Stochastic Simulation Algorithms for Iterative Solution of the Lamé Equation

In this paper, iterative stochastic simulation algorithms for the Lamé equation describing the displacements of an isotropic elastic body are constructed. Three different stochastic methods are proposed: the first one is based on a global algorithm of random walk on spheres to compute the solution a...

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Veröffentlicht in:Numerical analysis and applications Jg. 16; H. 4; S. 299 - 316
Hauptverfasser: Aksyuk, I. A., Kireeva, A. E., Sabelfeld, K. K., Smirnov, D. D.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Moscow Pleiades Publishing 01.12.2023
Springer Nature B.V
Schlagworte:
Algorithms
Approximation
Boundary conditions
Boundary value problems
Elastic bodies
Iterative methods
Iterative solution
Lame functions
Linear equations
Mathematics
Mathematics and Statistics
Numerical Analysis
Random variables
Random walk
global random walk algorithm
randomized algorithm for solving linear equations
grid-free stochastic algorithm
random walk on spheres
ISSN:1995-4239, 1995-4247
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Abstract In this paper, iterative stochastic simulation algorithms for the Lamé equation describing the displacements of an isotropic elastic body are constructed. Three different stochastic methods are proposed: the first one is based on a global algorithm of random walk on spheres to compute the solution and its derivatives for an anisotropic diffusion equation. It does not use grids and does not require large amounts of RAM. The second method is based on a randomized algorithm for solving large systems of linear equations and requires the introduction of a grid. The third method is also grid-based and uses a random walk algorithm. All three methods implement an iterative process, at each step of which anisotropic diffusion equations are solved. The paper provides a comparative analysis of the proposed methods and discusses the limits of applicability of each of them.
AbstractList In this paper, iterative stochastic simulation algorithms for the Lamé equation describing the displacements of an isotropic elastic body are constructed. Three different stochastic methods are proposed: the first one is based on a global algorithm of random walk on spheres to compute the solution and its derivatives for an anisotropic diffusion equation. It does not use grids and does not require large amounts of RAM. The second method is based on a randomized algorithm for solving large systems of linear equations and requires the introduction of a grid. The third method is also grid-based and uses a random walk algorithm. All three methods implement an iterative process, at each step of which anisotropic diffusion equations are solved. The paper provides a comparative analysis of the proposed methods and discusses the limits of applicability of each of them.
Author Kireeva, A. E.
Sabelfeld, K. K.
Aksyuk, I. A.
Smirnov, D. D.
Author_xml – sequence: 1
  givenname: I. A.
  surname: Aksyuk
  fullname: Aksyuk, I. A.
  email: i.aksyuk@g.nsu.ru
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  givenname: A. E.
  surname: Kireeva
  fullname: Kireeva, A. E.
  email: kireeva@ssd.sscc.ru
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  givenname: K. K.
  surname: Sabelfeld
  fullname: Sabelfeld, K. K.
  email: karl@osmf.sscc.ru
  organization: Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences
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  givenname: D. D.
  surname: Smirnov
  fullname: Smirnov, D. D.
  email: smirnovdd@mail.ru
  organization: Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences
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Cites_doi 10.1007/BF01010406
10.1090/S0025-5718-1979-0537973-X
10.1515/mcma-2016-0118
10.1002/mma.7861
10.1515/9783110315332
10.1016/j.aml.2021.107830
10.1515/mcma-2017-0113
10.1049/el:19740097
10.1214/aoms/1177728169
10.1515/mcma.1995.1.1.1
10.1016/s0020-7683(03)00364-0
10.1515/mcma-2022-2131
10.3103/S1063454108020064
10.1016/j.aml.2020.106466
10.1515/mcma-2021-2092
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Keywords global random walk algorithm
randomized algorithm for solving linear equations
grid-free stochastic algorithm
random walk on spheres
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Springer Nature B.V
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SabelfeldKarl K.KireevaAnastasyaA new Global Random Walk algorithm for calculation of the solution and its derivatives of elliptic equations with constant coefficients in an arbitrary set of pointsApplied Mathematics Letters2020107106466106466409934310.1016/j.aml.2020.1064661442.82005
SabelfeldK.K.Monte Carlo Methods in Boundary Value Problems1991BerlinSpringer
RouxStéphaneGeneralized Brownian motion and elasticityJournal of statistical physics19874820121391443510.1007/BF01010406
PolyaninA.D.Spravochnik po lineinym uravneniyam matematicheskoi fiziki (Handbook of Linear Equations of Mathematical Physics)2001MoscowFizmatlit0989.35001
Koshelev, A.I., Applying the Universal Iterative Process to Some Problems of Mechanics, Vest. Saint-Petersburg Univ. Ser. 1, 2008, no. 2, pp. 47–55.
