Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction

Ratowsky, R. P.; Fleck, J. A.

doi:10.1364/ol.16.000787

Cited by 45 publications

(12 citation statements)

References 6 publications

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“…The idea to use the exponential function of the Jacobian in a numerical integrator is by no means new, but it has mostly been regarded as rather impractical. Since the mid-eighties, Krylov subspace approximations to the action of the matrix exponential operator have, however, been found to be useful in Chemical Physics 16,20,22] and subsequently also in other elds 6,8,9,21,24,29]. On the numerical analysis side, the convergence of such Krylov approximations was studied in 4,5,13,26].…”

mentioning

confidence: 99%

Exponential Integrators for Large Systems of Differential Equations

Hochbruck¹,

Lubich²,

Selhofer³

1998

SIAM J. Sci. Comput.

454

478

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Abstract. We study the numerical integration of large sti systems of di erential equations by methods that use matrix-vector products with the exponential or a related function of the Jacobian. For large problems, these can be approximated by Krylov subspace methods, which typically converge faster than those for the solution of the linear systems arising in standard sti integrators. The exponential methods also o er favorable properties in the integration of di erential equations whose Jacobian has large imaginary eigenvalues. We derive methods up to order 4 which are exact for linear constant-coe cient equations. The implementation of the methods is discussed. Numerical experiments with reaction-di usion problems and a time-dependent Schr odinger equation are included.

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mentioning

confidence: 99%

Exponential Integrators for Large Systems of Differential Equations

Hochbruck¹,

Lubich²,

Selhofer³

1998

SIAM J. Sci. Comput.

454

478

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“…Fig. 3 but for the deviations of the Helmholtz finite-difference motivated to introduce the series of wide-angle procedures discussed in conjunction with (9) for which the propagation operator is constructed from positive integer powers of H and is therefore again represented by a banded matrix in the Lanczos basis. That such a procedure does not suffer from convergence difficulties is apparent from Figs.…”

Section: Resultsmentioning

confidence: 99%

“…Clearly, 60, 41, and 4 2 are orthonormal while for i > 2, orthonomality follows by induction. The products H43 can be evaluated with either a three-point finite-difference approximation for the second-derivative operator or, following previous work, the fast Fourier transform method [9].…”

Section: Methodsmentioning

confidence: 99%

“…The first nontrivial procedure is efficiently implemented through a synthesis of finite-difference and fast Fourier transform techniques but cannot adequately describe beams with half-widths greater than about 35" -40". This limitation is however circumvented with the Lanczos formalism, which projects the electric field and the propagation operator at a given refractive index cross section onto a restricted subspace comprised of a limited number of basis functions [8], [9]. The propagation problem is then solved exactly within the reduced subspace.…”

Section: Introductionmentioning

confidence: 99%

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A comparison of Lanczos electric field propagation methods

Hermansson¹,

Yevick

Bardyszewski

et al. 1992

J. Lightwave Technol.

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4bstract-We analyze the losses of a strongly guiding semiconductor rib waveguide Y-junction with the aid of various propagation algorithms based on Lanczos reduction. Although for our example Lanczos procedures adequately describe electric field propagation in the Fresnel approximation, the Helmholtz implementation converges unacceptably slowly. We therefore propose a finite-difference wide-angle approach which employs a power-series expansion of the forward propagation operator. The computational efficiency of our method is comparable to that of the Fresnel formalism with little or no reduction in accuracy from the Helmholtz technique.

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“…[6][7][8][9]. The intensity profiles for the f/20 optical system with a maximum in =iO is shown in Fig.…”

Section: Modeling Of Optical Limitersmentioning

confidence: 99%

<title>Numerical package for modeling optical limiters</title>

Law

Swartzlander

1996

SPIE Proceedings

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A numerical software package for (2+1)-dimensional simulation of optical limiters has been developed. The purpose of this effort is to provide the sensor protection community with a means to develop, understand, and optimize optical limiters. A graphics interface has been implemented to guide the user in both running the program and interpreting the results. Scientific visualization tools have been integrated with the package to facilitate a physical understanding of the numerical results.We have used it successfully to perform (2+1)-dimensional nonlinear propagation of a focused optical beam with arbitrary intensity profile, such as Gaussian or tophat profile, in a f/S system. The package can handle various nonlinear mechanisms, including intensity dependent refraction and absorption. Non-local nonlinearities such as thermal effects may be included as a modular subprogram. Temporal effects may also be included by simulating the integrated spatio-temporal evolution of the light and material properties, although this approach is memory-intensive and often requires the use of a supercomputer.

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Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction

Cited by 45 publications

References 6 publications

Exponential Integrators for Large Systems of Differential Equations

Exponential Integrators for Large Systems of Differential Equations

A comparison of Lanczos electric field propagation methods

<title>Numerical package for modeling optical limiters</title>

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