Abstract
Our objective is to calculate the derivatives of data corrupted by noise. This is a challenging task as even small amounts of noise can result in significant errors in the computation. This is mainly due to the randomness of the noise, which can result in high-frequency fluctuations. To overcome this challenge, we suggest an approach that involves approximating the data by eliminating high-frequency terms from the Fourier expansion of the given data with respect to the polynomial-exponential basis. This truncation method helps to regularize the issue, while the use of the polynomial-exponential basis ensures accuracy in the computation. We demonstrate the effectiveness of our approach through numerical examples in one and two dimensions.
REFERENCES
K. Ahnert and L. A. Segel, “Numerical differentiation of experimental data: local versus global methods,” Comput. Phys. Commun. 177, 764–774 (2007).
F. V. Breugel, J. N. Kutz, and B. W. Brunton, “Numerical differentiation of noisy data: A unifying multi-objective optimization framework,” IEEE Access 8, 196865–196877 (2020).
H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Mathematics and Its Applications (Kluwer, Dordrecht, 1996).
K. O. Friedrichs, “The identity of weak and strong extensions of differential operators,” Trans. Am. Math. Soc. 55, 132–151 (1944).
C. W. Groetsch, “Differentiation of approximately specified functions,” Am. Math. Mon. 98, 847–850 (1991).
M. Hanke and O. Sherzer, “Inverse problems light numerical differentiation,” Am. Math. Mon. 108, 512–521 (2001).
V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, L. Nguyen, A. Sullivan, and V. N. Astratov, “Convexification and experimental data for a 3D inverse scattering problem with the moving point source,” Inverse Probl. 36, 085007 (2020).
V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, L. Nguyen, A. Sullivan, and V. N. Astratov, “An inverse problem of a simultaneous reconstruction of the dielectric constant and conductivity from experimental backscattering data,” Inverse Probl. Sci. Eng. 29 (5), 712–735 (2021).
V. A. Khoa, M. V. Klibanov, and L. H. Nguyen, “Convexification for a 3D inverse scattering problem with the moving point source,” SIAM J. Imaging Sci. 13 (2), 871–904 (2020).
M. V. Klibanov, “Convexification of restricted Dirichlet to Neumann map,” J. Inverse Ill-Posed Probl. 25 (5), 669–685 (2017).
M. V. Klibanov, T. T. Le, and L. H. Nguyen, “Convergent numerical method for a linearized travel time tomography problem with incomplete data,” SIAM J. Sci. Comput. 42, B1173–B1192 (2020).
M. V. Klibanov and J. Li, Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data (De Gruyter, Berlin, 2021).
M. V. Klibanov and L. H. Nguyen, “PDE-based numerical method for a limited angle X-ray tomography,” Inverse Probl. 35, 045009 (2019).
M. V. Klibanov and A. Timonov, “A comparative study of two globally convergent numerical methods for acoustic tomography,” Commun. Anal. Comput. 1, 12–31 (2023).
I. Knowles and R. J. Renka, “Methods for numerical differentiation of noisy data,” Electron. J. Differ. Equat. Conf. 21, 235–246 (2014).
I. Knowles and R. Wallace, “A variational method for numerical differentiation,” Numer. Math. 70, 91–110 (1995).
T. T. Le and L. H. Nguyen, “The gradient descent method for the convexification to solve boundary value problems of quasi-linear PDEs and a coefficient inverse problem,” J. Sci. Comput. 91 (3), 74 (2022).
V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer Verlag, New York, 1984).
R. Ramlau, “Morozov’s discrepancy principle for Tikhonov regularization of nonlinear operators,” J. Numer. Funct. Anal. Opt. 23, 147–172 (2002).
A. G. Ramm and A. B. Smirnova, “On stable numerical differentiation,” Math. Comput. 70, 1131–1153 (2001).
C. H. Reinsch, “Smoothing by spline functions,” Numer. Math. 10, 177–183 (1967).
C. H. Reinsch, “Smoothing by spline functions. II,” Numer. Math. 16, 451–454 (1971).
O. Scherzer, “The use of Morozov’s discrepancy principle for Tikhonov regularization for solving nonlinear ill-posed problems,” SIAM J. Numer. Anal. 30, 1796–1838 (1993).
I. J. Schoenberg, “Spline functions and the problem of graduation,” Proc. Am. Math. Soc. 52, 497–950 (1964).
Funding
The works of PMN, TTL, and LHN were partially supported by National Science Foundation grant no. DMS-2208159, and by funds provided by the Faculty Research Grant program at UNC Charlotte Fund no. 111272, and by the CLAS small grant provided by the College of Liberal Arts & Sciences, UNC Charlotte.
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Nguyen, P.M., Le, T.T., Nguyen, L.H. et al. Numerical Differentiation by the Polynomial-Exponential Basis. J. Appl. Ind. Math. 17, 928–942 (2023). https://doi.org/10.1134/S1990478923040191
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DOI: https://doi.org/10.1134/S1990478923040191