Construction of approximation functionals for minimal splines

E. K. Kulikov   ; A. A. Makarov

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIV, Tome 504 (2021), pp. 136-156

Résumé

The paper presents formulas for constructing quadratic minimal splines, which explicitly depend on components of the generating vector function. Formulas for various approximation functionals for minimal splines used as coefficients in local approximation schemes are obtained. Examples of special cases of approximation constructions, which have a quasi-interpolation character, are provided. Results of numerical experiments on approximation of a circular arc by minimal splines are considered.

Bibliographie

[1] Yu. S. Volkov, Yu. N. Subbotin, “50 let zadache Shenberga o skhodimosti splain-interpolyatsii”, Tr. IMM UrO RAN, 20, no. 1, 2014, 52–67

[2] Yu. S. Volkov, V. T. Shevaldin, “Usloviya formosokhraneniya pri interpolyatsii splainami vtoroi stepeni po Subbotinu i po Marsdenu”, Tr. IMM UrO RAN, 18, no. 4, 2012, 145–152

[3] S. B. Stechkin, Yu. N. Subbotin, Splainy v vychislitelnoi matematike, M., 1976

[4] Yu. S. Zavyalov, B. I. Kvasov, V. L. Miroshnichenko, Metody splain-funktsii, M., 1980

[5] L. L. Schumaker, Spline functions: basic theory, John Wiley and Sons, Inc., New York, 1981 | Zbl

[6] Yu. K. Demyanovich, Lokalnaya approksimatsiya na mnogoobrazii i minimalnye splainy, Izd-vo S.-Peterb. un-ta, 1994

[7] C. de Boor, A Practical Guide to Splines, revised edition, Springer-Verlag, New York, 2001 | Zbl

[8] B. I. Kvasov, Metody izogeometricheskoi approksimatsii splainami, Fizmatlit, M., 2006

[9] I. G. Burova, Yu. K. Demyanovich, Minimalnye splainy i ikh prilozheniya, Izd-vo S.-Peterb. un-ta, 2010

[10] A. I. Grebennikov, Metod splainov i reshenie nekorrektnykh zadach v teorii priblizhenii, Izd-vo Mosk. un-ta, M., 1983

[11] T. Lyche, L. L. Shumaker, “Quasi-interpolants based on trigonometric splines”, J. Approx. Theor., 95 (1998), 280–309 | DOI | Zbl

[12] P. Sablonniere, “Quadratic spline quasi-interpolants on bounded domains of ${\mathbb R}^{d}$, $d = 1, 2, 3$”, Rend. Sem. Mat., 61:3 (2003), 229–246 | MR | Zbl

[13] T. Lyche, K. Mörken, Spline methods, Draft Centre of Mathematics for Applications, University of Oslo, 2005

[14] P. Constantini, C. Manni, F. Pelosi, M. Lucia Sampoli, “Quasi-interpolation in isogeometric analysis based on generalized $B$-splines”, Computer Aided Geom. Design, 27:8 (2010), 655–668

[15] A. A. Makarov, “Biortogonalnye sistemy funktsionalov i matritsy dekompozitsii dlya minimalnykh splainov”, Ukr. mat. visn., 9:2 (2012), 219–236 | Zbl

[16] Y. Jiang, Y. Xu, “B-spline Quasi-interpolation on Sparse Grids”, J. Complexity, 27:5 (2014), 466–488 | DOI

[17] W. Gao, Z. Wu, “A quasi-interpolation scheme for periodic data based on multiquadric trigonometric $B$-splines”, J. Comput. Appl. Math., 271 (2014), 20–30 | DOI | MR | Zbl

[18] M. Li, L. Chen, Q. Ma, “A meshfree quasi-interpolation method for solving burgers' equation”, Comput. Meth. Eng. Sci., 2014:3 (2014), 1–8

[19] R.- G. Yu, R.- H. Wang, C.- G. Zhu, “A numerical method for solving KdV equation with multilevel B-spline quasi-interpolation”, Appl. Anal., 92:8 (2013), 1682–1690 | DOI | MR | Zbl

[20] C. Dagnino, S. Remogna, P. Sablonniere, “On the solution of Fredholm integral equations based on spline quasi-interpolating projectors”, BIT, 54:4 (2014), 979–1008 | DOI | MR | Zbl

[21] M. Derakhshan, M. Zarebnia, “On the numerical treatment and analysis of two-dimensional Fredholm integral equations using quasi-interpolant”, Comput. Appl. Math., 39:106 (2020) | MR | Zbl

[22] P. Sablonniere, D. Shibih, T. Mohamed, “Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell–Sabin partitions”, BIT, 53:1 (2013), 145 ; 175–192 | DOI | Zbl

[23] L. Lu, “On polynomial approximation of circular arcs and helices”, Comput. Math. Appl., 63 (2012), 1192–1196 | DOI | MR | Zbl

[24] G. Jaklič, “Uniform approximation of a circle by a parametric polynomial curve”, Computer Aided Geom. Design, 41 (2016), 36–46 | DOI | MR | Zbl

[25] A. Rababah, “The best uniform cubic approximation of circular arcs with high accuracy”, Commun. Math. Appl., 7:1 (2016), 37–46 | MR

[26] C. Apprich, A. Dietrich, K. Höllig, E. Nava-Yazdani, “Cubic spline approximation of a circle with maximal smoothness and accuracy”, Computer Aided Geom. Design, 56 (2017), 1–3 | DOI | MR | Zbl

[27] A. A. Makarov, “An example of circular arc approximation by quadratic minimal splines”, Poincare J. Anal. Appl., 2018:2 (2018), 103–107 | Zbl

[28] Yu. K. Demyanovich, “Gladkost prostranstv splainov i vspleskovye razlozheniya”, Dokl. RAN, 401:4 (2005), 1–4

[29] Yu. K. Demyanovich, A. A. Makarov, “Neobkhodimye i dostatochnye usloviya neotritsatelnosti koordinatnykh trigonometricheskikh splainov vtorogo poryadka”, Vestn. S.-Peterb. un-ta. Ser. 1, 4(62):1 (2017), 9–16

[30] O. Kosogorov, A. Makarov, “On some piecewise quadratic spline functions”, Lect. Notes Comput. Sci., 10187, 2017, 448–455 | DOI | MR | Zbl

[31] A. A. Makarov, “O postroenii splainov maksimalnoi gladkosti”, Probl. matem. anal., 60 (2011), 25–38 | Zbl

[32] A. A. Makarov, “O dvoistvennykh funktsionalakh k minimalnym splainam”, Zap. nauchn. semin. POMI, 453, 2016, 198–218

[33] E. K. Kulikov, A. A. Makarov, “On de Boor–Fix type functionals for minimal splines”, Topics Class. Modern Anal., Applied and Numerical Harmonic Analysis, 2019, 211–225 | DOI

[34] E. K. Kulikov, A. A. Makarov, “Ob approksimatsii giperbolicheskimi splainami”, Zap. nauchn. semin. POMI, 472, 2018, 179–194

[35] E. K. Kulikov, A. A. Makarov, “O priblizhennom reshenii odnoi singulyarno vozmuschennoi kraevoi zadachi”, Diff. ur. prots. upr., 2020, no. 1, 91–102 | Zbl

[36] E. Kulikov, A. Makarov, “On biorthogonal approximation of solutions of some boundary value problems on Shishkin mesh”, AIP Conference Proceedings, 2302 (2020), 110005 | DOI

[37] E. K. Kulikov, A. A. Makarov, “O kvadratichnykh minimalnykh splainakh s kratnymi uzlami”, Zap. nauchn. semin. POMI, 482, 2019, 220–230