Extended Arithmetic Precision in Meshfree Calculations
Continuously differentiable radial basis functions (RBFs) are meshfree, converge faster as the dimensionality increases, and is theoretically spectrally convergent. When implemented on current single and double precision computers, such RBFs can suffer from ill-conditioning because the systems of equations needed to be solved to find the expansion coefficients are full. However, the Advanpix extended precision software package allows computer mathematics to resemble asymptotically ideal Platonic mathematics. Additionally, full systems with extended precision execute faster graphical processors units and field-programmable gate arrays because no branching is needed. Sparse equation systems are fast for iterative solvers in a very limited number of cases.
[1] Hardy, RL: "Multiquadric equations of topography and other irregular surfaces", J. Geophy. Res.1971, Vol. 76(8):pp.1905-1915.
[2] Hardy, RL, "Theory and application of the multiquadric biharmonic method: 25 years of discovery", Comput. Math. Appl. 1990,Vol. 19(8-9), pp.163-208.
[3] Madych, WR; Nelson, SA, ``Multivariate interpolation and conditionally positive definite functions, II'', 1990, Math. Comput. Vol. 54, pp 211-230.
[4] Driscoll, TA; Fornberg, “Interpolation in the limit of increasingly flat radial basis functions”, Comput. Math. Appl. 2002, Vol. 43, pp. 413-422.
[5] Kansa, EJ, "Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics I: Surface approximations and partial derivative estimates", Comput. Math. Applic., 1990, Vol. 19(8/9), pp.127-145.
[6] Kansa, EJ, "Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics II: Solutions to parabolic, hyperbolic, and elliptic partial differential equations", Comput. Math. Applic.,1990, Vol. 19(8/9), pp.147-161.
[7] Chen, W, Fu, ZJ, Chen, CS, ``Recent Advances in Radial Basis Function Collocation Methods'', ISBN 978-3-642-39572-7, Springer Briefs in Applied Sciences and Technology, 2014, Berlin.
[8] Holobrodko, P, https://www.advanpix.com/
[9] Kansa, EJ; Holoborodko, P, "On the ill-conditioned nature of C∞ -RBF strong collocation ", Eng. Anal. Bound. Elem. ,2017, Vol.78, pp.26-30.
[10] http://www.exforsys.com/tutorials/c-language/decision-making-and-branching-in-c.html/
[11] Coffer, HN; Stadtherr, MA, “Reliability of iterative linear equation solvers in chemical process simulation”, Comput. chem. Eng. 1996, Vol. 20, pp. 1123-1132.
[12] Fedoseyev, A; Friedman, M; Kansa, E, "Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary", Comput. Math. Applic. 2002, Vol. 43(3-5), pp.439-45.
[13] Wertz, J; Kansa, EJ, Ling, L. "The role of the multiquadric shape parameters in solving elliptic partial differential equations", Comput. Math. Applic. 2006, Vol. 51(8), pp 1335-1348.
[14] Huang, CS; Lee, CF; Cheng, AHD: "Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method", Eng. Anal. Bound. Elem. 2007, Vol. 31, pp.614-623.
[15] Huang, CS; Yen, HD; Cheng, AHD, "On the Increasingly Flat Radial Basis Function and Optimal Shape Parameter for the Solution of Elliptic PDEs", Eng. Anal. Bound. Elem. 2010, Vol.34, pp. 802-.809.
[16] Cheng, AHD, "Multiquadric and its shape parameter-a numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation," Eng. Anal .Bound. Elem. 2012, Vol. 36, pp. 220-239.
[17] Sarra, SA, "Radial basis function approximation methods with extended precision floating point arithmetic", Eng. Anal, Bound, Elem. 2011, Vol. 35, pp. 68-76.
[18] Kansa, EJ; Geiser, J: "Numerical solution to time-dependent 4D inviscid Burgers equations", 2013, Eng.Anal. Bound. Elem. Vol. 37, pp. 637-645.
[19] Rump, SM, "Verification methods: Rigorous results using floating-point arithmetic", Acta Numerica, 2010, Vol. 19: pp. 287-449.
[20] Higham, NJ, "Accuracy and Stability of Numerical Algorithms "(2 ed), 2002. SIAM, Philadelphia. .
[21] Brent, R; Zimmermann, P. "Modern Computer Arithmetic", Cambridge, 2010. Monographs on Comput. Appl. Math.
[22] Hida, YL; Li, XS; Bailey, DH, "Quad-Double Arithmetic: Algorithms, Implementation, and Application", 2000, Lawrence Berkeley National Laboratory Technical Report LBNL-46996.
