A review of radial kernel methods for the resolution of Fredholm integral equations of the second kind

Year 2024, Volume: 7 Issue: Special Issue: AT&A, 142 - 153, 16.12.2024

Roberto Cavoretto , Alessandra De Rossi , Domenico Mezzanotte

https://doi.org/10.33205/cma.1538581

Abstract

The paper presents an overview of the existing literature concerning radial kernel meshfree methods for the numerical treatment of second-kind Fredholm integral equations. More in detail, it briefly recalls radial basis function (RBF) interpolation and cubature rules to properly describe numerical methods for two-dimensional linear Fredholm equations. The RBF approach allows us to consider the case when the involved functions are not known analytically, but only as vectors of scattered data samples. The described methods do not require any background mesh and, hence, are also independent on the geometry of the domain.

Keywords

Radial basis functions, Fredholm integral equations, Meshfree methods, Scattered data

Supporting Institution

ICSC - Centro Nazionale di Ricerca in High-Performance Computing, Big Data and Quantum Computing; INdAM Research group GNCS.

Thanks

RITA "Research ITalian network on Approximation"; UMI Group TAA "Approximation Theory and Applications"; SIMAI Activity Group ANA&A "Numerical and Analytical Approximation of Data and Functions with Applications".

References

• T. Akbari, M. Esmaeilbeigi and D. Moazami: A stable meshless numerical scheme using hybrid kernels to solve linear Fredholm integral equations of the second kind and its applications, Math. Comput. Simulation, 220 (2024), 1–28.
• P. Assari, H. Adibi and M. Dehghan: A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method, Appl. Math. Model., 37 (22) (2013), 9269–9294.
• K. E. Atkinson: The Numerical Solution of Integral Equations of the second kind, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press (1997).
• K. E. Atkinson, F. Potra: The discrete Galerkin method for linear integral equations, IMA J. Numer. Anal., 9 (1989), 385–403.
• M. Bozzini, L. Lenarduzzi, M. Rossini and R. Schaback: Interpolation with variably scaled kernels, IMA J. Numer. Anal., 35 (2015), 199–219.
• M. D. Buhmann: Radial Basis Functions: Theory and Implementation, Cambridge Monogr. Appl. Comput. Math., vol. 12, Cambridge Univ. Press, Cambridge (2003).
• R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione and G. Santin: Partition of unity interpolation using stable kernel-based techniques, Appl. Numer. Math., 116 (2017), 95–107.
• R. Cavoretto, A. De Rossi: A two-stage adaptive scheme based on RBF collocation for solving elliptic PDEs, Comput. Math. Appl., 79 (11) (2020), 3206–3222.
• R. Cavoretto, A. De Rossi, F. Dell’Accio, F. Di Tommaso, N. Siar, A. Sommariva and M. Vianello: Numerical cubature on scattered data by adaptive interpolation, J. Comput. Appl. Math., 444 (2024), 115793.
• R. Cavoretto, A. De Rossi and S. Lancellotti: Bayesian approach for radial kernel parameter tuning, J. Comput. Appl. Math., 441 (2024), Article ID: 115716.
• R. Cavoretto, A. De Rossi, M. S. Mukhametzhanov and Y. D. Sergeyev: On the search of the shape parameter in radial basis functions using univariate global optimization methods, J. Global Optim., 79 (2021), 305–327.
• R. Cavoretto, A. De Rossi, A. Sommariva and M. Vianello: RBFCUB: A numerical package for near-optimal meshless cubature on general polygons, Appl. Math. Lett., 125 (2022), Article ID: 107704.
• R. Cavoretto, G. E. Fasshauer and M. McCourt: An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels, Numer. Algor., 68 (2015), 393–422.
• M. C. De Bonis, G. Mastroianni: Projection methods and condition numbers in uniform norm for Fredholm and Cauchy singular integral equations, SIAM J. Numer. Anal., 44 (4) (2006), 1351–1374.
• S. De Marchi, G. Santin: A new stable basis for radial basis function interpolation, J. Comput. Appl. Math., 253 (2013), 1–13.
• F. Dell’Accio, D. Mezzanotte, F. Nudo and D. Occorsio: Numerical approximation of Fredholm integral equation by the constrained mock-Chebyshev least squares operator, J. Comput. Appl. Math., 447 (2024), 115886.
• A. Doucet, A. M. Johansen and V. B. Tadi´c: On solving integral equations using Markov chain Monte Carlo methods, Appl. Math. Comput., 216 (2010), 2869–2880.
• R. Farengo, Y. C. Lee and P. N. Guzdar: An electromagnetic integral equation: application to microtearing modes, Phys. Fluids, 26 (1983), 3515–3523.
