메뉴 건너뛰기




Volumn 61, Issue 4, 2011, Pages 443-459

Numerical experiments on the condition number of the interpolation matrices for radial basis functions

Author keywords

Interpolation; Matrix condition number; Radial basis functions

Indexed keywords

CONDITION NUMBERS; GAUSSIANS; GRID-SPACINGS; INTERPOLATION MATRIX; LARGE POWER; MATRIX CONDITION NUMBERS; MULTIQUADRICS; NUMERICAL EXPERIMENTS; ORDERS OF MAGNITUDE; QUASI-RANDOM; RADIAL BASIS FUNCTIONS; RANDOM GRIDS; THIN PLATE SPLINE; UNIFORM GRIDS;

EID: 78651294406     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2010.11.009     Document Type: Article
Times cited : (52)

References (51)
  • 1
    • 0036467777 scopus 로고    scopus 로고
    • An operator splitting-radial basis function method for the solution transient nonlinear Poisson problems
    • K. Balakrishnan, R. Jureshkumar, and P.A. Ramachandran An operator splitting-radial basis function method for the solution transient nonlinear Poisson problems Comput. Math. Appl. 43 2002 289 304
    • (2002) Comput. Math. Appl. , vol.43 , pp. 289-304
    • Balakrishnan, K.1    Jureshkumar, R.2    Ramachandran, P.A.3
  • 2
    • 0038426086 scopus 로고
    • Eigenvalues of Euclidean distance matrices
    • K. Ball Eigenvalues of Euclidean distance matrices J. Approx. Theory 68 1992 74 82
    • (1992) J. Approx. Theory , vol.68 , pp. 74-82
    • Ball, K.1
  • 3
    • 84964038019 scopus 로고    scopus 로고
    • Spectral-like accuracy in space of a meshless vortex method
    • Advances in Meshfree Techniques: ECCOMAS Thematic Conference on Meshless Methods, ECCOMAS
    • L.A. Barba Spectral-like accuracy in space of a meshless vortex method V.M.A. Leitao, C.J.S. Alves, C.A. Duarte, Advances in Meshfree Techniques: ECCOMAS Thematic Conference on Meshless Methods, ECCOMAS Comput. Methods Appl. Sci. vol. 5 2007 Springer New York 187 197
    • (2007) Comput. Methods Appl. Sci. , vol.5 , pp. 187-197
    • Barba, L.A.1
  • 4
    • 72049126104 scopus 로고    scopus 로고
    • Global field interpolation for particle methods
    • L.A. Barba, and L.F. Rossi Global field interpolation for particle methods J. Comput. Phys. 229 2010 1292 1310
    • (2010) J. Comput. Phys. , vol.229 , pp. 1292-1310
    • Barba, L.A.1    Rossi, L.F.2
  • 5
    • 0000028195 scopus 로고
    • Norm estimates for inverses of Toeplitz distance matrices
    • B.J.C. Baxter Norm estimates for inverses of Toeplitz distance matrices J. Approx. Theory 70 1994 222 242
    • (1994) J. Approx. Theory , vol.70 , pp. 222-242
    • Baxter, B.J.C.1
  • 6
    • 78651331018 scopus 로고
    • On the asymptotic behaviour of the span of the translates of the multiquadric O(r)(2+c2)1/2 as c→b
    • B.J.C. Baxter On the asymptotic behaviour of the span of the translates of the multiquadric O (r ) = (r 2 + c 2 ) 1 / 2 as c → b Comput. Math. Appl. 24 1994 1 6
    • (1994) Comput. Math. Appl. , vol.24 , pp. 1-6
    • Baxter, B.J.C.1
  • 7
    • 0030269018 scopus 로고    scopus 로고
    • On shifted cardinal interpolation by Gaussians and multiquadrics
    • B.J.C. Baxter, and N. Sivakumar On shifted cardinal interpolation by Gaussians and multiquadrics J. Approx. Theory 87 1996 36 59
    • (1996) J. Approx. Theory , vol.87 , pp. 36-59
    • Baxter, B.J.C.1    Sivakumar, N.2