[23] Li, XS; Demmel, JW; Bailey, DH; Henry, G; Hida, Y; Iskandar, J; Kahan, W; Kang, SY; Kapur, A; Martin, MC;Thompson, BJ; Tung, TJ; Yoo, DJ, "Design, implementation and testing of extended and mixed precision BLAS.", 2002, ACM Trans. Math. Softw. Vol.1 ,pp.152-205.
[24] Higham, NJ, "Multi-precision world", SIAM News, Oct. 2017, pp 2-3.
[25] https://www.advanpix.com/2011/10/12/multiprecision-computation-eigenvalues-eigenvector/.
[26] Luh, LT, "The choice of the shape parameter--A friendly approach", Eng. Anal. Bound. Elem.2019, Vol. 98, pp. 103-109.
[27] Luh, LT, 'The mystery of the shape parameter III", Appl. Comput. Harmon. Anal. 2016, Vol. 40, pp. 186--199.
[28] Luh, LT, ‘’Solving Poisson’s equations by the MN curve approach”, 2019, preprint.
[29] Ling, L; Kansa, EJ, "Preconditioning for radial basis functions with domain decomposition methods", Math. & Comput. Model., 2004, Vol.40 pp 1413-1427.
[30] Koczka, G; Bauernfeind, T; Preis, K; Biro, O, "An Iterative Domain Decomposition Method for Solving Wave Propagation Problems", Electromagnetics, 2014, Vol. 34, Iss. 3-4.
[31] Hon, YC; Wu, Z: "Additive Schwarz domain decomposition with radial basis approximation", Int. J. Appl. Math. Stat. 2002, Vol.4: pp. 81-98.
[32] Li, J; Hon, YC, "Domain decomposition for radial basis meshless methods", Num. Meth. PDEs. 2004, Vol. 20, pp. 450-462.
[33] Ingber, MA; Chen, CS; Tanski, A, "A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations". Int. J. Num. Meth. Eng. 2004, Vol. 60: pp.2183-2201.
[34] Duan, Y; Tang, PF; Huang, TZ; Lai, SJ, "Coupling projection domain decomposition method and Kansa's method in electrostatic problems", Comput. Phys. Commun., 2009, Vol. 180, pp 200-214.
[35] Herrera, I; Rosas Medina Alberto, "The Derived-Vector Space Framework and Four General Purposes Massively Parallel DDM Algorithms". Eng. Anal. Bound. Elem. 2013, Vol. l37 (3) pp-646-657.
[36] Hernandez-Rosales, A; Power, H. "Non-overlapping domain decomposition algorithm for the Hermite radial basis function meshless collocation approach: applications to convection diffusion problems", J. Algo. & Comput. Tech., 2007, Vol. 1, pp 127-159.
[37] Kansa, EJ; Hon, YC, “Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations”, Comput. Math. Applic, . 2000, .Vol. 39, pp. 123-137.
[38] Fornberg, B, Flyer, N.”A Primer on Radial Basis Functions with Applications to the Geosciences”, 2015, SIAM, Philadelphia, PA.
[39] Gonzalez-Rodriguez, P; Bayona, V.; Moscoso, M; Kindelan, M, “Laurent series based RBF-FD method to avoid ill- conditioning”, Eng. Anal. Bound. Elem., 2015, Vol. 52, pp. 24–31.
[40] Lati, M; Dehghan, M, “The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations”, Eng. Anal. Bound. Elem.2015, Vol. 52, pp. 99-109.
[41] Wright. GB; Fornberg, B, “Stable computations with flat radial basis functions using vector-valued rational approximations”, J. Comput. Phys, 2017, Vol. 331, pp.137-156.
[42] Rashidinia, J; Fasshauer, G; Khasi, M “A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems”, Comput. Math. Applic., 2016, Vol. 72 , pp 178 – 193.
[43] Fasshauer, GE J; Zhang, JG, ”On choosing optimal shape parameters for rbf approximation”, Num. Algor., 2007, Vol. 45, pp. 345–368.
[44] Davis, TA, "Direct Methods for Sparse Linear Systems", 2006, SIAM, Philadelphia.
[45] Saad, Y, "Iterative Methods for Sparse Linear Systems", 2nd ed.2003, SIAM, Philadelphia.
[46] Higham, NJ, "Accuracy and Stability of Numerical Algorithms ", (2 ed), 2002. SIAM, Philadelphia.
[47] Sarra, SA; Kansa, EJ. "Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations", Advances in Computational Mechanics, 2009. Vol. 2, ISSN: 1940-5820.
[1] Hardy, RL: "Multiquadric equations of topography and other irregular surfaces", J. Geophy. Res.1971, Vol. 76(8):pp.1905-1915.
