Application of Fourier Truncation Method to Numerical Differentiation for Bivariate Functions
-
Evgeniya V. Semenova
,
Sergiy G. Solodky
Abstract
The problem of numerical differentiation for periodic bivariate functions with finite smoothness is studied. To achieve stable approximations, we investigate some variants of the Fourier truncation method. Estimates of the accuracy and volume of the used Fourier coefficients are found for the constructed methods. We perform numerical experiments that confirm correctness of our theoretical conclusions.
References
[1] S. Ahn, U. J. Choi and A. G. Ramm, A scheme for stable numerical differentiation, J. Comput. Appl. Math. 186 (2006), no. 2, 325–334. 10.1016/j.cam.2005.02.002Search in Google Scholar
[2] K. I. Babenko, Approximation of periodic functions of many variables by trigonometric polynomials (in Russian), Dokl. Akad. Nauk SSSR 132 (1960), 247–250. Search in Google Scholar
[3] F. Cobos, T. Kühn and W. Sickel, Optimal approximation of multivariate periodic Sobolev functions in the sup-norm, J. Funct. Anal. 270 (2016), no. 11, 4196–4212. 10.1016/j.jfa.2016.03.018Search in Google Scholar
[4] D. Dũng, V. Temlyakov and T. Ullrich, Hyperbolic Cross Approximation, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2018. 10.1007/978-3-319-92240-9Search in Google Scholar
[5] T. F. Dolgopolova and V. K. Ivanov, On numerical differentiation, USSR Comput. Math. Math. Phys. 6 (1966), no. 3, 223–232. 10.1016/0041-5553(66)90145-5Search in Google Scholar
[6] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000. Search in Google Scholar
[7] C. W. Groetsch, Optimal order of accuracy in Vasin’s method for differentiation of noisy functions, J. Optim. Theory Appl. 74 (1992), no. 2, 373–378. 10.1007/BF00940901Search in Google Scholar
[8] M. Hanke and O. Scherzer, Inverse problems light: Numerical differentiation, Amer. Math. Monthly 108 (2001), no. 6, 512–521. 10.1080/00029890.2001.11919778Search in Google Scholar
[9] O. V. Lepskiĭ, A problem of adaptive estimation in Gaussian white noise, Theory Probab. Appl. 35 (1990), no. 3, 454–466. 10.1090/advsov/012/04Search in Google Scholar
[10] Z. Meng, Z. Zhao, D. Mei and Y. Zhou, Numerical differentiation for two-dimensional functions by a Fourier extension method, Inverse Probl. Sci. Eng. 28 (2020), no. 1, 126–143. 10.1080/17415977.2019.1661410Search in Google Scholar
[11] G. L. Mileĭko and S. G. Solodkiĭ, Hyperbolic cross and the complexity of various classes of linear ill-posed problems, Ukrainian Math. J. 69 (2017), no. 7, 1107–1122. 10.1007/s11253-017-1418-3Search in Google Scholar
[12] G. Nakamura, S. Wang and Y. Wang, Numerical differentiation for the second order derivatives of functions of two variables, J. Comput. Appl. Math. 212 (2008), no. 2, 341–358. 10.1016/j.cam.2006.11.035Search in Google Scholar
[13] S. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal. 43 (2005), no. 5, 2060–2076. 10.1137/S0036142903433819Search in Google Scholar
[14] S. V. Pereverzev, Optimization of projection methods for solving ill-posed problems, Computing 55 (1995), no. 2, 113–124. 10.1007/BF02238096Search in Google Scholar
[15] S. V. Pereverzev and S. G. Solodkiĭ, Optimal discretization of ill-posed problems, Ukrainian Math. J. 52 (2000), no. 1, 115–132. 10.1007/BF02514141Search in Google Scholar
[16] Z. Qian, C.-L. Fu, X.-T. Xiong and T. Wei, Fourier truncation method for high order numerical derivatives, Appl. Math. Comput. 181 (2006), no. 2, 940–948. 10.1016/j.amc.2006.01.057Search in Google Scholar
[17] A. G. Ramm, Numerical differentiation, Izv. Vyssh. Uchebn. Zaved. Mat. 11 (1968), 131–134. Search in Google Scholar
[18] K. Sharipov, On the recovery of continuous functions from noisy Fourier coefficients, J. Numer. Appl. Math. 109 (2012), 116–124. 10.2478/cmam-2011-0004Search in Google Scholar