  • 9
    • 77953136806 scopus 로고    scopus 로고
    • Sensitivity of RBF interpolation on an otherwise uniform grid with a point omitted or slightly shifted
    • J.P. Boyd, and L.R. Bridge Sensitivity of RBF interpolation on an otherwise uniform grid with a point omitted or slightly shifted Appl. Numer. Math. 60 2010 659 672
    • (2010) Appl. Numer. Math. , vol.60 , pp. 659-672
    • Boyd, J.P.1    Bridge, L.R.2
  • 10
    • 85095815155 scopus 로고    scopus 로고
    • Radial basis functions
    • M.D. Buhmann Radial basis functions Acta Numer. 9 2000 1 38
    • (2000) Acta Numer. , vol.9 , pp. 1-38
    • Buhmann, M.D.1
  • 11
    • 15844363062 scopus 로고    scopus 로고
    • Radial Basis Functions: Theory and Implementations
    • Cambridge Univ. Press
    • M.D. Buhmann Radial Basis Functions: Theory and Implementations Cambridge Monogr. Appl. Comput. Math. vol. 12 2003 Cambridge Univ. Press
    • (2003) Cambridge Monogr. Appl. Comput. Math. , vol.12
    • Buhmann, M.D.1
  • 12
    • 1942539307 scopus 로고    scopus 로고
    • Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions
    • T. Cecil, J. Qian, and S. Osher Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions J. Comput. Phys. 196 2004 327 347
    • (2004) J. Comput. Phys. , vol.196 , pp. 327-347
    • Cecil, T.1    Qian, J.2    Osher, S.3
  • 14
    • 33644584897 scopus 로고    scopus 로고
    • Unsymmetric and symmetric meshless schemes for the unsteady convection-diffusion equation
    • P.P. Chinchapatnam, K. Djidjeli, and P.B. Nair Unsymmetric and symmetric meshless schemes for the unsteady convection-diffusion equation Comput. Methods Appl. Mech. Engrg. 195 2006 2432 2453
    • (2006) Comput. Methods Appl. Mech. Engrg. , vol.195 , pp. 2432-2453
    • Chinchapatnam, P.P.1    Djidjeli, K.2    Nair, P.B.3
  • 15
    • 34248187646 scopus 로고    scopus 로고
    • Adaptive residual subsampling methods for radial basis function interpolation and collocation problems
    • T.A. Driscoll, and A. Heryudono Adaptive residual subsampling methods for radial basis function interpolation and collocation problems Comput. Math. Appl. 53 2007 927 939
    • (2007) Comput. Math. Appl. , vol.53 , pp. 927-939
    • Driscoll, T.A.1    Heryudono, A.2
  • 16
    • 0041593773 scopus 로고    scopus 로고
    • Solving differential equations with radial basis functions: Multilevel methods and smoothing
    • G.E. Fasshauer Solving differential equations with radial basis functions: Multilevel methods and smoothing Adv. Comput. Math. 11 1999 139 159
    • (1999) Adv. Comput. Math. , vol.11 , pp. 139-159
    • Fasshauer, G.E.1
  • 17
    • 0036467952 scopus 로고    scopus 로고
    • Newton iteration with multiquadrics for the solution of nonlinear PDEs
    • G.E. Fasshauer Newton iteration with multiquadrics for the solution of nonlinear PDEs Comput. Math. Appl. 43 2002 423 438
    • (2002) Comput. Math. Appl. , vol.43 , pp. 423-438
    • Fasshauer, G.E.1
  • 18
    • 49549087872 scopus 로고    scopus 로고
    • Meshfree Approximation Methods with MATLAB
    • World Scientific Publishing Company Singapore
    • G.F. Fasshauer Meshfree Approximation Methods with MATLAB Interdiscip. Math. Sci. 2007 World Scientific Publishing Company Singapore
    • (2007) Interdiscip. Math. Sci.