BudaevB.V.BogyD.B.Probabilistic approach to the Lamé equations of linear elastostaticsInternational Journal of Solids and Structures20034062856306201180910.1016/s0020-7683(03)00364-01053.74006
KupradzeV.D.GegeliaT.G.BasheleishviliM.O.BurchuladzeT.V.Trekhmernye zadachi teorii uprugosti (Three-Dimensional Problems of Elasticity Theory)1976MoscowNauka
O’LearyDianne P.StewartG.W.VandergraftJames S.Estimating the largest eigenvalue of a positive definite matrixMathematics of Computation1979331289129253797310.1090/S0025-5718-1979-0537973-X0435.65026
SabelfeldKarl K.Random walk on spheres method for solving drift-diffusion problemsMonte Carlo Methods and Applications201622199223357492710.1515/mcma-2016-01181354.65265
ErmakovS.M.MikhailovG.A.Statisticheskoe modelirovanie (Statistical Modeling)1982MoscowNauka
MullerMervin E.Some continuous Monte Carlo methods for the Dirichlet problemThe Annals of Mathematical Statistics1956275695898878610.1214/aoms/11777281690075.28902
WalkerAlastair J.New fast method for generating discrete random numbers with arbitrary frequency distributionsElectronics Letters19741012712810.1049/el:19740097
SabelfeldKarl K.KireevaK.A Global Random Walk on Spheres Algorithm for Calculating the Solution and Its Derivatives of Drift-Diffusion-Reaction EquationsMathematical Methods in the Applied Sciences20224514201431436812710.1002/mma.7861
ShalimovaIrina A.SabelfeldKarl K.Development and implementation of branching random walk on spheres algorithms for solving the 2D elastostatics Lamé equationMonte Carlo Methods and Applications2023297993455450010.1515/mcma-2022-21311518.65008
SabelfeldK.K.A new randomized vector algorithm for iterative solution of large linear systemsApplied Mathematics Letters2022126107830107830434926810.1016/j.aml.2021.1078301483.65052
SabelfeldK.K.TalayD.Integral Formulation of the Boundary Value Problems and the Method of Random Walk on SpheresMonte Carlo Meth. Appl.19951134136879710.1515/mcma.1995.1.1.10824.65127
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StakgoldI.Green’s Functions and Boundary Value Problems1979NYWiley0421.34027
SabelfeldKarl K.SmirnovDmitriiA global random walk on grid algorithm for second order elliptic equationsMonte Carlo Methods and Applications202127211225430865110.1515/mcma-2021-209207419515
LurieA.I.Teoriya uprugosti (Theory of Elasticity)1970MoscowNauka
SabelfeldKarl K.Random walk on spheres algorithm for solving transient drift-diffusion-reaction problemsMonte Carlo Methods and Applications201723189212369206710.1515/mcma-2017-01131375.65007
