Towards stability of radial basis function based cubature formulas

Cubature formulas (CFs) based on radial basis functions (RBFs) have become an important tool for multivariate numerical integration of scattered data. Although numerous works have been published on such RBF-CFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for RBF-CFs. In particular, we prove stability for RBF-CFs based on compactly supported RBFs under certain conditions on the shape parameter and the data points. Moreover, it is shown that asymptotic stability of many RBF-CFs is independent of polynomial terms, which are often included in RBF approximations. While our findings provide some novel conditions for stability of RBF-CFs, the present work also demonstrates that there are still many gaps to fill in future investigations.

[1]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[2]  Lloyd N. Trefethen,et al.  Cubature, Approximation, and Isotropy in the Hypercube , 2017, SIAM Rev..

[3]  L. M. M. van den Bos,et al.  Adaptive sampling-based quadrature rules for efficient Bayesian prediction , 2019, J. Comput. Phys..

[4]  Andrew D. Back,et al.  Radial Basis Functions , 2001 .

[5]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .

[6]  A. W. Wymore,et al.  Numerical Evaluation of Multiple Integrals I , 2010 .

[7]  Siraj-ul-Islam,et al.  Numerical integration of multi-dimensional highly oscillatory, gentle oscillatory and non-oscillatory integrands based on wavelets and radial basis functions , 2012 .

[8]  Bahman Mehri,et al.  Lebesgue function for multivariate interpolation by radial basis functions , 2007, Appl. Math. Comput..

[9]  유재철,et al.  Randomization , 2020, Randomization, Bootstrap and Monte Carlo Methods in Biology.

[10]  Frances Y. Kuo,et al.  High-dimensional integration: The quasi-Monte Carlo way*† , 2013, Acta Numerica.

[11]  Hermann Engles,et al.  Numerical quadrature and cubature , 1980 .

[12]  Jan Glaubitz,et al.  Stable High Order Quadrature Rules for Scattered Data and General Weight Functions , 2020, SIAM J. Numer. Anal..

[13]  Bengt Fornberg,et al.  Numerical quadrature over smooth surfaces with boundaries , 2018, J. Comput. Phys..

[14]  Alvise Sommariva,et al.  INTEGRATION BY RBF OVER THE SPHERE , 2005 .

[15]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[16]  Alvise Sommariva,et al.  Meshless cubature by Green's formula , 2006, Appl. Math. Comput..

[17]  B. Fornberg,et al.  Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions , 2003 .

[18]  Bengt Fornberg,et al.  On spherical harmonics based numerical quadrature over the surface of a sphere , 2014, Advances in Computational Mathematics.

[19]  Jan Glaubitz,et al.  Stable High-Order Cubature Formulas for Experimental Data , 2020, J. Comput. Phys..

[20]  S. De Marchi,et al.  On Optimal Center Locations for Radial Basis Function Interpolation: Computational Aspects , 2022 .

[21]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[22]  Daan Huybrechs,et al.  Stable high-order quadrature rules with equidistant points , 2009, J. Comput. Appl. Math..

[23]  Chang Shu,et al.  Integrated radial basis functions‐based differential quadrature method and its performance , 2007 .

[24]  T. Driscoll,et al.  Interpolation in the limit of increasingly flat radial basis functions , 2002 .

[25]  A. Iske.  On the Approximation Order and Numerical Stability of Local Lagrange Interpolation by Polyharmonic Splines , 2003 .

[26]  Marco Vianello,et al.  Bivariate Lagrange interpolation at the Padua points: the ideal theory approach , 2007, Numerische Mathematik.

[27]  I. P. Mysovskih.  Approximate Calculation of Integrals , 1969 .

[28]  Carsten Franke,et al.  Solving partial differential equations by collocation using radial basis functions , 1998, Appl. Math. Comput..

[29]  Victor Bayona,et al.  An insight into RBF-FD approximations augmented with polynomials , 2019, Comput. Math. Appl..

[30]  Bengt Fornberg,et al.  Stable computations with flat radial basis functions using vector-valued rational approximations , 2016, J. Comput. Phys..

[31]  M. Urner.  Scattered Data Approximation , 2016 .

[32]  R. Cooke.  Real and Complex Analysis , 2011 .

[33]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[34]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[35]  E. Kansa,et al.  Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations , 2000 .

[36]  M. L. Watts.  Radial Basis Function Based Quadrature over Smooth Surfaces , 2016 .

[37]  Alvise Sommariva,et al.  Numerical Cubature on Scattered Data by Radial Basis Functions , 2005, Computing.

[38]  Christoph W. Ueberhuber,et al.  Computational Integration , 2018, An Introduction to Scientific, Symbolic, and Graphical Computation.

[39]  Alvise Sommariva,et al.  Meshless cubature over the disk using thin-plate splines , 2008 .

[40]  Lloyd N. Trefethen,et al.  Exactness of quadrature formulas , 2021, SIAM Rev..

[41]  B. Fornberg,et al.  A numerical study of some radial basis function based solution methods for elliptic PDEs , 2003 .

[42]  Marco Vianello,et al.  Bivariate Lagrange interpolation at the Padua points: The generating curve approach , 2006, J. Approx. Theory.

