×

Passive approximation and optimization using B-splines. (English) Zbl 1416.41008

A combined method from approximation theory and B-splines, together with modern efficient methods of convex optimisation, is used to solve so-called passive approximation problems. Largest lower bounds for the approximation error are given. Herglotz functions of a certain Hölder continuity can be used to approach this problem. Their continuity is required in suitably small neighbourhoods of the approximation domain measured in weighted \(L^p\)-norms. The Herglotz functions themselves are approximated by uniform expansions in B-splines (which are compactly supported piecewise polynomial functions with minimal support for a required smoothness without being trivial). For this, it is an important observation that uniform B-splines provide a dense set within the Hölder space of Herglotz functions.

MSC:

41A15 Spline approximation
26A16 Lipschitz (Hölder) classes
65K10 Numerical optimization and variational techniques
78M50 Optimization problems in optics and electromagnetic theory

Software:

CVX
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] N. I. Akhiezer, {\it The Classical Moment Problem}, Oliver and Boyd, Edinburgh, 1965. · Zbl 0135.33803
[2] N. Aronszajn and W. Donoghue, {\it A supplement to the paper on exponential representations of analytic functions in the upper half-plane with positive imaginary part}, J. Anal. Math., 12 (1964), pp. 113-127. · Zbl 0138.29601
[3] N. Aronszajn and W. F. Donoghue, {\it On exponential representations of analytic functions in the upper half-plane with positive imaginary part}, J. Anal. Math., 5 (1956), p. 321. · Zbl 0138.29601
[4] L. Baratchart, J. Leblond, and J. R. Partington, {\it Hardy approximation to \(L^∞\) functions on subsets of the circle}, Constr. Approx., 12 (1996), pp. 423-435. · Zbl 0853.30022
[5] E. J. Beltrami and M. R. Wohlers, {\it Distributions and the Boundary Values of Analytic Functions}, Academic Press, New York, 1966. · Zbl 0186.19202
[6] A. Bernland, A. Luger, and M. Gustafsson, {\it Sum rules and constraints on passive systems}, J. Phys. A, 44 (2011), 145205. · Zbl 1222.30031
[7] C. F. Bohren and D. R. Huffman, {\it Absorption and Scattering of Light by Small Particles}, John Wiley & Sons, New York, 1983.
[8] S. Boyd and L. Vandenberghe, {\it Convex Optimization}, Cambridge University Press, Cambridge, UK, 2004. · Zbl 1058.90049
[9] M. Cassier and G. W. Milton, {\it Bounds on Herglotz functions and fundamental limits of broadband passive quasi-static cloaking}, J. Math. Phys., 58 (2017), 071504. · Zbl 1375.30046
[10] C. Dagnino and E. Santi, {\it On the convergence of spline product quadratures for Cauchy principal value integrals}, J. Comput. Appl. Math., 36 (1991), pp. 181-187. · Zbl 0738.65007
[11] G. Dahlquist and \AA. Björck, {\it Numerical Methods}, Prentice-Hall, Englewood Cliffs, NJ, 1974. · Zbl 1153.65001
[12] C. de Boor, {\it On uniform approximation by splines}, J. Approx. Theory, 1 (1968), pp. 219-235. · Zbl 0193.02502
[13] C. de Boor, {\it On calculating with B-splines}, J. Approx. Theory, 6 (1972), pp. 50-62. · Zbl 0239.41006
[14] C. De Boor, {\it A Practical Guide to Splines}, Appl. Math. Sci. 27, Springer-Verlag, New York, 2001. · Zbl 0987.65015
[15] J. Domsta, {\it Approximation by spline interpolating bases}, Stud. Math., 58 (1976), pp. 223-237. · Zbl 0345.41005
[16] R. M. Fano, {\it Theoretical limitations on the broadband matching of arbitrary impedances}, J. Franklin Inst., 249 (1950), pp. 57-83 and 139-154.
[17] F. Gesztesy and E. Tsekanovskii, {\it On matrix-valued Herglotz functions}, Math. Nachr., 218 (2000), pp. 61-138. · Zbl 0961.30027
[18] M. Grant and S. Boyd, {\it CVX: A System for Disciplined Convex Programming, Release 2.0,} CVX Research, Austin, TX, 2012.
[19] M. Gustafsson, {\it Sum rules for lossless antennas}, IET Microwaves Antennas Propagation, 4 (2010), pp. 501-511.
[20] M. Gustafsson, M. Cismasu, and S. Nordebo, {\it Absorption efficiency and physical bounds on antennas}, Int. J. Antennas Propagation, 2010 (2010), pp. 1-7.
[21] M. Gustafsson and D. Sjöberg, {\it Sum rules and physical bounds on passive metamaterials}, New J. Phys., 12 (2010), 043046. · Zbl 1375.83021
[22] M. Gustafsson and D. Sjöberg, {\it Physical bounds and sum rules for high-impedance surfaces}, IEEE Trans. Antennas and Propagatation, 59 (2011), pp. 2196-2204.
[23] M. Gustafsson, I. Vakili, S. E. B. Keskin, D. Sjöberg, and C. Larsson, {\it Optical theorem and forward scattering sum rule for periodic structures}, IEEE Trans. Antennas and Propagation, 60 (2012), pp. 3818-3826. · Zbl 1369.78196
[24] M. W. Haakestad and J. Skaar, {\it Causality and kramers-kronig relations for waveguides}, Optics Express, 13 (2005), pp. 9922-9934.