    • Fasshauer, G.F.1
  • 19
    • 0034134865 scopus 로고    scopus 로고
    • Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method
    • A.I. Fedoseyev, M.J. Friedman, and E.J. Kansa Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method Internat. J. Bifurcation Chaos 10 2000 481 492
    • (2000) Internat. J. Bifurcation Chaos , vol.10 , pp. 481-492
    • Fedoseyev, A.I.1    Friedman, M.J.2    Kansa, E.J.3
  • 20
    • 0036468097 scopus 로고    scopus 로고
    • Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary
    • A.I. Fedoseyev, M.J. Friedman, and E.J. Kansa Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary Comput. Math. Appl. 43 2002 439 455
    • (2002) Comput. Math. Appl. , vol.43 , pp. 439-455
    • Fedoseyev, A.I.1    Friedman, M.J.2    Kansa, E.J.3
  • 21
    • 34548390120 scopus 로고    scopus 로고
    • Transport schemes on a sphere using radial basis functions
    • N. Flyer, and G.B. Wright Transport schemes on a sphere using radial basis functions J. Comput. Phys. 226 2007 1059 1084
    • (2007) J. Comput. Phys. , vol.226 , pp. 1059-1084
    • Flyer, N.1    Wright, G.B.2
  • 22
    • 67149139433 scopus 로고    scopus 로고
    • A radial basis function method for the shallow water equations on a sphere
    • N. Flyer, and G.B. Wright A radial basis function method for the shallow water equations on a sphere Proc. Roy. Soc. A 465 2009 1949 1976
    • (2009) Proc. Roy. Soc. A , vol.465 , pp. 1949-1976
    • Flyer, N.1    Wright, G.B.2
  • 23
    • 76549122679 scopus 로고    scopus 로고
    • Comparisons between pseudospectral and radial basis function derivative approximations
    • B. Fornberg, N. Flyer, and J.M. Russell Comparisons between pseudospectral and radial basis function derivative approximations IMA J. Numer. Anal. 30 2010 149 172
    • (2010) IMA J. Numer. Anal. , vol.30 , pp. 149-172
    • Fornberg, B.1    Flyer, N.2    Russell, J.M.3
  • 24
    • 38549125605 scopus 로고    scopus 로고
    • A stable algorithm for flat radial basis functions on a sphere
    • B. Fornberg, and C. Piret A stable algorithm for flat radial basis functions on a sphere SIAM J. Sci. Comput. 30 2007 60 80
    • (2007) SIAM J. Sci. Comput. , vol.30 , pp. 60-80
    • Fornberg, B.1    Piret, C.2
  • 25
    • 14644431037 scopus 로고    scopus 로고
    • Stable computation of multiquadric interpolants for all values of the shape parameter
    • B. Fornberg, and G. Wright Stable computation of multiquadric interpolants for all values of the shape parameter Comput. Math. Appl. 48 2004 853 867
    • (2004) Comput. Math. Appl. , vol.48 , pp. 853-867
    • Fornberg, B.1    Wright, G.2
  • 26
    • 34250838673 scopus 로고    scopus 로고
    • The Runge phenomenon and spatially variable shape parameters in RBF interpolation
    • B. Fornberg, and J. Zuev The Runge phenomenon and spatially variable shape parameters in RBF interpolation Comput. Math. Appl. 54 2007 379 398
    • (2007) Comput. Math. Appl. , vol.54 , pp. 379-398
    • Fornberg, B.1    Zuev, J.2
  • 27
    • 0002245065 scopus 로고    scopus 로고
    • Solving partial differential equations by collocation using radial basis functions
    • C. Franke, and R. Schaback Solving partial differential equations by collocation using radial basis functions Appl. Math. Comput. 93 1998 73 91
    • (1998) Appl. Math. Comput. , vol.93 , pp. 73-91
    • Franke, C.1    Schaback, R.2
  • 28
    • 0002020770 scopus 로고
    • On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals
    • J. Halton On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals Numer. Math. 2 1960 84 90
    • (1960) Numer. Math. , vol.2 , pp. 84-90
