American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations
  • DCDS Home
  • This Issue
  • Next Article
    An invariant set generated by the domain topology for parabolic semiflows with small diffusion
November  2007, 18(4): 627-636. doi: 10.3934/dcds.2007.18.627

Rapid perturbational calculations for the Helmholtz equation in two dimensions

Sang-Yeun Shim 1, , Marcos Capistran 2, and  Yu Chen 1,

1. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1110, United States, United States
2. 

Centro de Investigación en Matemáticas, CIMAT A. C., Jalisco s/n, Mineral de Valenciana, C.P. 36240, Guanajuato, Guanajuato, Mexico

Received  May 2006 Revised  April 2007 Published  May 2007

Existing approaches to the solution of the inverse scattering problems in two and three dimensions rely on linearization of the Helmholtz equation, which requires the knowledge of the Fr\'echet derivative of the far field with respect to the index of refraction. We present an efficient algorithm for this perturbational calculation in two dimensions. Our method is based on the merging and splitting procedures already established for the solution of the Lippmann-Schwinger equation [2], [3], [4]. For an $m$-by-$m$ wavelength problem, the algorithm obtains perturbations to scattered waves for $m$ distinct incident waves in $O(m^3)$ steps.
Keywords: scattering matrix, merging formulae, fast direct algorithms..
Mathematics Subject Classification: Primary: 65R20, 49N45; Secondary: 45Q0.
Citation: Sang-Yeun Shim, Marcos Capistran, Yu Chen. Rapid perturbational calculations for the Helmholtz equation in two dimensions. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 627-636. doi: 10.3934/dcds.2007.18.627
[1]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027
[2]

Cornelis van der Mee. Direct scattering of AKNS systems with $L^2$ potentials. Conference Publications, 2015, 2015 (special) : 1089-1097. doi: 10.3934/proc.2015.1089
[3]

Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Fast algorithms for the approximation of a traffic flow model on networks. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 427-448. doi: 10.3934/dcdsb.2006.6.427
[4]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757
[5]

Ai-Guo Wu, Ying Zhang, Hui-Jie Sun. Parametric Smith iterative algorithms for discrete Lyapunov matrix equations. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019093
[6]

Lu Tan, Ling Li, Senjian An, Zhenkuan Pan. Nonlinear diffusion based image segmentation using two fast algorithms. Mathematical Foundations of Computing, 2019, 2 (2) : 149-168. doi: 10.3934/mfc.2019011
[7]

Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems & Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111
[8]

Gemma Huguet, Rafael de la Llave, Yannick Sire. Computation of whiskered invariant tori and their associated manifolds: New fast algorithms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1309-1353. doi: 10.3934/dcds.2012.32.1309
[9]

Jeremy L. Marzuola. Dispersive estimates using scattering theory for matrix Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 995-1035. doi: 10.3934/dcds.2011.30.995
[10]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042
[11]

Gusein Sh. Guseinov. Spectral method for deriving multivariate Poisson summation formulae. Communications on Pure & Applied Analysis, 2013, 12 (1) : 359-373. doi: 10.3934/cpaa.2013.12.359
[12]

Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435
[13]

Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226
[14]

Shi-Liang Wu, Cheng-Hsiung Hsu. Entire solutions with merging fronts to a bistable periodic lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2329-2346. doi: 10.3934/dcds.2016.36.2329
[15]

Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055
[16]

Thomas Hillen, Jeffery Zielinski, Kevin J. Painter. Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2513-2536. doi: 10.3934/dcdsb.2013.18.2513
[17]

Wenxue Huang, Xiaofeng Li, Yuanyi Pan. Increase statistical reliability without losing predictive power by merging classes and adding variables. Big Data & Information Analytics, 2016, 1 (4) : 341-347. doi: 10.3934/bdia.2016014
[18]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116
[19]

Marc Bocquet, Julien Brajard, Alberto Carrassi, Laurent Bertino. Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization. Foundations of Data Science, 2020, 2 (1) : 55-80. doi: 10.3934/fods.2020004
[20]

Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences