Welcome to my personal homepage! I am the leader of the Junior Research Group: Waves in Periodic Structures. You can also find information for the project as an associate project AP7 of CRC 1173.

You can find my latest research and teaching activities below.

News

I become a KIT Associate Fellow from June 2021.
I become a KIT Junior Research Group Leader (KIT-Nachwuchsgruppenleiterin) from May 2021.

Research interest

• Analysis of PDEs: asymptotic behaviour, spectral theory, (multivariable) complex analysis, operator theory, perturbation theory
• Numerical analysis of PDEs: boundary element method, finite element method, spectral method, eigenvalue problems, quasi-adaptive method
• Scattering theory: transparent boundary conditions, perfectly matched layers
• High performance computing: parallel computing, preconditioners
• Inverse problems: fast imaging methods (linear sampling method, factorization method, monotonicity method, etc.), PDE constrained optimization, regularization
• Stochastic PDEs: numerical simulation, multi-scale method

Project

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

Publications

1. R. Zhang, A spectral decomposition method to approximate DtN maps in complicated waveguides. Accepted by SIAM J. Numer. Anal., 2023. https://arxiv.org/pdf/2207.12690.pdf
2. R. Zhang, Numerical methods for scattering problems from multi-layers with different periodicities. Numer. Methods Partial Differential Eq., 39, 1778-1798, 2023. https://onlinelibrary.wiley.com/doi/epdf/10.1002/num.22954
3. R. Zhang, High Order Complex Contour Discretization Methods to Simulate Scattering Problems in Locally Perturbed Periodic Waveguides, SIAM J. Sci. Comput., 44(5), B1257-B1281, 2022. https://doi.org/10.1137/21M1421532
4. R. Zhang, Exponential convergence of perfectly matched layers for scattering problems with periodic surfaces, SIAM J. Numer. Anal., 60(2), 804-823, 2022. https://doi.org/10.1137/21M1439043
5. 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
6. 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
7. 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
8. 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
9. 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
10. 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
11. 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
12. 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
13. 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
14. 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
15. 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
16. 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
17. 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
18. 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
19. 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
20. 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
21. 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
22. 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
23. 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
24. 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
25. 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
26. 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

Preprints

1. T. Arens, R. Griesmaier, R. Zhang, Monotonicity-based shape reconstruction for an inverse scattering problem in a waveguide. Preprint. https://www.waves.kit.edu/downloads/CRC1173_Preprint_2022-69.pdf
2. T. Arens, R. Zhang, A nonuniform mesh method for wave scattered by periodic surfaces. Preprint. https://arxiv.org/pdf/2203.05792.pdf
3. R. Zhang, Higher order convergence of perfectly matched layers in 3D bi-periodic surface scattering problems. Preprint. https://arxiv.org/pdf/2211.01222.pdf
4. R. Zhang, Fast convergence for of perfectly matched layers for scattering with periodic surfaces: the exceptional case. Preprint. https://arxiv.org/pdf/2211.01229.pdf