Research Article
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Year 2023, , 38 - 54, 15.03.2023
https://doi.org/10.33205/cma.1247239

Abstract

References

  • R. A. Adams, J. Fournier: Sobolev Spaces, Elsevier, London, (2003).
  • M. G. Armentano: Error estimates in Sobolev spaces for moving least square approximations, SIAM J. Numer. Anal., 39 (2001), 38–51.
  • M. G. Armentano, R. G. Duran: Error estimates for moving least square approximations, Appl. Numer. Math., 37 (2001), 397–416.
  • G. E. Backus, J. F. Gilbert: Numerical applications of a formalism for geophysical inverse problems, Geophys. J. Int., 13 (1967), 247–276.
  • V. Bayona: Comparison of moving least squares and RBF+poly for interpolation and derivative approximation, J. Sci. Comput., 81 (2019), 486–512.
  • V. Bayona, N. Flyer, B. Fornberg and G. A. Barnett: On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs, J. Comput. Phys., 332 (2017), 257–273.
  • S. C. Bernard, L. R. Scott: The mathematical theory of finite element methods, Springer, New York, (2003).
  • L. Bos, K. Salkauskas: Moving least-squares are Backus-Gilbert optimal, J. Approx. Theory, 59 (1989), 267–275.
  • M. Bozzini, L. Lenarduzzi, M. Rossini and R. Schaback: Interpolation with variably scaled kernels, SIAM J. Numer. Anal., 35 (2015), 199–219.
  • S. Cuomo, A. Galletti, G. Giunta and A. Starace: Surface reconstruction from scattered point via RBF interpolation on GPU, Federated Conference on Computer Science and Information Systems, Krakow (Poland) (2013), 433–440.
  • S. De Marchi, W. Erb, F. Marchetti, E. Perracchione and M. Rossini: Shape-Driven interpolation with discontinuous kernels: Error analysis, edges extraction and application in magnetic particle imaging, J. Sci. Comput., 42 (2020), 472–491.
  • S. De Marchi, F. Marchetti and E. Perracchione: Jumping with variably scaled discontinuous kernels, BIT Numer. Math., 60 (2019), 441–463.
  • G. E. Fasshauer: Meshfree Approximation Methods, World Scientific Publishing: Singapore, (2007).
  • G. E. Fasshauer, M. J. McCourt: Kernel based approximation methods using MATLAB, World Scientific Publishing: Singapore, (2015).
  • S. Guastavino, F. Benvenuto: Convergence rates of spectral regularization methods: A comparison between ill-posed inverse problems and statistical kernel learning, SIAM J. Numer. Anal., 58 (6) (2020), 3504–3529.
  • P. Lancaster, K. Salkuaskas: Surfaces generated by moving least squares methods, Math. Comput., 37 (1981), 141–158.
  • D. Levin: The approximation power of moving least-squares. Math. Comp., 67 (1998), 1517–1531.
  • L. B. Lucy: A numerical approach to the testing of the fission hypothesis, AJ, 82 (1982), 1013–1024.
  • D. Mirzaei: Analysis of moving least square approximation revisited, J. Comput. Appl. Math., 282 (2015), 237-250.
  • D. Mirzaei, R. Schaback: Direct meshless local Petrov–Galerkin (DMLPG) method: A generalized MLS approximation, Appl. Numer. Math., 68 (2013), 73–82.
  • D. Mirzaei, R. Schaback and M. Dehghan: On generalized moving least squares and diffuse derivatives, SIAM J. Numer. Anal., 32 (2012), 983–1000.
  • F. J. Narcowich, J. D. Ward and H. Wendland: Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comput., 78 (2015), 743–763.
  • V. P. Nguyen, T. Rabczuk, S. Bordas and M. Duflot: Meshless methods: A review and computer implementation aspects, Math. Comput. Simul., 79 (3) (2008), 763–813.
  • C. Rieger, B. Zwicknagl: Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning, Adv. Comput. Math., 32 (1) (2010), 103–129.
  • D. Shepard: A two-dimensional interpolation function for irregularly-spaced data, Proceedings of the 1968 23rd ACM national conference, New York, (U.S.A) (1968), 27–29.
  • H.Wendland: Local polynomial reproduction and moving least squares approximation, SIAM J. Numer. Anal., 21 (2001), 285–300.
  • H. Wendland: Scattered Data Approximation, Cambridge University Press: Cambridge (2005).