[19] S. G. Solodky and K. K. Sharipov, Summation of smooth functions of two variables with perturbed Fourier coefficients, J. Inverse Ill-Posed Probl. 23 (2015), no. 3, 287–297. 10.1515/jiip-2013-0076Search in Google Scholar
[20] S. G. Solodky and S. A. Stasyuk, Estimates of efficiency for two methods of stable numerical summation of smooth functions, J. Complexity 56 (2020), Article ID 101422. 10.1016/j.jco.2019.101422Search in Google Scholar
[21] V. Temlyakov, Multivariate Approximation, Cambridge Monogr. Appl. Comput. Math. 32, Cambridge University, Cambridge, 2018. 10.1017/9781108689687Search in Google Scholar
[22] V. V. Vasin, Regularization of a numerical differentiation problem (in Russian), Ural. Gos. Univ. Mat. Zap. 7 (1969/1970), no. 2, 29–33. Search in Google Scholar
[23] Z. Zhao, Z. Meng, L. Zhao, L. You and O. Xie, A stabilized algorithm for multi-dimensional numerical differentiation, J. Algorithms Comput. Technol. 10 (2016), no. 2, 73–81. 10.1177/1748301816640450Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Abstract
The problem of numerical differentiation for periodic bivariate functions with finite smoothness is studied. To achieve stable approximations, we investigate some variants of the Fourier truncation method. Estimates of the accuracy and volume of the used Fourier coefficients are found for the constructed methods. We perform numerical experiments that confirm correctness of our theoretical conclusions.
References
[1] S. Ahn, U. J. Choi and A. G. Ramm, A scheme for stable numerical differentiation, J. Comput. Appl. Math. 186 (2006), no. 2, 325–334. 10.1016/j.cam.2005.02.002Search in Google Scholar
[2] K. I. Babenko, Approximation of periodic functions of many variables by trigonometric polynomials (in Russian), Dokl. Akad. Nauk SSSR 132 (1960), 247–250. Search in Google Scholar
[3] F. Cobos, T. Kühn and W. Sickel, Optimal approximation of multivariate periodic Sobolev functions in the sup-norm, J. Funct. Anal. 270 (2016), no. 11, 4196–4212. 10.1016/j.jfa.2016.03.018Search in Google Scholar
[4] D. Dũng, V. Temlyakov and T. Ullrich, Hyperbolic Cross Approximation, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2018. 10.1007/978-3-319-92240-9Search in Google Scholar
[5] T. F. Dolgopolova and V. K. Ivanov, On numerical differentiation, USSR Comput. Math. Math. Phys. 6 (1966), no. 3, 223–232. 10.1016/0041-5553(66)90145-5Search in Google Scholar
[6] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000. Search in Google Scholar
[7] C. W. Groetsch, Optimal order of accuracy in Vasin’s method for differentiation of noisy functions, J. Optim. Theory Appl. 74 (1992), no. 2, 373–378. 10.1007/BF00940901Search in Google Scholar
[8] M. Hanke and O. Scherzer, Inverse problems light: Numerical differentiation, Amer. Math. Monthly 108 (2001), no. 6, 512–521. 10.1080/00029890.2001.11919778Search in Google Scholar
[9] O. V. Lepskiĭ, A problem of adaptive estimation in Gaussian white noise, Theory Probab. Appl. 35 (1990), no. 3, 454–466. 10.1090/advsov/012/04Search in Google Scholar
[10] Z. Meng, Z. Zhao, D. Mei and Y. Zhou, Numerical differentiation for two-dimensional functions by a Fourier extension method, Inverse Probl. Sci. Eng. 28 (2020), no. 1, 126–143. 10.1080/17415977.2019.1661410Search in Google Scholar
[11] G. L. Mileĭko and S. G. Solodkiĭ, Hyperbolic cross and the complexity of various classes of linear ill-posed problems, Ukrainian Math. J. 69 (2017), no. 7, 1107–1122. 10.1007/s11253-017-1418-3Search in Google Scholar
[12] G. Nakamura, S. Wang and Y. Wang, Numerical differentiation for the second order derivatives of functions of two variables, J. Comput. Appl. Math. 212 (2008), no. 2, 341–358. 10.1016/j.cam.2006.11.035Search in Google Scholar
[13] S. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal. 43 (2005), no. 5, 2060–2076. 10.1137/S0036142903433819Search in Google Scholar
[14] S. V. Pereverzev, Optimization of projection methods for solving ill-posed problems, Computing 55 (1995), no. 2, 113–124. 10.1007/BF02238096Search in Google Scholar
