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Engström, Christian

Open this publication in new window or tab >>Accumulation of complex eigenvalues of a class of analytic operator functions### Engström, Christian

### Torshage, Axel

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##### Abstract [en]

##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-146118 (URN)10.1016/j.jfa.2018.03.019 (DOI)000434877800008 ()2-s2.0-85045081595 (Scopus ID)
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##### Funder

Swedish Research Council, 621-2012-3863
Available from: 2018-04-01 Created: 2018-04-01 Last updated: 2018-09-27Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of electromagnetic field theory is given.

Open this publication in new window or tab >>Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map### Araujo-Cabarcas, Juan Carlos

### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Jarlebring, Elias

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, p. 177-192Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Amsterdam: Elsevier, 2018
##### Keywords

Nonlinear eigenvalue problems, Helmholtz problem, Scattering resonances, Dirichlet-to-Neumann map, Arnoldi's method, Matrix functions
##### National Category

Computational Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-138325 (URN)10.1016/j.cam.2017.08.012 (DOI)000415783000014 ()
#####

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Available from: 2017-08-21 Created: 2017-08-21 Last updated: 2019-05-20Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Royal Institute of Technology.

We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method (TIAR). Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

Open this publication in new window or tab >>Spectral properties of conservative, dispersive, and absorptive photonic crystals### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Torshage, Axel

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: GAMM - Mitteilungen, Vol. 41, p. 1-16, article id e201800009Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Weinheim: Wiley-VCH Verlagsgesellschaft, 2018
##### Keywords

accumulation of eigenvalues, non-linear spectral problem, variational principle
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-152784 (URN)10.1002/gamm.201800009 (DOI)
#####

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Available from: 2018-10-24 Created: 2018-10-24 Last updated: 2018-11-09Bibliographically approved

Mathematisches Institut, Universität Bern, Bern, Switzerland.

This article reviews both recent progress on the mathematics of dispersive and absorptive photonic crystals and well-established results on conservative photonic crystals. The focus is on properties of the photonic band structures and we also provide results that are of importance for the understanding of lossy metal-dielectric photonic crystals.

Open this publication in new window or tab >>Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Torshage, Axel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 88, no 2, p. 151-184Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Non-linear spectral problem, Numerical range, Pseudospectra, Resolvent estimate
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-137726 (URN)10.1007/s00020-017-2378-6 (DOI)000405016300001 ()
#####

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##### Funder

Swedish Research Council, 621-2012-3863
Available from: 2017-07-07 Created: 2017-07-07 Last updated: 2018-06-09Bibliographically approved

In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the finite dimensional case. Moreover, the new enclosure is minimal given only the numerical ranges of the operator coefficients and many characteristics of the numerical range can be obtained by investigating the enclosure. We introduce a pseudonumerical range and study an enclosure of this set. This enclosure provides a computable upper bound of the norm of the resolvent.

Open this publication in new window or tab >>On equivalence and linearization of operator matrix functions with unbounded entries### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Torshage, Axel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 89, no 4, p. 465-492Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Equivalence after extension, Block operator matrices, Operator functions, Spectrum
##### National Category

Mathematical Analysis
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-142058 (URN)10.1007/s00020-017-2415-5 (DOI)000416537600001 ()
#####

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Available from: 2017-11-17 Created: 2017-11-17 Last updated: 2018-06-09Bibliographically approved

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.

Open this publication in new window or tab >>On spurious solutions in finite element approximations of resonances in open systems### Araujo-Cabarcas, Juan Carlos

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 74, no 10, p. 2385-2402Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2017
##### Keywords

Scattering resonances, Lippmann–Schwinger equation, Nonlinear eigenvalue problems, Acoustic resonator, Dielectric resonator, Bragg resonator
##### National Category

Computational Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-138096 (URN)10.1016/j.camwa.2017.07.020 (DOI)000415908400013 ()
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##### Funder

Swedish Research Council, 621-2012-3863
Available from: 2017-08-09 Created: 2017-08-09 Last updated: 2019-05-20Bibliographically approved

In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann–Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases.

