Solving Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method

Kurz, Stefan; Schöps, Sebastian; Unger, Gerhard; Wolf, Felix

doi:10.1002/mma.7447

Cited by 5 publications

(4 citation statements)

References 33 publications

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“…These linear systems can be independently solved so that the eigensolvers have good scalability, as demonstrated in [21,19]. For this reason, complex moment-based eigensolvers have attracted considerable attention, particularly in physics [26,21], materials science [19,9,22], power systems [34], data science [16] and so on. Currently, there are several methods, including direct extensions of Sakurai and Sugiura's approach [28,11,10,12,14,18,15], the FEAST eigensolver [26] developed by Polizzi, and its improvements [30,6,21].…”

Section: Complex Moment-based Matrix Eigensolversmentioning

confidence: 99%

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Complex moment-based methods for differential eigenvalue problems

Imakura¹,

Morikuni²,

Takayasu³

2022

Preprint

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This paper considers computing partial eigenpairs of differential eigenvalue problems (DEPs) such that eigenvalues are in a certain region on the complex plane. Recently, based on a "solve-then-discretize" paradigm, an operator analogue of the FEAST method has been proposed for DEPs without discretization of the coefficient operators. Compared to conventional "discretize-then-solve" approaches that discretize the operators and solve the resulting matrix problem, the operator analogue of FEAST exhibits much higher accuracy; however, it involves solving a large number of ordinary differential equations (ODEs). In this paper, to reduce the computational costs, we propose operation analogues of Sakurai-Sugiura-type complex moment-based eigensolvers for DEPs using higher-order complex moments and analyze the error bound of the proposed methods. We show that the number of ODEs to be solved can be reduced by a factor of the degree of complex moments without degrading accuracy, which is verified by numerical results. Numerical results demonstrate that the proposed methods are over five times faster compared with the operator analogue of FEAST for several DEPs while maintaining almost the same high accuracy. This study is expected to promote the "solve-then-discretize" paradigm for solving DEPs and contribute to faster and more accurate solutions in real-world applications.

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Section: Complex Moment-based Matrix Eigensolversmentioning

confidence: 99%

“…with boundary conditions, where A and B are linear, ordinary differential operators and Ω is a prescribed simply connected open set. This type of problems appears in various fields such as physics [26,21] and materials science [19,9,22]. Here, λ i and u i is an eigenvalue and the corresponding eigenfunction, respectively.…”

Section: Introductionmentioning

confidence: 99%

Complex moment-based methods for differential eigenvalue problems

Imakura¹,

Morikuni²,

Takayasu³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…These linear systems can be independently solved so that the eigensolvers have good scalability, as demonstrated in [2,3]. For this reason, complex moment-based eigensolvers have attracted considerable attention, particularly in physics [1,2], materials science [3][4][5], power systems [17], data science [18], and so on. Currently, there are several methods, including direct extensions of Sakurai and Sugiura's approach [19][20][21][22][23][24][25], the FEAST eigensolver [1] developed by Polizzi, and its improvements [2,26,27].…”

Section: Complex Moment-based Matrix Eigensolversmentioning

confidence: 99%

“…with boundary conditions, where A and B are linear, ordinary differential operators acting on functions from a Hilbert space H and is a prescribed simply connected open set. This type of problems appears in various fields such as physics [1,2] and materials science [3][4][5]. Here, λ i and u i is an eigenvalue and the corresponding eigenfunction, respectively.…”

Section: Introductionmentioning

confidence: 99%

Complex moment-based methods for differential eigenvalue problems

2022

View full text Add to dashboard Cite

This paper considers computing partial eigenpairs of differential eigenvalue problems (DEPs) such that eigenvalues are in a certain region on the complex plane. Recently, based on a “solve-then-discretize” paradigm, an operator analogue of the FEAST method has been proposed for DEPs without discretization of the coefficient operators. Compared to conventional “discretize-then-solve” approaches that discretize the operators and solve the resulting matrix problem, the operator analogue of FEAST exhibits much higher accuracy; however, it involves solving a large number of ordinary differential equations (ODEs). In this paper, to reduce the computational costs, we propose operation analogues of Sakurai–Sugiura-type complex moment-based eigensolvers for DEPs using higher-order complex moments and analyze the error bound of the proposed methods. We show that the number of ODEs to be solved can be reduced by a factor of the degree of complex moments without degrading accuracy, which is verified by numerical results. Numerical results demonstrate that the proposed methods are over five times faster compared with the operator analogue of FEAST for several DEPs while maintaining almost the same high accuracy. This study is expected to promote the “solve-then-discretize” paradigm for solving DEPs and contribute to faster and more accurate solutions in real-world applications.

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Solving acoustic scattering problems by the isogeometric boundary element method

Dölz,

Harbrecht,

Multerer

2024

Engineering with Computers

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We solve acoustic scattering problems by means of the isogeometric boundary integral equation method. In order to avoid spurious modes, we apply the combined field integral equations for either sound-hard scatterers or sound-soft scatterers. These integral equations are discretized by Galerkin’s method, which especially enables the mathematically correct regularization of the hypersingular integral operator. In order to circumvent densely populated system matrices, we employ the isogeometric embedded fast multipole method, which is based on interpolation of the kernel function under consideration on the reference domain, rather than in space. To overcome the prohibitive cost of the potential evaluation in case of many evaluation points, we also accelerate the potential evaluation by a fast multipole method which interpolates in space. The result is a frequency stable algorithm that scales essentially linear in the number of degrees of freedom and potential points. Numerical experiments are performed which show the feasibility and the performance of the approach.

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Solving Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method

Abstract: We solve Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method. We discuss the analytic properties of the discretization, outline the implementation, and showcase numerical examples.

Cited by 5 publications

References 33 publications

Complex moment-based methods for differential eigenvalue problems

Complex moment-based methods for differential eigenvalue problems

Complex moment-based methods for differential eigenvalue problems

Solving acoustic scattering problems by the isogeometric boundary element method

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