Dr. Ruming Zhang
- by appointment
- Kollegiengebäude Mathematik (20.30)
You can find my latest research and teaching activities below.
|Winter Semester 2021/22||Advanced Mathematics III||Lecture|
|Summer Semester 2021||Seminar (Numerical Methods for Scattering Problems)||Seminar|
|Winter Semester 2020/21||Wave propagation in periodic structures||Lecture|
|Winter Semester 2018/19||Inverse Problems||Lecture|
I become a KIT Associate Fellow from June 2021.
I become a KIT Junior Research Group Leader (KIT-Nachwuchsgruppenleiterin) from May 2021.
DFG research grant (No. 433126998)
Higher order numerical methods for acoustic scattering problems with locally perturbed periodic structures
This project is devoted to the investigation of time-harmonic acoustic scattering problems with locally perturbed periodic inhomogeneous layers above impenetrable plates in three dimensional spaces. The scattering problems are modelled by Helmholtz equations in unbounded domains, both the theoretical analysis and the numerical solution of which are very challenging. The main tool involved in this project is the Floquet-Bloch transform, which has been proven to be very powerful for scattering problems with periodic structures in two dimensional spaces. The first objective is to analyze continuity and regularity of the Bloch transformed field with respect to the quasi-periodicity parameter, where the Dirichlet-to-Neumann map plays an important role. The second goal is to propose a high order numerical method for scattering problems with periodic layers, based on the regularity results established for the quasi-periodic Bloch transformed problems. In contrast to the 2D case, the singularities of the Bloch transformed fields are no longer localized in a finite number of points, but cover a union of "singular circles". Thus a straightforward extension of the high order numerical method for the 2D case may not be appropriate for the 3D case, and new ideas will be required. The third goal is to develop an efficient numerical method for locally perturbed periodic layers. Either a coupled finite element method or a discretization of the Lippmann-Schwinger equation will be applied. https://gepris.dfg.de/gepris/projekt/433126998?language=en
- Direct scattering problems in unbounded structures.
- Variational approach of the scattering problems in perturbed periodic structures, via the Floquet-Bloch transform.
- Limiting absorption principle in scattering problems in periodic waveguides.
- Integral equation methods for scattering problems from periodic or rough surfaces.
- Finite element methods for scattering problems from perturbed periodic surfaces or waveguides.
- Inverse problems in perturbed periodic structures
- Optimization and regularization methods for inverse scattering problems in periodic structures.
- Fast imaging methods for inverse scattering problems: linear sampling method, factorization method, etc.
- Fast algorithms for evaluations of quasi-periodic Green's functions.
- Numerical method for evaluations of eigenvalues in a linear system.
- R. Zhang, Numerical method for scattering problems in periodic waveguides, Numer. Math., 148(4), 959-996, 2021. https://doi.org/10.1007/s00211-021-01229-0
- R. Zhang, Numerical Methods for Plane Waves Scattered by Locally Perturbed Periodic Surfaces. Accepted by Comput. Math. Appl., 2021. https://arXiv.org/abs/1801.01761
- R. Zhang, Spectrum decomposition of translation operators in periodic waveguide, SIAM J. Appl. Math. 81(1), 233-257, 2021. https://doi.org/10.1137/19M1290942
- X. Liu, R. Zhang, Near-field imaging of locally perturbed periodic surfaces, Inverse Problems 35(11), Special Issue in Memory of Professor Armin Lechleiter, 114003, 2019. https://doi.org/10.1088/1361-6420/ab2e8f