    • Halton, J.1
  • 31
    • 15544362032 scopus 로고    scopus 로고
    • Multiresolution Methods in Scattered Data Modelling
    • Springer Heidelberg
    • A. Iske Multiresolution Methods in Scattered Data Modelling Lect. Notes Comput. Sci. Eng. vol. 37 2004 Springer Heidelberg
    • (2004) Lect. Notes Comput. Sci. Eng. , vol.37
    • Iske, A.1
  • 32
    • 0025210711 scopus 로고
    • Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics-II. Solutions to hyperbolic, parabolic, and elliptic partial differential equations
    • E.J. Kansa Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics-II. Solutions to hyperbolic, parabolic, and elliptic partial differential equations Comput. Math. Appl. 19 1990 147 161
    • (1990) Comput. Math. Appl. , vol.19 , pp. 147-161
    • Kansa, E.J.1
  • 33
    • 3142729350 scopus 로고    scopus 로고
    • A volumetric integral radial basis function method for time-dependent partial differential equations: I. Formulation
    • E.J. Kansa, H. Power, G.E. Fasshauer, and L. Ling A volumetric integral radial basis function method for time-dependent partial differential equations: I. Formulation Engrg. Anal. Boundary Elements 28 2004 1191 1206
    • (2004) Engrg. Anal. Boundary Elements , vol.28 , pp. 1191-1206
    • Kansa, E.J.1    Power, H.2    Fasshauer, G.E.3    Ling, L.4
  • 34
    • 0242667184 scopus 로고    scopus 로고
    • A numerical study of some radial basis function based solution methods for elliptic PDEs
    • E. Larsson, and B. Fornberg A numerical study of some radial basis function based solution methods for elliptic PDEs Comput. Math. Appl. 46 2003 891 902
    • (2003) Comput. Math. Appl. , vol.46 , pp. 891-902
    • Larsson, E.1    Fornberg, B.2
  • 35
    • 17544362394 scopus 로고    scopus 로고
    • Approximation of parabolic PDEs on spheres using spherical basis functions
    • Q.T. Le Gia Approximation of parabolic PDEs on spheres using spherical basis functions Adv. Comput. Math. 22 2005 377 397
    • (2005) Adv. Comput. Math. , vol.22 , pp. 377-397
    • Le Gia, Q.T.1
  • 37
    • 33745746200 scopus 로고    scopus 로고
    • A finite volume method based on radial basis functions for two-dimensional nonlinear diffusion equations
    • T.J. Moroney, and I.W. Turner A finite volume method based on radial basis functions for two-dimensional nonlinear diffusion equations Appl. Math. Model. 30 2006 1118 1133
    • (2006) Appl. Math. Model. , vol.30 , pp. 1118-1133
    • Moroney, T.J.1    Turner, I.W.2
  • 38
    • 34547502438 scopus 로고    scopus 로고
    • A three-dimensional finite volume method based on radial basis functions for the accurate computational modeling of nonlinear diffusion equations
    • T.J. Moroney, and I.W. Turner A three-dimensional finite volume method based on radial basis functions for the accurate computational modeling of nonlinear diffusion equations J. Comput. Phys. 225 2007 1409 1426
    • (2007) J. Comput. Phys. , vol.225 , pp. 1409-1426
    • Moroney, T.J.1    Turner, I.W.2
  • 39
    • 38249012211 scopus 로고
    • Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices
    • F.J. Narcowich, and J.D. Ward Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices J. Fourier Anal. Appl. 69 1992 84 109
    • (1992) J. Fourier Anal. Appl. , vol.69 , pp. 84-109
    • Narcowich, F.J.1    Ward, J.D.2
  • 40
    • 33745923525 scopus 로고    scopus 로고
    • Computing eigenmodes of elliptic operators using radial basis functions
    • R.B. Platte, and T.A. Driscoll Computing eigenmodes of elliptic operators using radial basis functions Comput. Math. Appl. 48 2006 1251 1268
