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Passive Approximation and Optimization Using B-Splines
Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för fysik och elektroteknik (IFE).
Lund University.
KTH Royal Instute of Technology.
Stockholm University.
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2019 (Engelska)Ingår i: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 79, nr 1, s. 436-458Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

A passive approximation problem is formulated where the target function is an arbitrary complex-valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The norm used is a weighted L-p-norm where 1 <= p <= infinity. The approximating functions are Herglotz functions generated by a measure with Holder continuous density in an arbitrary neighborhood of the approximation domain. Hence, the imaginary and the real parts of the approximating functions are Holder continuous functions given by the density of the measure and its Hilbert transform, respectively. In practice, it is useful to employ finite B-spline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finite-dimensional convex optimization problem which is amenable for numerical solution. A constructive proof is given here showing that the convex cone of approximating functions generated by finite uniform B-spline expansions of fixed arbitrary order (linear, quadratic, cubic, etc.) is dense in the convex cone of Herglotz functions which are locally Holder continuous in a neighborhood of the approximation domain, as mentioned above. As an illustration, typical physical application examples are included regarding the passive approximation and optimization of a linear system having metamaterial characteristics, as well as passive realization of optimal absorption of a dielectric small sphere over a finite bandwidth.

Ort, förlag, år, upplaga, sidor
2019. Vol. 79, nr 1, s. 436-458
Nyckelord [en]
approximation, Herglotz functions, B-splines, passive systems, convex optimization, sum rules
Nationell ämneskategori
Matematik
Forskningsämne
Matematik, Tillämpad matematik
Identifikatorer
URN: urn:nbn:se:lnu:diva-81228DOI: 10.1137/17M1161026ISI: 000460127100021Scopus ID: 2-s2.0-85063407473OAI: oai:DiVA.org:lnu-81228DiVA, id: diva2:1298160
Tillgänglig från: 2019-03-22 Skapad: 2019-03-22 Senast uppdaterad: 2019-11-21Bibliografiskt granskad
Ingår i avhandling
1. Optimization and Physical Bounds for Passive and Non-passive Systems
Öppna denna publikation i ny flik eller fönster >>Optimization and Physical Bounds for Passive and Non-passive Systems
2019 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

Physical bounds in electromagnetic field theory have been of interest for more than a decade. Considering electromagnetic structures from the system theory perspective, as systems satisfying linearity, time-invariance, causality and passivity, it is possible to characterize their transfer functions via Herglotz functions. Herglotz functions are useful in modeling of passive systems with applications in mathematical physics, engineering, and modeling of wave phenomena in materials and scattering. Physical bounds on passive systems can be derived in the form of sum rules, which are based on low- and high-frequency asymptotics of the corresponding Herglotz functions. These bounds provide an insight into factors limiting the performance of a given system, as well as the knowledge about possibilities to improve a desired system from a design point of view. However, the asymptotics of the Herglotz functions do not always exist for a given system, and thus a new method for determination of physical bounds is required. In Papers I–II of this thesis, a rigorous mathematical framework for a convex optimization approach based on general weighted Lp-norms, 1≤p≤∞, is introduced. The developed framework is used to approximate a desired system response, and to determine an optimal performance in realization of a system satisfying the target requirement. The approximation is carried out using Herglotz functions, B-splines, and convex optimization. 

Papers III–IV of this thesis concern modeling and determination of optimal performance bounds for causal, but not passive systems. To model them, a new class of functions, the quasi-Herglotz functions, is introduced. The new functions are defined as differences of two Herglotz functions and preserve the majority of the properties of Herglotz functions useful for the mathematical framework based on convex optimization. We consider modeling of gain media with desired properties as a causal system, which can be active over certain frequencies or  frequency intervals.  Here, sum rules can also be used under certain assumptions.

In Papers V–VII of this thesis, the optical theorem for scatterers immersed in lossy media is revisited. Two versions of the optical theorem are derived: one based on internal equivalent currents and the other based on external fields in terms of a T-matrix formalism, respectively. The theorems are exploited to derive fundamental bounds on absorption by using elementary optimization techniques. The theory has a potential impact in applications where the surrounding losses cannot be neglected, e.g., in medicine, plasmonic photothermal therapy, radio frequency absorption of gold nanoparticle suspensions, etc.  In addition to this, a new method for detection of electrophoretic resonances in a material with Drude-type of dispersion, which is placed in a straight waveguide, is proposed.

Ort, förlag, år, upplaga, sidor
Växjö, Sweden: Linnaeus University Press, 2019. s. 217
Serie
Linnaeus University Dissertations ; 373/2019
Nyckelord
Convex optimization, physical bounds, Herglotz functions, quasi-Herglotz functions, passive systems, non-passive systems, approximation, absorption in lossy media
Nationell ämneskategori
Annan elektroteknik och elektronik
Forskningsämne
Fysik, Vågor och signaler
Identifikatorer
urn:nbn:se:lnu:diva-90223 (URN)978-91-89081-23-9 (ISBN)978-91-89081-24-6 (ISBN)
Disputation
2019-12-13, Newton, Hus C, Växjö, 09:15 (Engelska)
Opponent
Handledare
Forskningsfinansiär
Stiftelsen för strategisk forskning (SSF), AM13-0011
Tillgänglig från: 2019-11-22 Skapad: 2019-11-21 Senast uppdaterad: 2019-11-22Bibliografiskt granskad

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