In this talk, we consider inverse scattering by unbounded rough surfaces which is to reconstruct the scattering surface from the scattered field measured on a horizontal straight line segment at a finite distance abov...
In this talk, we consider inverse scattering by unbounded rough surfaces which is to reconstruct the scattering surface from the scattered field measured on a horizontal straight line segment at a finite distance above the rough surface. An unbounded rough surface means a nonlocal perturbation of an infinite plane surface such that the whole surface lies within a finite distance of the original plane. We propose a direct imaging method to reconstruct the rough surface from the scattered nearfield Cauchy data generated by acoustic point sources. A theoretical analysis is also provided for the imaging function. Numerical experiments are presented to show that the direct imaging algorithm is fast, accurate and very robust with respect to noise in the data. We will extend the results to both the case of inverse elastic scattering by an unbounded rigid rough surface and the plane wave incidence *** the latter case with incident plane waves, we use the measured scattered near-field data instead of the Cauchy data. This talk is based on joint works with Xiaoli Liu and Haiwen Zhang.
We present and analyze a second order in time variable step BDF2 numerical scheme for the Cahn-Hilliard equation. The construction relies on second order backward difference, convex-splitting technique, and viscous re...
We present and analyze a second order in time variable step BDF2 numerical scheme for the Cahn-Hilliard equation. The construction relies on second order backward difference, convex-splitting technique, and viscous regularizing at the discrete *** show that the scheme is unconditionally stable and uniquely solvable. In addition,under mild restriction on the ratio of adjacent time steps, optimal second-order in time convergence rate is established. The proof involves a novel generalized discrete Gronwall type inequality. So far as we know, this is the first rigorous proof of second order convergence for variable step BDF2 scheme, even in the linear case,without severe restriction on the ratio of adjacent time-steps. Results of our numerical experiments corroborate with our theoretical analysis. This is a joint work with Wenbin Chen, Yue Yan and Zhuying Zhang.
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