• R. Farnoosh, M. Ebrahimi: Monte Carlo method for solving Fredholm integral equations of the second kind, Appl. Math. Comput., 195 (1) (2008), 309–315.
• G. E. Fasshauer: Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, Vol. 6, World Scientific Publishing Co., Singapore (2007).
• G. E. Fasshauer, M. J. McCourt: Kernel-based Approximation Methods using MATLAB, Interdisciplinary Mathematical Sciences, Vol. 19, World Scientific Publishing Co., Singapore (2015).
• G. E. Fasshauer, M. J. McCourt: Stable evaluation of Gaussian radial basis function interpolants, SIAM J. Sci. Comput., 34 (2012), A737–A762.
• G. E. Fasshauer, J. G. Zhang: On choosing “optimal” shape parameters for RBF approximation, Numer. Algorithms, 45 (2007), 345–368.
• B. Fornberg, E. Larsson and N. Flyer: Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33 (2011), 869–892.
• H. Fatahi, J. Saberi-Nadjafi and E. Shivanian: A new spectral meshless radial point interpolation (SMRPI) method for the two-dimensional Fredholm integral equations on general domains with error analysis, J. Comput. Appl. Math., 294 (2016), 196–209.
• B. Fornberg, C. Piret: A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput., 30 (2007), 60–80.
• B. Fornberg, J. Zuev: The Runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput. Math. Appl., 54 (2007), 379–398.
• Y. Guan, T. Fang, D. Zhang and C. Jin: Solving Fredholm Integral Equations Using Deep Learning, Int. J. Appl. Comput. Math., 8 (2022), Article ID: 87.
• G. Han, R. Wang: Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations, J. Comput. Appl. Math., 139 (1) (2002), 49–63.
• P. C. Hansen, T. K. Jensen: Large-scale methods in image deblurring, Lect. Notes. Comput. Sci., 4699 (2007), 24–35.
• J. T. Kajiya: The rendering equation, Proceedings of the 13th annual conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’86), ACM Press, New York (USA) (1986), 143–150.
• A. Keller: Instant radiosity, Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’97), ACM Press, New York (USA) (1997), 49–56.
• J. Kennedy, R. Eberhart: Particle swarm optimization. Proceedings of IEEE International Conference on Neural Networks, 4 (1995), 1942–1948.
• E. Larsson, R. Schaback: Scaling of radial basis functions, IMA J. Numer. Analysis, 44 (2) (2024), 1130–1152.
• P. A. Martin, L. Farina: Radiation of water waves by a heaving submerged horizontal disc, J. Fluid. Mech., 337 (1997), 365–379.
• D. Mezzanotte, D. Occorsio and M. G. Russo: Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind, Mathematics, 9 (2021), 2652.
• A. Noorizadegan, C.-S. Chen, R. Cavoretto and A. De Rossi: Efficient truncated randomized SVD for mesh-free kernel methods, Comput. Math. Appl., 164 (2024), 12–20.
• D. Occorsio, M. G. Russo: Numerical methods for Fredholm integral equations on the square, Appl. Math. Comput., 218 (2011), 2318–2333.
• D. Occorsio, M. G. Russo: Nyström Methods for Fredholm Integral Equations Using Equispaced Points, Filomat, 28 (1) (2014), 49–63.
• D. Occorsio, M. G. Russo and W. Themistocklakis: Some numerical applications of generalized Bernstein operators, Constr. Math. Anal., 4 (2) (2021), 186–214.
• J. Radlow: A two-dimensional singular integral equation of diffraction theory, Bull. Am. Math. Soc., 70 (1964), 596–599.
• S. Rippa: An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math., 11 (1999), 193–210.
• M. Rossini: Variably scaled kernels: an overview, Dolomites Res. Notes Approx., 15 (2022), 61–72.
• R. Schaback: Native Hilbert spaces for radial basis functions I, New Developments in Approximation Theory. ISNM International Series of Numerical Mathematics, vol 132. Birkhäuser, Basel (1999).
• H. Schäfer, E. Sternin, R. Stannarius, M. Arndt, and F. Kremer: Novel Approach to the Analysis of Broadband Dielectric Spectra, Phys. Rev. Lett., 76 (1996), 2177–2180.
• A. Sommariva, M. Vianello: Numerical Cubature on Scattered Data by Radial Basis Functions, Computing, 76 (2006), 295–310.
• H. Wendland: Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., vol. 17, Cambridge Univ. Press, Cambridge (2005).
• T. Wenzel, G. Santin and B. Haasdonk: Analysis of Target Data-Dependent Greedy Kernel Algorithms: Convergence Rates for f−, f · P− and f/P−Greedy, Constr. Approx., 57 (2023), 45–74.
• E. Zappala, A. H. d. O. Fonseca, J. O. Caro, A. H. Moberly, M. J. Higley, J. Cardin and D. van Dijk: Learning integral operators via neural integral equations, Nat. Mach. Intell., 6 (2024), 1046–1062.