Moving least squares approximation using variably scaled discontinuous weight function

Year 2023, , 38 - 54, 15.03.2023
https://doi.org/10.33205/cma.1247239

Abstract

Functions with discontinuities appear in many applications such as image reconstruction, signal processing, optimal control problems, interface problems, engineering applications and so on. Accurate approximation and interpolation of these functions are therefore of great importance. In this paper, we design a moving least-squares approach for scattered data approximation that incorporates the discontinuities in the weight functions. The idea is to control the influence of the data sites on the approximant, not only with regards to their distance from the evaluation point, but also with respect to the discontinuity of the underlying function. We also provide an error estimate on a suitable piecewise Sobolev Space. The numerical experiments are in compliance with the convergence rate derived theoretically.

References

  • R. A. Adams, J. Fournier: Sobolev Spaces, Elsevier, London, (2003).
  • M. G. Armentano: Error estimates in Sobolev spaces for moving least square approximations, SIAM J. Numer. Anal., 39 (2001), 38–51.
  • M. G. Armentano, R. G. Duran: Error estimates for moving least square approximations, Appl. Numer. Math., 37 (2001), 397–416.
  • G. E. Backus, J. F. Gilbert: Numerical applications of a formalism for geophysical inverse problems, Geophys. J. Int., 13 (1967), 247–276.
  • V. Bayona: Comparison of moving least squares and RBF+poly for interpolation and derivative approximation, J. Sci. Comput., 81 (2019), 486–512.
  • V. Bayona, N. Flyer, B. Fornberg and G. A. Barnett: On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs, J. Comput. Phys., 332 (2017), 257–273.
  • S. C. Bernard, L. R. Scott: The mathematical theory of finite element methods, Springer, New York, (2003).
  • L. Bos, K. Salkauskas: Moving least-squares are Backus-Gilbert optimal, J. Approx. Theory, 59 (1989), 267–275.
  • M. Bozzini, L. Lenarduzzi, M. Rossini and R. Schaback: Interpolation with variably scaled kernels, SIAM J. Numer. Anal., 35 (2015), 199–219.
  • S. Cuomo, A. Galletti, G. Giunta and A. Starace: Surface reconstruction from scattered point via RBF interpolation on GPU, Federated Conference on Computer Science and Information Systems, Krakow (Poland) (2013), 433–440.
  • S. De Marchi, W. Erb, F. Marchetti, E. Perracchione and M. Rossini: Shape-Driven interpolation with discontinuous kernels: Error analysis, edges extraction and application in magnetic particle imaging, J. Sci. Comput., 42 (2020), 472–491.
  • S. De Marchi, F. Marchetti and E. Perracchione: Jumping with variably scaled discontinuous kernels, BIT Numer. Math., 60 (2019), 441–463.
  • G. E. Fasshauer: Meshfree Approximation Methods, World Scientific Publishing: Singapore, (2007).
  • G. E. Fasshauer, M. J. McCourt: Kernel based approximation methods using MATLAB, World Scientific Publishing: Singapore, (2015).
  • S. Guastavino, F. Benvenuto: Convergence rates of spectral regularization methods: A comparison between ill-posed inverse problems and statistical kernel learning, SIAM J. Numer. Anal., 58 (6) (2020), 3504–3529.
  • P. Lancaster, K. Salkuaskas: Surfaces generated by moving least squares methods, Math. Comput., 37 (1981), 141–158.
  • D. Levin: The approximation power of moving least-squares. Math. Comp., 67 (1998), 1517–1531.
  • L. B. Lucy: A numerical approach to the testing of the fission hypothesis, AJ, 82 (1982), 1013–1024.
  • D. Mirzaei: Analysis of moving least square approximation revisited, J. Comput. Appl. Math., 282 (2015), 237-250.
  • D. Mirzaei, R. Schaback: Direct meshless local Petrov–Galerkin (DMLPG) method: A generalized MLS approximation, Appl. Numer. Math., 68 (2013), 73–82.
  • D. Mirzaei, R. Schaback and M. Dehghan: On generalized moving least squares and diffuse derivatives, SIAM J. Numer. Anal., 32 (2012), 983–1000.
  • F. J. Narcowich, J. D. Ward and H. Wendland: Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comput., 78 (2015), 743–763.
  • V. P. Nguyen, T. Rabczuk, S. Bordas and M. Duflot: Meshless methods: A review and computer implementation aspects, Math. Comput. Simul., 79 (3) (2008), 763–813.
  • C. Rieger, B. Zwicknagl: Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning, Adv. Comput. Math., 32 (1) (2010), 103–129.
  • D. Shepard: A two-dimensional interpolation function for irregularly-spaced data, Proceedings of the 1968 23rd ACM national conference, New York, (U.S.A) (1968), 27–29.
  • H.Wendland: Local polynomial reproduction and moving least squares approximation, SIAM J. Numer. Anal., 21 (2001), 285–300.
  • H. Wendland: Scattered Data Approximation, Cambridge University Press: Cambridge (2005).
There are 27 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Articles
Authors

Mohammad Karimnejad Esfahani 0000-0002-8532-8819

Stefano De Marchı 0000-0002-2832-8476

Francesco Marchetti 0000-0003-1087-7589

Publication Date March 15, 2023
Published in Issue Year 2023

Cite

APA Karimnejad Esfahani, M., De Marchı, S., & Marchetti, F. (2023). Moving least squares approximation using variably scaled discontinuous weight function. Constructive Mathematical Analysis, 6(1), 38-54. https://doi.org/10.33205/cma.1247239
AMA Karimnejad Esfahani M, De Marchı S, Marchetti F. Moving least squares approximation using variably scaled discontinuous weight function. CMA. March 2023;6(1):38-54. doi:10.33205/cma.1247239
Chicago Karimnejad Esfahani, Mohammad, Stefano De Marchı, and Francesco Marchetti. “Moving Least Squares Approximation Using Variably Scaled Discontinuous Weight Function”. Constructive Mathematical Analysis 6, no. 1 (March 2023): 38-54. https://doi.org/10.33205/cma.1247239.
EndNote Karimnejad Esfahani M, De Marchı S, Marchetti F (March 1, 2023) Moving least squares approximation using variably scaled discontinuous weight function. Constructive Mathematical Analysis 6 1 38–54.
IEEE M. Karimnejad Esfahani, S. De Marchı, and F. Marchetti, “Moving least squares approximation using variably scaled discontinuous weight function”, CMA, vol. 6, no. 1, pp. 38–54, 2023, doi: 10.33205/cma.1247239.
ISNAD Karimnejad Esfahani, Mohammad et al. “Moving Least Squares Approximation Using Variably Scaled Discontinuous Weight Function”. Constructive Mathematical Analysis 6/1 (March 2023), 38-54. https://doi.org/10.33205/cma.1247239.
JAMA Karimnejad Esfahani M, De Marchı S, Marchetti F. Moving least squares approximation using variably scaled discontinuous weight function. CMA. 2023;6:38–54.
MLA Karimnejad Esfahani, Mohammad et al. “Moving Least Squares Approximation Using Variably Scaled Discontinuous Weight Function”. Constructive Mathematical Analysis, vol. 6, no. 1, 2023, pp. 38-54, doi:10.33205/cma.1247239.
Vancouver Karimnejad Esfahani M, De Marchı S, Marchetti F. Moving least squares approximation using variably scaled discontinuous weight function. CMA. 2023;6(1):38-54.