In the context of numerical approximation of partial differential equations, meshless methods are recent developments, which have been reported to be relatively straightforward, yet provide better convergence and accuracy as compared to the conventional mesh-based approaches, for some specific problems like stress–strain analysis and modeling in a deforming media. Among several proposed schemes, strong-form collocation schemes using radial basis functions are easy-to-program, require no mesh in the domain or at the boundary, avoid numerical integration, and have similar formulations for all dimensions. Some of the radial basiss functions like the Gaussian, multiquadric and inverse-multiquadric are infinitely smooth. The standard global meshless formulations based on such radial basis functions are often found to lead towards ill-conditioned systems of linear equations. For such formulations, we investigate the degree of ill-conditioning against degrees of freedom for different radial basis functions. Also, the convergence of four iterative methods and pseudoinverse approaches has been tested for the solution of such ill-conditioned systems. In order to compute the pseudoinverse, two different approaches, i.e., singular value decomposition and full rank Cholesky factorization have been used. A set of numerical experiments, performed here, demonstrate that pseudoinverse approach based on singular value decomposition performs better than the iterative methods. Although, pseudoinverse computation via full rank Cholesky factorization is faster, it is not recommended in global meshless collocation schemes due to poor convergence at relatively large degrees of freedom.
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Beatson, R., Cherrie, J., Mouat, C.: Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration. Adv. Comput. Math. 11(2–3), 253–270 (1999)
Beatson, R.K., Light, W.A., Billings, S.: Fast solution of the radial basis function interpolation equations: domain decomposition methods. SIAM J. Sci. Comput. 22(5), 1717–1740 (2000)
Chen, W., Fu, Z., Chen, C.: Recent Advances in Radial Basis Function Collocation Methods. Springer, Berlin (2014)
Courrieu, P.: Fast computation of Moore–Penrose inverse matrices. Neural Inf. Process. Lett. Rev. 8(2), 25–29 (2005)
Dehghan, M., Shokri, A.: A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions. Comput. Math. Appl. 54(1), 136–146 (2007)
Demirkaya, G., Wafo, S.C., Ilegbusi, O.: Direct solution of Navier–Stokes equations by radial basis functions. Appl. Math. Model. 32(9), 1848–1858 (2008)
Fasshauer, G., McCourt, M.: The uncertainty principle—an unfortunate misconception. In: Kernel-Based Approximation Methods Using MATLAB, pp. 199–201 (2016)
Fasshauer, G.F.: Meshfree Approximation Methods with MATLAB. World Scientific Publishing Co., Inc., River Edge (2007)
Ferreira, A., Roque, C., Jorge, R.: Static and free vibration analysis of composite shells by radial basis functions. Eng. Anal. Bound. Elem. 30(9), 719–733 (2006)
Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33(2), 869–892 (2011)
Fornberg, B., Wright, G.: Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl. 48(5–6), 853–867 (2004)
Franke, R.: A Critical Comparison of Some Methods for Interpolation of Scattered Data. Defense Technical Information Center (1979)
Hansen, P.C.: Regularization tools: a matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6(1), 1–35 (1994)
Hardy, Rolland L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76(8), 2156–2202 (1971)
Kansa, E.J.: Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19(8–9), 147–161 (1990)
Kansa, E.J., Hon, Y.C.: Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput. Math. Appl. 39(7–8), 123–137 (2000)
Kindelan, M., Bernal, F., González-Rodríguez, P., Moscoso, M.: Application of the RBF meshless method to the solution of the radiative transport equation. J. Comput. Phys. 229(5), 1897–1908 (2010)
Ling, L., Kansa, E.J.: Preconditioning for radial basis functions with domain decomposition methods. Math. Comput. Model. 40(13), 1413–1427 (2004)
Moore, E.H.: On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26(2), 394–395 (1920)
Penrose, R.: A generalized inverse for matrices. Math. Proc. Camb. Philos. Soc. 51(3), 406–413 (1955)
Sajavicius, S.: Optimization, conditioning and accuracy of radial basis function method for partial differential equations with nonlocal boundary conditions—a case of two-dimensional Poisson equation. Eng. Anal. Bound. Elem. 37(4), 788–804 (2013)
Sarler, B.: Towards a mesh-free computation of transport phenomena. Eng. Anal. Bound. Elem. 26(9), 731–738 (2002)
Sarler, B.: From global to local radial basis function collocation method for transport phenomena. In: Leitão, V., Alves, C., Armando Duarte, C. (eds.) Advances in Meshfree Techniques, Computational Methods in Applied Sciences, vol. 5, pp. 257–282. Springer, Dordrecht (2007)
Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3(3), 251–264 (2013)
Strang, G.: Computational Science and Engineering. Wellesley-Cambridge Press, Wellesley (2007)
ul Islam, S., Aziz, I., Ahmad, M.: Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions. Comput. Math. Appl. 69(3), 180–205 (2015)
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Mishra, P.K., Nath, S.K. On the Convergence of Iterative Methods and Pseudoinverse Approaches in Global Meshless Collocation. Int. J. Appl. Comput. Math 3, 3987–4000 (2017). https://doi.org/10.1007/s40819-016-0272-6
- Radial basis function
- Generalized inverse
- Iterative methods
- Meshless methods
Mathematics Subject Classification