    • (2006) Comput. Math. Appl. , vol.48 , pp. 1251-1268
    • Platte, R.B.1    Driscoll, T.A.2
  • 41
    • 0000202520 scopus 로고
    • The deferred approach to the limit. Part I-Single lattice
    • L.F. Richardson The deferred approach to the limit. Part I-Single lattice Philos. Trans. R. Soc. 226 1927 299 349
    • (1927) Philos. Trans. R. Soc. , vol.226 , pp. 299-349
    • Richardson, L.F.1
  • 42
    • 0347632817 scopus 로고
    • The deferred approach to the limit. Part I-Single lattice
    • L.F. Richardson The deferred approach to the limit. Part I-Single lattice O.M. Ashford, H. Charnock, P.G. Drazin, J.C.R. Hunt, P. Smoker, I. Sutherland, Collected Papers of Lewis Fry Richardson 1993 Cambridge Univ. Press New York 625 678
    • (1993) Collected Papers of Lewis Fry Richardson , pp. 625-678
    • Richardson, L.F.1
  • 43
    • 0013465435 scopus 로고
    • Lower bounds for norms of inverses of interpolation matrices for radial basis functions
    • R. Schaback Lower bounds for norms of inverses of interpolation matrices for radial basis functions J. Approx. Theory 79 1994 287 306
    • (1994) J. Approx. Theory , vol.79 , pp. 287-306
    • Schaback, R.1
  • 44
    • 65749310935 scopus 로고
    • Error estimates and condition numbers for radial basis function interpolation
    • R. Schaback Error estimates and condition numbers for radial basis function interpolation Constr. Approx. 3 1995 251 264
    • (1995) Constr. Approx. , vol.3 , pp. 251-264
    • Schaback, R.1
  • 45
    • 33646731541 scopus 로고    scopus 로고
    • Kernel techniques: From machine learning to meshless methods
    • R. Schaback, and H. Wendland Kernel techniques: From machine learning to meshless methods Acta Numer. 15 2006 543 639
    • (2006) Acta Numer. , vol.15 , pp. 543-639
    • Schaback, R.1    Wendland, H.2
  • 46
    • 0000246322 scopus 로고    scopus 로고
    • Application of the multiquadric method for the solution of elliptic partial differential equations
    • M. Sharan, E.J. Kansa, and S. Gupta Application of the multiquadric method for the solution of elliptic partial differential equations Appl. Math. Comput. 84 1997 275 302
    • (1997) Appl. Math. Comput. , vol.84 , pp. 275-302
    • Sharan, M.1    Kansa, E.J.2    Gupta, S.3
  • 47
    • 0003424374 scopus 로고    scopus 로고
    • Society for Industrial and Applied Mathematics (SIAM) Philadelphia
    • L.N. Trefethen, and D. Bau III Numerical Linear Algebra 1997 Society for Industrial and Applied Mathematics (SIAM) Philadelphia
    • (1997) Numerical Linear Algebra
    • Trefethen, L.N.1    Bau Iii, D.2
  • 48
    • 0033465031 scopus 로고    scopus 로고
    • Meshless Galerkin methods using radial basis functions
    • H. Wendland Meshless Galerkin methods using radial basis functions Math. Comp. 68 1999 1521 1531
    • (1999) Math. Comp. , vol.68 , pp. 1521-1531
    • Wendland, H.1
  • 50
    • 0037170718 scopus 로고    scopus 로고
    • Compactly supported radial basis functions for shallow water equations
    • S.M. Wong, Y.C. Hon, and M.A. Golberg Compactly supported radial basis functions for shallow water equations Appl. Math. Comput. 127 2002 79 101
    • (2002) Appl. Math. Comput. , vol.127 , pp. 79-101
    • Wong, S.M.1    Hon, Y.C.2    Golberg, M.A.3
  • 51
    • 0003544101 scopus 로고    scopus 로고
    • Handbook of Function and Generalized Function Transformations
    • CRC Press Boca Raton, FL
    • A.I. Zayed Handbook of Function and Generalized Function Transformations Math. Sci. Ref. Ser. vol. 3 1996 CRC Press Boca Raton, FL
    • (1996) Math. Sci. Ref. Ser. , vol.3
    • Zayed, A.I.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.