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 and Continuous Dynamical Systems, 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 and 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 and 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 and Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757
[5]

Deyue Zhang, Yue Wu, Yinglin Wang, Yukun Guo. A direct imaging method for the exterior and interior inverse scattering problems. Inverse Problems and Imaging, 2022, 16 (5) : 1299-1323. doi: 10.3934/ipi.2022025
[6]

Ai-Guo Wu, Ying Zhang, Hui-Jie Sun. Parametric Smith iterative algorithms for discrete Lyapunov matrix equations. Journal of Industrial and Management Optimization, 2020, 16 (6) : 3047-3063. doi: 10.3934/jimo.2019093
[7]

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
[8]

Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems and Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111
[9]

Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems and Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067
[10]

Gemma Huguet, Rafael de la Llave, Yannick Sire. Computation of whiskered invariant tori and their associated manifolds: New fast algorithms. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1309-1353. doi: 10.3934/dcds.2012.32.1309
[11]

Jeremy L. Marzuola. Dispersive estimates using scattering theory for matrix Hamiltonian equations. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 995-1035. doi: 10.3934/dcds.2011.30.995
[12]

Guanghui Hu, Andrea Mantile, Mourad Sini, Tao Yin. Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles. Inverse Problems and Imaging, 2020, 14 (6) : 1025-1056. doi: 10.3934/ipi.2020054
[13]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems and Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042
[14]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems and Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076
[15]

Franziska Nestler, Martin Stoll, Theresa Wagner. Learning in high-dimensional feature spaces using ANOVA-based fast matrix-vector multiplication. Foundations of Data Science, 2022, 4 (3) : 423-440. doi: 10.3934/fods.2022012
[16]

Gusein Sh. Guseinov. Spectral method for deriving multivariate Poisson summation formulae. Communications on Pure and Applied Analysis, 2013, 12 (1) : 359-373. doi: 10.3934/cpaa.2013.12.359
[17]

Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435
[18]

Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226
[19]

Caroline Hillairet, Ying Jiao, Anthony Réveillac. Pricing formulae for derivatives in insurance using Malliavin calculus. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 7-. doi: 10.1186/s41546-018-0028-9
[20]

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

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy