-
Previous Article
Full identification of acoustic sources with multiple frequencies and boundary measurements
- IPI Home
- This Issue
-
Next Article
Vector ellipsoidal harmonics and neuronal current decomposition in the brain
A time-domain probe method for three-dimensional rough surface reconstructions
| 1. | Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Berkshire, RG6 6AX, United Kingdom, United Kingdom |
In particular, we discuss the reconstruction of the shape of a rough surface in three dimensions from time-domain measurements of the scattered field. In practise, measurement data is collected where the incident field is given by a pulse. We formulate the time-domain field reconstruction problem equivalently via frequency-domain integral equations or via a retarded boundary integral equation based on results of Bamberger, Ha-Duong, Lubich. In contrast to pure frequency domain methods here we use a time-domain characterization of the unknown shape for its reconstruction.
Our paper will describe the Time-Domain Probe Method and relate it to previous frequency-domain approaches on sampling and probe methods by Colton, Kirsch, Ikehata, Potthast, Luke, Sylvester et al. The approach significantly extends recent work of Chandler-Wilde and Lines (2005) and Luke and Potthast (2006) on the time-domain point source method. We provide a complete convergence analysis for the method for the rough surface scattering case and provide numerical simulations and examples.
| [1] |
Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems and Imaging, 2021, 15 (3) : 499-517. doi: 10.3934/ipi.2021002 |
| [2] |
Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations and Control Theory, 2014, 3 (2) : 331-348. doi: 10.3934/eect.2014.3.331 |
| [3] |
Chan Liu, Jin Wen, Zhidong Zhang. Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation. Inverse Problems and Imaging, 2020, 14 (6) : 1001-1024. doi: 10.3934/ipi.2020053 |
| [4] |
Bernard Ducomet. Asymptotics for 1D flows with time-dependent external fields. Conference Publications, 2007, 2007 (Special) : 323-333. doi: 10.3934/proc.2007.2007.323 |
| [5] |
Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222 |
| [6] |
Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations and Control Theory, 2022, 11 (2) : 515-536. doi: 10.3934/eect.2021011 |
| [7] |
Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control and Related Fields, 2013, 3 (2) : 143-160. doi: 10.3934/mcrf.2013.3.143 |
| [8] |
Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068 |
| [9] |
Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035 |
| [10] |
Fengjuan Meng, Meihua Yang, Chengkui Zhong. Attractors for wave equations with nonlinear damping on time-dependent space. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 205-225. doi: 10.3934/dcdsb.2016.21.205 |
| [11] |
Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178 |
| [12] |
Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4921-4941. doi: 10.3934/dcds.2021062 |
| [13] |
Matti Viikinkoski, Mikko Kaasalainen. Shape reconstruction from images: Pixel fields and Fourier transform. Inverse Problems and Imaging, 2014, 8 (3) : 885-900. doi: 10.3934/ipi.2014.8.885 |
| [14] |
Mikko Kaasalainen. Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction. Inverse Problems and Imaging, 2011, 5 (1) : 37-57. doi: 10.3934/ipi.2011.5.37 |
| [15] |
Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic and Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 |
| [16] |
Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 1-21. doi: 10.3934/dcdss.2021006 |
| [17] |
Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure and Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003 |
| [18] |
Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations and Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022 |
| [19] |
Roman Chapko. On a Hybrid method for shape reconstruction of a buried object in an elastostatic half plane. Inverse Problems and Imaging, 2009, 3 (2) : 199-210. doi: 10.3934/ipi.2009.3.199 |
| [20] |
Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]