[15] S. V. Pereverzev and S. G. Solodkiĭ, Optimal discretization of ill-posed problems, Ukrainian Math. J. 52 (2000), no. 1, 115–132. 10.1007/BF02514141Search in Google Scholar
[16] Z. Qian, C.-L. Fu, X.-T. Xiong and T. Wei, Fourier truncation method for high order numerical derivatives, Appl. Math. Comput. 181 (2006), no. 2, 940–948. 10.1016/j.amc.2006.01.057Search in Google Scholar
[17] A. G. Ramm, Numerical differentiation, Izv. Vyssh. Uchebn. Zaved. Mat. 11 (1968), 131–134. Search in Google Scholar
[18] K. Sharipov, On the recovery of continuous functions from noisy Fourier coefficients, J. Numer. Appl. Math. 109 (2012), 116–124. 10.2478/cmam-2011-0004Search in Google Scholar
[19] S. G. Solodky and K. K. Sharipov, Summation of smooth functions of two variables with perturbed Fourier coefficients, J. Inverse Ill-Posed Probl. 23 (2015), no. 3, 287–297. 10.1515/jiip-2013-0076Search in Google Scholar
[20] S. G. Solodky and S. A. Stasyuk, Estimates of efficiency for two methods of stable numerical summation of smooth functions, J. Complexity 56 (2020), Article ID 101422. 10.1016/j.jco.2019.101422Search in Google Scholar
[21] V. Temlyakov, Multivariate Approximation, Cambridge Monogr. Appl. Comput. Math. 32, Cambridge University, Cambridge, 2018. 10.1017/9781108689687Search in Google Scholar
[22] V. V. Vasin, Regularization of a numerical differentiation problem (in Russian), Ural. Gos. Univ. Mat. Zap. 7 (1969/1970), no. 2, 29–33. Search in Google Scholar
[23] Z. Zhao, Z. Meng, L. Zhao, L. You and O. Xie, A stabilized algorithm for multi-dimensional numerical differentiation, J. Algorithms Comput. Technol. 10 (2016), no. 2, 73–81. 10.1177/1748301816640450Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- A General Error Estimate For Parabolic Variational Inequalities
- Functional A Posteriori Error Estimates for the Parabolic Obstacle Problem
- Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations
- Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data
- Quasi-Newton Iterative Solution of Non-Orthotropic Elliptic Problems in 3D with Boundary Nonlinearity
- Improving Regularization Techniques for Incompressible Fluid Flows via Defect Correction
- Fully Discrete Finite Element Approximation of the MHD Flow
- Sparse Data-Driven Quadrature Rules via ℓ p -Quasi-Norm Minimization
- Identification of Matrix Diffusion Coefficient in a Parabolic PDE
- A Finite Element Method for Two-Phase Flow with Material Viscous Interface
- On the Finite Element Approximation of Fourth-Order Singularly Perturbed Eigenvalue Problems
- Application of Fourier Truncation Method to Numerical Differentiation for Bivariate Functions
- Factorized Schemes for First and Second Order Evolution Equations with Fractional Powers of Operators
Articles in the same Issue
- Frontmatter
- A General Error Estimate For Parabolic Variational Inequalities
- Functional A Posteriori Error Estimates for the Parabolic Obstacle Problem
- Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations
- Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data
- Quasi-Newton Iterative Solution of Non-Orthotropic Elliptic Problems in 3D with Boundary Nonlinearity
- Improving Regularization Techniques for Incompressible Fluid Flows via Defect Correction
- Fully Discrete Finite Element Approximation of the MHD Flow
- Sparse Data-Driven Quadrature Rules via ℓ p -Quasi-Norm Minimization
- Identification of Matrix Diffusion Coefficient in a Parabolic PDE
- A Finite Element Method for Two-Phase Flow with Material Viscous Interface
- On the Finite Element Approximation of Fourth-Order Singularly Perturbed Eigenvalue Problems
- Application of Fourier Truncation Method to Numerical Differentiation for Bivariate Functions
- Factorized Schemes for First and Second Order Evolution Equations with Fractional Powers of Operators
