tayachi. "Boundary conditions and Schwarz waveform relaxation method for linear viscous Shallow Water equations in hydrodynamics." SMAI journal of computational mathematics 3 (September 14, 2017): 117–37. http://dx.doi.org/10.5802/smai-jcm.22.

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47

Frommer, Andreas, Reinhard Nabben, and Daniel B. Szyld. "Convergence of Stationary Iterative Methods for Hermitian Semidefinite Linear Systems and Applications to Schwarz Methods." SIAM Journal on Matrix Analysis and Applications 30, no. 2 (2008): 925–38. http://dx.doi.org/10.1137/080714038.

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48

Heuer, Norbert. "Additive schwarz methods for indefinite hypersingular integral equations in R3- the p-version." Applicable Analysis 72, no. 3-4 (1999): 411–37. http://dx.doi.org/10.1080/00036819908840750.

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49

de Gee, Maarten. "Linear Multistep Methods for Functional Differential Equations." Mathematics of Computation 48, no. 178 (1987): 633. http://dx.doi.org/10.2307/2007833.

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50

Chen, Yong-Lin. "Iterative methods for solving restricted linear equations." Applied Mathematics and Computation 86, no. 2-3 (1997): 171–84. http://dx.doi.org/10.1016/s0096-3003(96)00180-4.

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