[25] L. Hörmander, {\it The Analysis of Linear Partial Differential Operators} I, Grundlehren Math. Wiss. 256, Springer-Verlag, Berlin, 1983. · Zbl 0521.35001
[26] M. Hutton and B. Friedland, {\it Routh approximations for reducing order of linear, time-invariant systems}, IEEE Trans. Automat. Control, 20 (1975), pp. 329-337. · Zbl 0301.93026
[27] Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, {\it Passive Approximation and Optimization with B-Splines}, Tech. report URN: urn:nbn:se:lnu:diva-63878, Department of Physics and Electrical Engineering, Linn\ae us University, Växjö, Sweden, 2017. · Zbl 1416.41008
[28] Y. Ivanenko and S. Nordebo, {\it Approximation of dielectric spectroscopy data with Herglotz functions on the real line and convex optimization}, in Proceedings of the International Conference on Electromagnetics in Advanced Applications, 2016, pp. 863-866.
[29] B. L. G. Jonsson, C. I. Kolitsidas, and N. Hussain, {\it Array antenna limitations}, Antennas Wireless Propagation Lett., 12 (2013), pp. 1539-1542.
[30] I. S. Kac and M. G. Krein, {\it R-functions - Analytic functions mapping the upper halfplane into itself}, Amer. Math. Soc. Transl., 103 (1974), pp. 1-18. · Zbl 0291.34016
[31] F. W. King, {\it Hilbert Transforms,} Cambridge University Press, Cambridge, UK, 2009.
[32] M. G. Krein and P. Y. Nudel’man, {\it Approximation of \(L_2(ω_1,ω_2)\) functions by minimum-energy transfer functions of linear systems}, Problemy Peredachi Informatsii, 11 (1975), pp. 37-60. · Zbl 0328.41018
[33] R. Kress, {\it Linear Integral Equations}, 2nd ed., Springer-Verlag, Berlin, 1999. · Zbl 0920.45001
[34] G. Kristensson, {\it Scattering of Electromagnetic Waves by Obstacles}, SciTech Publishing, Edison, NJ, 2016. · Zbl 1365.78001
[35] G. W. Milton, D. J. Eyre, and J. V. Mantese, {\it Finite frequency range Kramers-Kronig relations: Bounds on the dispersion}, Phys. Rev. Lett., 79 (1997), pp. 3062-3065.
[36] G. Monegato, {\it The numerical evaluation of one-dimensional Cauchy principal value integrals}, Computing, 29 (1982), pp. 337-354. · Zbl 0485.65017
[37] M. Nedic, C. Ehrenborg, Y. Ivanenko, A. Ludvig-Osipov, S. Nordebo, A. Luger, B. L. G. Jonsson, D. Sjöberg, and M. Gustafsson, {\it Herglotz functions and applications in electromagnetics}, in Advances in Mathematical Methods for Electromagnetics, K. Kobayashi and P. Smith, eds., IET, London, 2019.
[38] S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, {\it On the physical limitations for radio frequency absorption in gold nanoparticle suspensions}, J. Phys. D Appl. Phys., 50 (2017), pp. 1-12.
[39] S. Nordebo, M. Gustafsson, B. Nilsson, and D. Sjöberg, {\it Optimal realizations of passive structures}, IEEE Trans. Antennas and Propagation, 62 (2014), pp. 4686-4694. · Zbl 1371.78337
[40] H. M. Nussenzveig, {\it Causality and Dispersion Relations}, Academic Press, London, 1972.
[41] K. N. Rozanov, {\it Ultimate thickness to bandwidth ratio of radar absorbers}, IEEE Trans. Antennas and Propagation, 48 (2000), pp. 1230-1234.
[42] W. Rudin, {\it Real and Complex Analysis}, McGraw-Hill, New York, 1987. · Zbl 0925.00005
[43] B. Simon, {\it Harmonic Analysis. A Comprehensive Course in Analysis, Part} 3, AMS, Providence, RI, 2015. · Zbl 1334.00002
[44] C. Sohl, M. Gustafsson, and G. Kristensson, {\it Physical limitations on broadband scattering by heterogeneous obstacles}, J. Phys. A, 40 (2007), pp. 11165-11182. · Zbl 1121.78006
[45] S. Steshenko and F. Capolino, {\it Single dipole approximation for modeling collections of nanoscatterers}, in Metamaterials Handbook: Theory and Phenomena of Metamaterials, F. Capolino, ed., CRC Press, Boca Raton, FL, 2009.
[46] G. Szegö, {\it Orthogonal Polynomials,} Amer. Math. Soc. Colloq. Publ. 23, AMS, Providence, RI, 1975.
[47] S. Tretyakov, {\it Maximizing absorption and scattering by dipole particles}, Plasmonics, 9 (2014), pp. 935-944.
[48] D. C. Tzarouchis, P. Ylä-Oijala, and A. Sihvola, {\it Unveiling the scattering behavior of small spheres}, Phys. Rev. B, 94 (2016), 140301.
[49] I. Vakili, M. Gustafsson, D. Sjöberg, R. Seviour, M. Nilsson, and S. Nordebo, {\it Sum rules for parallel-plate waveguides: Experimental results and theory}, IEEE Trans. Microwave Theory Techniques, 62 (2014), pp. 2574-2582.
[50] B. Wahlberg and P. Mäkilä, {\it On approximation of stable linear dynamical systems using Laguerre and Kautz functions}, Automatica, 32 (1996), pp. 693-708. · Zbl 0856.93017
[51] A. Welters, Y. Avniel, and S. G. Johnson, {\it Speed-of-light limitations in passive linear media}, Phys. Rev. A, 90 (2014), 023847.
[52] M. Wohlers and E. Beltrami, {\it Distribution theory as the basis of generalized passive-network analysis}, IEEE Trans. Circuit Theory, 12 (1965), pp. 164-170.
[53] A. H. Zemanian, {\it Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications}, McGraw-Hill, New York, 1965. · Zbl 0127.07201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.