A RBF partition of unity collocation method based on a ï‌nite diï‌€erence scheme to solve parabolic stochastic partial differential equations

Authors
Mohsen Esmaeilbeigi
Omid Chatrabgoun
Maryam Shafa
10.52547/mmr.7.3.413
Abstract
Meshfree methods based on radial basis functions (RBFs) are popular tools for the numerical solution of parabolic stochastic partial differential equations (PSPDEs). However, the RBF collocation methods in the global view have some disadvantages for the numerical solution of PSPDEs. Condition number in the resulting dense linear systems indicates that the meshless method using global RBFs may be unstable at each realization to solve PSPDEs. In order to avoid numerical instabilities in the global view, we are interested in the use of RBF methods in the local view for the numerical solution of PSPDEs. In this paper, the RBF partition of unity collocation method based on a finite difference scheme for the Gaussian random field (RBF-PU-FD) as a localized RBF approximation presented to deal with these issues. For this purpose, we simulate the Gaussian field with a spatial covariance structure at a finite collection of collocation points. The matrices formed during the RBF-PU-FD method will be sparse and, hence, will not suffer from ill-conditioning and high computational cost. For the test problems, we perform 1000 realizations and statistical criterions such as mean, standard deviation, lower bound and upper bound of prediction are evaluated using the Monte-Carlo method.

./files/site1/files/%D8%A7%D8%B3%D9%85%D8%A7%D8%B9%DB%8C%D9%84_%D8%A8%DB%8C%DA%AF%DB%8C_-%DA%86%DA%A9%DB%8C%D8%AF%D9%87_%D8%A7%D9%86%DA%AF%D9%84%DB%8C%D8%B3%DB%8C(2).pdf
Keywords
stochastic partial differential equations
meshless methods
RBF collocation
partition of unity method
[1] K.K. Sabelfeld, I. Shalimova, Forward and backward stochastic Lagrangian

models for turbulent transport and the well-mixed condition, Monte Carlo

Methods Appl. 7 (2001) 369-382.

[2] I. Babuska, B. Andersson, P.J. Smith, K. Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput Methods Appl Mech Eng. 172 (1999) 1-4.

[3] I. Elishakoff, Whys and hows in uncertainty modelling, Springer-Verlag, Vienna, 1999.

[4] I. Elishakoff, Y. Ren, The bird’s eye view on finite element method for structures with large stochastic variations, Comput Methods Appl Mech Eng. 168 (1999) 51-61.

[5] P-L. Chow, Stochastic partial differential equations, Chapman and Hall/CRC applied mathematics and nonlinear science series, Boca Raton, FL: Chapman and Hall/CRC, 2007.

[6] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge, UK: Cambridge University Press, 1992.

[7] JB. Walsh, An introduction to stochastic partial differential equations, Lecture notes in mathematics, vol. 1180, Springer-Verlag, 1986, pp. 265-439.

[8] EJ. Allen, S.J. Novosel, Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stoch Stoch Rep. 64 (1998) 117-142.

[9] Y, Cao, H. Yang, L. Yin, Finite element methods for semilinear elliptic stochastic partial differential equations, Numer Math. 106 (2007) 181-198.

[10] M. Kamrani, S.M. Hosseini, Spectral collocation method for stochastic Burgers equation driven by additive noise, Math Comput Simulat. 82 (2012) 1630-1644.

[11] M. Dehghan, M. Shirzadi, Numerical solution of stochastic elliptic partial differential equations using the meshless method of radial basis functions, Eng. Anal. Bound. Elem. 50 (2015) 291-303.

[12] GE. Fasshauer, Q. Ye, Kernel-based collocation methods versus Galerkin finite element methods for approximating elliptic stochastic partial differential equations, Meshfree methods for partial differential equations VI, lecture notes in computational science and engineering. 89 (2013) 155-170.

[13] H. Rafieayan Zadeh, M. Mohammadi, E. Babolian, Solving a class of PDEs by a local reproducing kernel method with an adaptive residual subsampling technique, CMES. 108 (2015) 375-396.

[14] M. Mohammadi, F. Saberi Zafarghandi, E. Babolian, and S. Javadi. A local reproducing kernel method accompanied by some different edge improvement techniques: application to the Burgers’ equation, Iran. J. Sci. Technol. Trans. A Sci. 42 (2018) 857-871.

[15] F. Saberi Zafarghandi, M. Mohammadi, E. Babolian, S. Javadi, A localized Newton basis functions meshless method for the numerical solution of the 2D non-linear coupled Burgers’ equations, Internat. J. Numer. Methods Heat Fluid Flow. 27 (2016) 2582-2602.

[16] G. Garmanjani, R. Cavoretto, M. Esmaeilbeigi, A RBF partition of unity collocation method based on finite difference for initial-boundary value problems, Comput. Math. Appl. 75 (2018) 4066-4090.

[17] A.M. Davie, J.G. Gaines, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math Comput. 70 (2001) 121-34.

[18] Q. Du, T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises, SIAM J Numer Anal. 40 (2002) 1421-1445.

[19] M.D. Buhmann, Radial Basis Functions: Theory and Implementation, Cambridge Monogr. Appl. Comput. Math., vol. 12, Cambridge Univ. Press, Cambridge, 2003.

[20] G.E. Fasshauer, Meshfree Approximation Methods with Matlab, World Scientifc, Singapore, 2007.

[21] C.K. Lee, X. Liu, S.C. Fan, Local multiquadric approximation for solving

boundary value problems, Computational Mechanics. 30 (2003) 396-409.

[22] O. Davydov, D.T. Oanh, Adaptive meshless centres and RBF stencils for

Poisson equation, Journal of Computational Physics. 230 (2011) 287-304.

[23] H. Wendland, Fast evaluation of radial basis functions: Methods based on partition of unity, in: Approximation Theory X: Wavelets, Splines, and Applications, C.K. Chui, L.L. Schumaker, J. Stockler, eds., Vanderbilt Univ. Press, Nashville, TN, 2002, pp. 473-483.

[24] H. Wendland, Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., vol. 17, Cambridge Univ. Press, Cambridge, 2005.

[25] I. Cialenco, GE. Fasshauer, Q. Ye, Approximation of stochastic partial differential equations by a kernel-based collocation method, Int. J. Comput. Math. 89 (2012) 2543-2561.

[26] A. Safdari-Vaighani, A. Heryudono, E. Larsson, A radial basis function partition of unity collocation method for convection-diffusion equations, J. Sci. Comput. 64 (2015) 341-367.

[27] M. Dehghan, M. Shirzadi, Meshless simulation of stochastic advection-diffusion equations based on radial basis functions, Eng. Anal. Bound. Elem. 53 (2015) 18-26.

[28] A. Heryudono, E. Larsson, A. Ramage, L.V. Sydow, Preconditioning for radial basis function partition of unity methods, J. Sci. Comput. 67 (2016) 1089-1109.
Mathematical Research
Volume 7, Issue 3 - Serial Number 18
Autumn 2021
Pages 413-442