[2] Hardy, RL, "Theory and application of the multiquadric biharmonic method: 25 years of discovery", Comput. Math. Appl. 1990,Vol. 19(8-9), pp.163-208.
[3] Madych, WR; Nelson, SA, ``Multivariate interpolation and conditionally positive definite functions, II'', 1990, Math. Comput. Vol. 54, pp 211-230.
[4] Driscoll, TA; Fornberg, “Interpolation in the limit of increasingly flat radial basis functions”, Comput. Math. Appl. 2002, Vol. 43, pp. 413-422.
[5] Kansa, EJ, "Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics I: Surface approximations and partial derivative estimates", Comput. Math. Applic., 1990, Vol. 19(8/9), pp.127-145.
[6] Kansa, EJ, "Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics II: Solutions to parabolic, hyperbolic, and elliptic partial differential equations", Comput. Math. Applic.,1990, Vol. 19(8/9), pp.147-161.
[7] Chen, W, Fu, ZJ, Chen, CS, ``Recent Advances in Radial Basis Function Collocation Methods'', ISBN 978-3-642-39572-7, Springer Briefs in Applied Sciences and Technology, 2014, Berlin.
[8] Holobrodko, P, https://www.advanpix.com/
[9] Kansa, EJ; Holoborodko, P, "On the ill-conditioned nature of C∞ -RBF strong collocation ", Eng. Anal. Bound. Elem. ,2017, Vol.78, pp.26-30.
[10] http://www.exforsys.com/tutorials/c-language/decision-making-and-branching-in-c.html/
[11] Coffer, HN; Stadtherr, MA, “Reliability of iterative linear equation solvers in chemical process simulation”, Comput. chem. Eng. 1996, Vol. 20, pp. 1123-1132.
[12] Fedoseyev, A; Friedman, M; Kansa, E, "Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary", Comput. Math. Applic. 2002, Vol. 43(3-5), pp.439-45.
[13] Wertz, J; Kansa, EJ, Ling, L. "The role of the multiquadric shape parameters in solving elliptic partial differential equations", Comput. Math. Applic. 2006, Vol. 51(8), pp 1335-1348.
[14] Huang, CS; Lee, CF; Cheng, AHD: "Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method", Eng. Anal. Bound. Elem. 2007, Vol. 31, pp.614-623.
[15] Huang, CS; Yen, HD; Cheng, AHD, "On the Increasingly Flat Radial Basis Function and Optimal Shape Parameter for the Solution of Elliptic PDEs", Eng. Anal. Bound. Elem. 2010, Vol.34, pp. 802-.809.
[16] Cheng, AHD, "Multiquadric and its shape parameter-a numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation," Eng. Anal .Bound. Elem. 2012, Vol. 36, pp. 220-239.
[17] Sarra, SA, "Radial basis function approximation methods with extended precision floating point arithmetic", Eng. Anal, Bound, Elem. 2011, Vol. 35, pp. 68-76.
[18] Kansa, EJ; Geiser, J: "Numerical solution to time-dependent 4D inviscid Burgers equations", 2013, Eng.Anal. Bound. Elem. Vol. 37, pp. 637-645.
[19] Rump, SM, "Verification methods: Rigorous results using floating-point arithmetic", Acta Numerica, 2010, Vol. 19: pp. 287-449.
[20] Higham, NJ, "Accuracy and Stability of Numerical Algorithms "(2 ed), 2002. SIAM, Philadelphia. .
[21] Brent, R; Zimmermann, P. "Modern Computer Arithmetic", Cambridge, 2010. Monographs on Comput. Appl. Math.
[22] Hida, YL; Li, XS; Bailey, DH, "Quad-Double Arithmetic: Algorithms, Implementation, and Application", 2000, Lawrence Berkeley National Laboratory Technical Report LBNL-46996.
[23] Li, XS; Demmel, JW; Bailey, DH; Henry, G; Hida, Y; Iskandar, J; Kahan, W; Kang, SY; Kapur, A; Martin, MC;Thompson, BJ; Tung, TJ; Yoo, DJ, "Design, implementation and testing of extended and mixed precision BLAS.", 2002, ACM Trans. Math. Softw. Vol.1 ,pp.152-205.
[24] Higham, NJ, "Multi-precision world", SIAM News, Oct. 2017, pp 2-3.
[25] https://www.advanpix.com/2011/10/12/multiprecision-computation-eigenvalues-eigenvector/.
[26] Luh, LT, "The choice of the shape parameter--A friendly approach", Eng. Anal. Bound. Elem.2019, Vol. 98, pp. 103-109.