[43]  Bengt Fornberg,et al.  A primer on radial basis functions with applications to the geosciences , 2015, CBMS-NSF regional conference series in applied mathematics.

[44]  H. Weyl.  Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

[45]  Marcel Bauer,et al.  Numerical Methods for Partial Differential Equations , 1994 .

[46]  Jan Glaubitz.  Construction and application of provable positive and exact cubature formulas , 2021, ArXiv.

[47]  Piecewise Polynomial , 2014, Computer Vision, A Reference Guide.

[48]  Ronald Cools,et al.  An encyclopaedia of cubature formulas , 2003, J. Complex..

[49]  Jan Glaubitz,et al.  Towards Stable Radial Basis Function Methods for Linear Advection Problems , 2021, Comput. Math. Appl..

[50]  Michael J. McCourt,et al.  Stable Evaluation of Gaussian Radial Basis Function Interpolants , 2012, SIAM J. Sci. Comput..

[51]  R. Schaback.  Multivariate Interpolation by Polynomials and Radial Basis Functions , 2005 .

[52]  Bengt Fornberg,et al.  Solving PDEs with radial basis functions * , 2015, Acta Numerica.

[53]  James Clerk Maxwell,et al.  On Approximate Multiple Integration between Limits by Summation , 2011 .

[54]  Armin Iske,et al.  Armin Iske * Scattered Data Approximation by Positive Definite Kernel Functions , 2012 .

[55]  Helmut Brass,et al.  Quadrature Theory: The Theory of Numerical Integration on a Compact Interval , 2011 .

[56]  Alan Genz,et al.  Testing multidimensional integration routines , 1984 .

[57]  Robert Schaback,et al.  A Newton basis for Kernel spaces , 2009, J. Approx. Theory.

[58]  I. P. Mysovskikh.  THE APPROXIMATION OF MULTIPLE INTEGRALS BY USING INTERPOLATORY CUBATURE FORMULAE , 1980 .

[59]  Bayram Ali Ibrahimoglu,et al.  Lebesgue functions and Lebesgue constants in polynomial interpolation , 2016, Journal of Inequalities and Applications.

[60]  Jean-François Richard,et al.  Methods of Numerical Integration , 2000 .

[61]  Robert Schaback,et al.  Bases for kernel-based spaces , 2011, J. Comput. Appl. Math..

[62]  Fabio Nobile,et al.  Stable high-order randomized cubature formulae in arbitrary dimension , 2018, J. Approx. Theory.

[64]  Martin D. Buhmann,et al.  Radial Basis Functions: Theory and Implementations: Preface , 2003 .

[65]  Elisabeth Larsson,et al.  Stable Computations with Gaussian Radial Basis Functions , 2011, SIAM J. Sci. Comput..

[66]  Robert Schaback,et al.  Error estimates and condition numbers for radial basis function interpolation , 1995, Adv. Comput. Math..

[67]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[68]  M. Vianello,et al.  RBF Moment computation and meshless cubature on general polygonal regions , 2021, Appl. Math. Comput..

[69]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[70]  Kevin P. Murphy,et al.  Machine learning - a probabilistic perspective , 2012, Adaptive computation and machine learning series.

[71]  Stefano De Marchi,et al.  A new stable basis for radial basis function interpolation , 2013, J. Comput. Appl. Math..

[72]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[73]  Anne Gelb,et al.  Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions , 2021, J. Sci. Comput..

[74]  Gerald B. Folland,et al.  How to Integrate A Polynomial Over A Sphere , 2001, Am. Math. Mon..

[75]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[76]  Ronald Cools,et al.  Constructing cubature formulae: the science behind the art , 1997, Acta Numerica.

[77]  Daniel W. Lozier,et al.  NIST Digital Library of Mathematical Functions , 2003, Annals of Mathematics and Artificial Intelligence.

[78]  Joseph D. Ward,et al.  Kernel based quadrature on spheres and other homogeneous spaces , 2012, Numerische Mathematik.

[79]  Bengt Fornberg,et al.  Stable calculation of Gaussian-based RBF-FD stencils , 2013, Comput. Math. Appl..

[80]  B. Fornberg,et al.  Some observations regarding interpolants in the limit of flat radial basis functions , 2003 .

[81]  Louis J. Wicker,et al.  Enhancing finite differences with radial basis functions: Experiments on the Navier-Stokes equations , 2016, J. Comput. Phys..

[82]  A. Iske,et al.  On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions , 1996 .

[83]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[84]  J. Lasserre.  Simple formula for integration of polynomials on a simplex , 2019, BIT Numerical Mathematics.

[85]  Victor Bayona,et al.  Comparison of Moving Least Squares and RBF+poly for Interpolation and Derivative Approximation , 2019, Journal of Scientific Computing.

[86]  P. Gruber,et al.  Funktionen von beschränkter Variation in der Theorie der Gleichverteilung , 1990 .

[87]  I. P. Mysovskikh.  Cubature formulas that are exact for trigonometric polynomials , 1998 .

[88]  J A Reeger,et al.  Numerical quadrature over smooth, closed surfaces , 2016, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[89]  Robert Schaback,et al.  Stability of kernel-based interpolation , 2010, Adv. Comput. Math..