- R. Zhang, High order numerical method for scattering from locally perturbed periodic surfaces, SIAM J. Sci. Comput., 40(4), A2286-A2314, 2018. https://doi.org/10.1137/17M1144945
- B. Zhang, R. Zhang, An FFT-based algorithm for efficient computation of quasiperiodic Green's functions for the Helmholtz and Maxwell's equations, SIAM J. Sci. Comput., 40(3), B915-B941, 2018. https://doi.org/10.1137/18M1165621
- A. Lechleiter, R. Zhang, The reconstruction of a local perturbation in periodic structures, Inverse Problems, 34, 035006, 2018. https://doi.org/10.1088/1361-6420/aaa7b1
- R. Zhang, J. Sun, The reconstruction of obstacles in a waveguide using finite elements, J. Comput. Math., 36(1), 29-46, 2018. http://doi.org/10.4208/jcm.1610-m2016-0559
- R. Zhang, B. Zhang, A new integral equation formulation for scattering by penetrable diffraction gratings, J. Comput. Math., 36(1), 110-127, 2018. http://doi.org/10.4208/jcm.1612-m2016-0501
- R. Zhang, J. Sun, C. Zheng, Reconstruction of a penetrable obstacle in periodic waveguides, Comput. Math. Appl., 74(11), 2739 - 2751, 2017. https://doi.org/10.1016/j.camwa.2017.08.028
- A. Lechleiter, R. Zhang, A Floquet-Bloch transform based numerical method for scattering from locally perturbed periodic surfaces, SIAM J. Sci. Comput., 39(5), B819-B839, 2017. https://doi.org/10.1137/16M1104111
- A. Lechleiter, R. Zhang, Non-periodic acoustic and electromagnetic scattering from periodic structures in 3D, Comput. Math. Appl., 74(11), 2723 - 2738, 2017. https://doi.org/10.1016/j.camwa.2017.08.042
- M. Li, R. Zhang, Near-field imaging of sound-soft obstacles in periodic waveguides, Inverse Problems and Imaging, 11(6), 1091-1105, 2017. http://doi.org/10.3934/ipi.2017050
- A. Lechleiter, R. Zhang, A convergent numerical scheme for scattering of aperiodic waves from periodic surfaces based on the Floquet-Bloch transform, SIAM J. Numer. Anal., 55(2), 713-736, 2017. https://doi.org/10.1137/16M1067524
- R. Huang, A. Struthers, J. Sun, R. Zhang, Recursive integral method for transmission eigenvalues, J. Comput. Phys., 327, 830-840, 2016. https://doi.org/10.1016/j.jcp.2016.10.001
- G. Sun, R. Zhang, A sampling method for the reconstruction of a periodic interface in a layered medium, Inverse Problems, 32, 075005, 2016. https://doi.org/10.1088/0266-5611/32/7/075005
- J. Yang, B. Zhang, R. Zhang, Near-field imaging of periodic interfaces in multilayered media, Inverse Problems, 32, 035010, 2016. https://doi.org/10.1088/0266-5611/32/3/035010
- J. Li, G. Sun, R. Zhang, The numerical solution of scattering by infinite rough interfaces based on the integral equation method, Comput. Math. Appl., 71(7), 1491-1502, 2016. https://doi.org/10.1016/j.camwa.2016.02.031
- R. Zhang, J. Sun, Efficient finite element method for grating profile reconstruction, J. Comput. Phys., 302(1), 405-419, 2015. https://doi.org/10.1016/j.jcp.2015.09.016
- R. Zhang, B. Zhang, Near-field imaging of periodic inhomogeneous media, Inverse Problems, 30, 045004, 2014. https://doi.org/10.1088/0266-5611/30/4/045004
- J. Yang, B. Zhang, R. Zhang, Reconstruction of penetrable grating profiles, Inverse Problems and Imaging, 7(4), 1393-1407, 2013. https://doi.org/10.3934/ipi.2013.7.1393
- J. Yang, B. Zhang, R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28, 035004, 2012. https://doi.org/10.1088/0266-5611/28/3/035004
- R. Zhang, Numerical methods for scattering problems from multi-layers with different periodicities. Submitted to Numer. Methods Partial Differential Eq. https://arXiv.org/abs/1806.05063
- R. Zhang, High order methods to simulate scattering problems in locally perturbed periodic waveguides. Submitted to SISC. https://arxiv.org/submit/3753573
- R. Zhang, Exponential convergence of perfectly matched layers for scattering problems with periodic surfaces. Submitted to SINUM. Preprint. https://arxiv.org/abs/2107.02032