[27] Luh, LT, 'The mystery of the shape parameter III", Appl. Comput. Harmon. Anal. 2016, Vol. 40, pp. 186--199.
[28] Luh, LT, ‘’Solving Poisson’s equations by the MN curve approach”, 2019, preprint.
[29] Ling, L; Kansa, EJ, "Preconditioning for radial basis functions with domain decomposition methods", Math. & Comput. Model., 2004, Vol.40 pp 1413-1427.
[30] Koczka, G; Bauernfeind, T; Preis, K; Biro, O, "An Iterative Domain Decomposition Method for Solving Wave Propagation Problems", Electromagnetics, 2014, Vol. 34, Iss. 3-4.
[31] Hon, YC; Wu, Z: "Additive Schwarz domain decomposition with radial basis approximation", Int. J. Appl. Math. Stat. 2002, Vol.4: pp. 81-98.
[32] Li, J; Hon, YC, "Domain decomposition for radial basis meshless methods", Num. Meth. PDEs. 2004, Vol. 20, pp. 450-462.
[33] Ingber, MA; Chen, CS; Tanski, A, "A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations". Int. J. Num. Meth. Eng. 2004, Vol. 60: pp.2183-2201.
[34] Duan, Y; Tang, PF; Huang, TZ; Lai, SJ, "Coupling projection domain decomposition method and Kansa's method in electrostatic problems", Comput. Phys. Commun., 2009, Vol. 180, pp 200-214.
[35] Herrera, I; Rosas Medina Alberto, "The Derived-Vector Space Framework and Four General Purposes Massively Parallel DDM Algorithms". Eng. Anal. Bound. Elem. 2013, Vol. l37 (3) pp-646-657.
[36] Hernandez-Rosales, A; Power, H. "Non-overlapping domain decomposition algorithm for the Hermite radial basis function meshless collocation approach: applications to convection diffusion problems", J. Algo. & Comput. Tech., 2007, Vol. 1, pp 127-159.
[37] Kansa, EJ; Hon, YC, “Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations”, Comput. Math. Applic, . 2000, .Vol. 39, pp. 123-137.
[38] Fornberg, B, Flyer, N.”A Primer on Radial Basis Functions with Applications to the Geosciences”, 2015, SIAM, Philadelphia, PA.
[39] Gonzalez-Rodriguez, P; Bayona, V.; Moscoso, M; Kindelan, M, “Laurent series based RBF-FD method to avoid ill- conditioning”, Eng. Anal. Bound. Elem., 2015, Vol. 52, pp. 24–31.
[40] Lati, M; Dehghan, M, “The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations”, Eng. Anal. Bound. Elem.2015, Vol. 52, pp. 99-109.
[41] Wright. GB; Fornberg, B, “Stable computations with flat radial basis functions using vector-valued rational approximations”, J. Comput. Phys, 2017, Vol. 331, pp.137-156.
[42] Rashidinia, J; Fasshauer, G; Khasi, M “A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems”, Comput. Math. Applic., 2016, Vol. 72 , pp 178 – 193.
[43] Fasshauer, GE J; Zhang, JG, ”On choosing optimal shape parameters for rbf approximation”, Num. Algor., 2007, Vol. 45, pp. 345–368.
[44] Davis, TA, "Direct Methods for Sparse Linear Systems", 2006, SIAM, Philadelphia.
[45] Saad, Y, "Iterative Methods for Sparse Linear Systems", 2nd ed.2003, SIAM, Philadelphia.
[46] Higham, NJ, "Accuracy and Stability of Numerical Algorithms ", (2 ed), 2002. SIAM, Philadelphia.
[47] Sarra, SA; Kansa, EJ. "Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations", Advances in Computational Mechanics, 2009. Vol. 2, ISSN: 1940-5820.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:79827", author = "Edward J. Kansa and Pavel Holoborodko", title = "Extended Arithmetic Precision in Meshfree Calculations", abstract = "Continuously differentiable radial basis functions (RBFs) are meshfree, converge faster as the dimensionality increases, and is theoretically spectrally convergent. When implemented on current single and double precision computers, such RBFs can suffer from ill-conditioning because the systems of equations needed to be solved to find the expansion coefficients are full. However, the Advanpix extended precision software package allows computer mathematics to resemble asymptotically ideal Platonic mathematics. Additionally, full systems with extended precision execute faster graphical processors units and field-programmable gate arrays because no branching is needed. Sparse equation systems are fast for iterative solvers in a very limited number of cases.
", keywords = "Meshless spectrally convergent, partial differential equations, extended arithmetic precision, no branching.", volume = "14", number = "9", pages = "101-6", }