SabelfeldKarl K.ShalimovaIrina A.Spherical and Plane Integral Operators for PDEs: Construction, Analysis, and Applications2013BerlinDE GRUYTER10.1515/97831103153321316.47001
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K.K. Sabelfeld (223_CR22) 1995; 1
223_CR1
B.V. Budaev (223_CR8) 2003; 40
E.B. Dynkin (223_CR9) 1961
K.K. Sabelfeld (223_CR14) 2022; 126
Mervin E. Muller (223_CR11) 1956; 27
Karl K. Sabelfeld (223_CR16) 2016; 22
223_CR3
Karl K. Sabelfeld (223_CR18) 2020; 107
Dianne P. O’Leary (223_CR12) 1979; 33
V.D. Kupradze (223_CR4) 1976
Karl K. Sabelfeld (223_CR19) 2022; 45
S.M. Ermakov (223_CR2) 1982
A.I. Lurie (223_CR5) 1970
Stéphane Roux (223_CR13) 1987; 48
A.D. Polyanin (223_CR6) 2001
Irina A. Shalimova (223_CR23) 2023; 29
I. Stakgold (223_CR24) 1979
References_xml – reference: SabelfeldKarl K.KireevaAnastasyaA new Global Random Walk algorithm for calculation of the solution and its derivatives of elliptic equations with constant coefficients in an arbitrary set of pointsApplied Mathematics Letters2020107106466106466409934310.1016/j.aml.2020.1064661442.82005
– reference: SabelfeldKarl K.Random walk on spheres algorithm for solving transient drift-diffusion-reaction problemsMonte Carlo Methods and Applications201723189212369206710.1515/mcma-2017-01131375.65007
– reference: Koshelev, A.I., Applying the Universal Iterative Process to Some Problems of Mechanics, Vest. Saint-Petersburg Univ. Ser. 1, 2008, no. 2, pp. 47–55.
– reference: DynkinE.B.The Theory of Markov Processes1961NYPergamon Press0091.13605
– reference: SabelfeldKarl K.Random walk on spheres method for solving drift-diffusion problemsMonte Carlo Methods and Applications201622199223357492710.1515/mcma-2016-01181354.65265
– reference: PolyaninA.D.Spravochnik po lineinym uravneniyam matematicheskoi fiziki (Handbook of Linear Equations of Mathematical Physics)2001MoscowFizmatlit0989.35001
– reference: SabelfeldK.K.TalayD.Integral Formulation of the Boundary Value Problems and the Method of Random Walk on SpheresMonte Carlo Meth. Appl.19951134136879710.1515/mcma.1995.1.1.10824.65127
– reference: KupradzeV.D.GegeliaT.G.BasheleishviliM.O.BurchuladzeT.V.Trekhmernye zadachi teorii uprugosti (Three-Dimensional Problems of Elasticity Theory)1976MoscowNauka
– reference: SabelfeldKarl K.SmirnovDmitriiA global random walk on grid algorithm for second order elliptic equationsMonte Carlo Methods and Applications202127211225430865110.1515/mcma-2021-209207419515
– reference: Joint Supercomputer Center of the Russian Academy of Sciences; http://www.jscc.ru.
– reference: MullerMervin E.Some continuous Monte Carlo methods for the Dirichlet problemThe Annals of Mathematical Statistics1956275695898878610.1214/aoms/11777281690075.28902
– reference: StakgoldI.Green’s Functions and Boundary Value Problems1979NYWiley0421.34027
– reference: ErmakovS.M.MikhailovG.A.Statisticheskoe modelirovanie (Statistical Modeling)1982MoscowNauka
– reference: RouxStéphaneGeneralized Brownian motion and elasticityJournal of statistical physics19874820121391443510.1007/BF01010406
– reference: LurieA.I.Teoriya uprugosti (Theory of Elasticity)1970MoscowNauka
– reference: SabelfeldK.K.Monte Carlo Methods in Boundary Value Problems1991BerlinSpringer
– reference: O’LearyDianne P.StewartG.W.VandergraftJames S.Estimating the largest eigenvalue of a positive definite matrixMathematics of Computation1979331289129253797310.1090/S0025-5718-1979-0537973-X0435.65026
– reference: SabelfeldKarl K.KireevaK.A Global Random Walk on Spheres Algorithm for Calculating the Solution and Its Derivatives of Drift-Diffusion-Reaction EquationsMathematical Methods in the Applied Sciences20224514201431436812710.1002/mma.7861
– reference: WalkerAlastair J.New fast method for generating discrete random numbers with arbitrary frequency distributionsElectronics Letters19741012712810.1049/el:19740097
– reference: SabelfeldKarl K.ShalimovaIrina A.Spherical and Plane Integral Operators for PDEs: Construction, Analysis, and Applications2013BerlinDE GRUYTER10.1515/97831103153321316.47001
– reference: ShalimovaIrina A.SabelfeldKarl K.Development and implementation of branching random walk on spheres algorithms for solving the 2D elastostatics Lamé equationMonte Carlo Methods and Applications2023297993455450010.1515/mcma-2022-21311518.65008
– reference: StarchenkoA.V.BertsunV.N.Metody parallelnykh vychislenii. Uchebnik. Ser. Superkomp’yuternoe obrazovanie (Parallel Computing Methods. Textbook. Ser. “Supercomputer Education,”)2013TomskTomsk State University
– reference: Eremeev, V.A., Reshenie sistem lineinykh algebraicheskikh uravnenii dlya bolshikh razrezhennykh matrits. Uchebno-metodicheskoe posobie (Solving Systems of Linear Algebraic Equations for Large Sparse Matrices. Teaching Manual), Rostov-on-Don, 2008.
– reference: BudaevB.V.BogyD.B.Probabilistic approach to the Lamé equations of linear elastostaticsInternational Journal of Solids and Structures20034062856306201180910.1016/s0020-7683(03)00364-01053.74006
– reference: SabelfeldK.K.A new randomized vector algorithm for iterative solution of large linear systemsApplied Mathematics Letters2022126107830107830434926810.1016/j.aml.2021.1078301483.65052
– volume-title: Trekhmernye zadachi teorii uprugosti (Three-Dimensional Problems of Elasticity Theory)
  year: 1976
  ident: 223_CR4
– volume: 48
  start-page: 201
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  publication-title: Journal of statistical physics
  doi: 10.1007/BF01010406
– volume: 33
  start-page: 1289
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  publication-title: Mathematics of Computation
  doi: 10.1090/S0025-5718-1979-0537973-X
– volume-title: Spravochnik po lineinym uravneniyam matematicheskoi fiziki (Handbook of Linear Equations of Mathematical Physics)
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– ident: 223_CR1
– volume: 22
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  publication-title: Monte Carlo Methods and Applications
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– volume: 45
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  publication-title: Mathematical Methods in the Applied Sciences
  doi: 10.1002/mma.7861
– volume-title: Spherical and Plane Integral Operators for PDEs: Construction, Analysis, and Applications
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  doi: 10.1515/9783110315332
– volume: 126
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  publication-title: Applied Mathematics Letters
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  ident: 223_CR25
  publication-title: Electronics Letters
  doi: 10.1049/el:19740097
– volume-title: Teoriya uprugosti (Theory of Elasticity)
  year: 1970
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– volume-title: Green’s Functions and Boundary Value Problems
  year: 1979
  ident: 223_CR24
– volume: 27
  start-page: 569
  year: 1956
  ident: 223_CR11
  publication-title: The Annals of Mathematical Statistics
  doi: 10.1214/aoms/1177728169
– volume-title: Monte Carlo Methods in Boundary Value Problems
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Snippet In this paper, iterative stochastic simulation algorithms for the Lamé equation describing the displacements of an isotropic elastic body are constructed....
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StartPage 299
SubjectTerms Algorithms
Approximation
Boundary conditions
Boundary value problems
Elastic bodies
Iterative methods
Iterative solution
Lame functions
Linear equations
Mathematics
Mathematics and Statistics
Numerical Analysis
Random variables
Random walk
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Title Stochastic Simulation Algorithms for Iterative Solution of the Lamé Equation
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