Open this publication in new window or tab >>Rational eigenvalue problems and applications to photonic crystals### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Langer, Heinz

### Tretter, Christiane

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 445, no 1, p. 240-279Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

San Diego: Elsevier, 2017
##### Keywords

Non-linear spectral problem, Eigenvalue, Variational principle, Spectral gap, Photonic crystal, Finite ement method
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-137724 (URN)10.1016/j.jmaa.2016.07.048 (DOI)000384385300012 ()
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##### Funder

Swedish Research Council, 621-2012-3863
Available from: 2017-07-07 Created: 2017-07-07 Last updated: 2018-06-09Bibliographically approved

Vienna University of Technology.

Universität Bern.

We establish new analytic results for a general class of rational spectral problems. They arise e.g. in modelling photonic crystals whose capability to control the flow of light depends on specific features of the eigenvalues. Our results comprise a complete spectral analysis including variational principles and two-sided bounds for all eigenvalues, as well as numerical implementations. They apply to the eigenvalues between the poles where classical variational principles fail completely. In the application to multi-pole Lorentz models of permittivity functions we show, in particular, that our abstract two-sided eigenvalue estimates are optimal and we derive explicit bounds on the band gap above a Lorentz pole. A high order finite element method (FEM) is used to compute the two-sided bounds for a selection of eigenvalues for several concrete Lorentz models, e.g. polaritonic materials and multi-pole models.

Open this publication in new window or tab >>Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Giani, Stefano

### Grubisic, Luka

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2016 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 72, no 4, p. 952-973Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Nonlinear eigenvalue problems, Numerical methods, Invariant pairs
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-126328 (URN)10.1016/j.camwa.2016.06.001 (DOI)000381532400010 ()
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Available from: 2016-10-27 Created: 2016-10-03 Last updated: 2018-06-09Bibliographically approved

We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.

Open this publication in new window or tab >>A Subspace Iteration Algorithm for Fredholm Valued Functions### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Grubisic, Luka

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: Mathematical problems in engineering (Print), ISSN 1024-123X, E-ISSN 1563-5147, Vol. 2015, article id 459895Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Hindawi Publishing Corporation, 2015
##### National Category

Computational Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-111101 (URN)10.1155/2015/459895 (DOI)000364859000001 ()
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Available from: 2015-11-05 Created: 2015-11-05 Last updated: 2018-06-07Bibliographically approved

University of Zagreb.

We present an algorithm for approximating an eigensubspace of a spectral component of an analytic Fredholm valued function. Our approach is based on numerical contour integration and the analytic Fredholm theorem. The presented method can be seen as a variant of the FEAST algorithm for infinite dimensional nonlinear eigenvalue problems. Numerical experiments illustrate the performance of the algorithm for polynomial and rational eigenvalue problems.

Open this publication in new window or tab >>Topology and shape optimization of plasmonic nano-antennas### Wadbro, Eddie

### Engström, Christian

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 293, p. 155-169Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2015
##### Keywords

Nano-antennas, Plasmonics, Topology optimization, Shape optimization
##### National Category

Mechanical Engineering Mathematics
##### Identifiers

urn:nbn:se:umu:diva-109945 (URN)10.1016/j.cma.2015.04.011 (DOI)000361475900008 ()
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Available from: 2015-10-14 Created: 2015-10-09 Last updated: 2018-06-07Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Metallic nano-antennas are devices used to concentrate the energy in light into regions that are much smaller than the wavelength. These structures are currently used to develop new measurement and printing techniques, such as optical microscopy with sub-wavelength resolution, and high-resolution lithography. Here, we analyze and design a nano-antenna in a two-dimensional setting with the source being a planar TE-polarized wave. The design problem is to place silver and air in a pre-specified design region to maximize the electric energy in a small given target region. At optical frequencies silver exhibits extreme dielectric properties, having permittivity with a negative real part. We prove existence and uniqueness of solutions to the governing nonstandard Helmholtz equation with absorbing boundary conditions. To solve the design optimization problem, we develop a two-stage procedure. The first stage uses a material distribution parameterization and aims at finding a conceptual design without imposing any a priori information about the number of shapes of components comprising the nano-antenna. The second design stage uses a domain variation approach and aims at finding a precise shape. Both of the above design problems are formulated as non-linear mathematical programming problems that are solved using the method of moving asymptotes. The final designs perform very well and the electric energy in the target region is several orders of magnitude larger than when there is only air in the design region. The performance of the optimized designs is verified with a high order interior penalty method.