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We consider the numerical simulation of the acoustic wave equations arising from seismic applications, for which staggered grid finite difference methods are popular choices due to their simplicity and efficiency. We relax the uniform grid restriction on finite difference methods and allow the grids to be block-wise uniform with nonconforming interfaces. In doing so, variations in the wave speeds of the subterranean media can be accounted for more efficiently. Staggered grid finite difference operators satisfying the summation-by-parts (SBP) property are devised to approximate the spatial derivatives appearing in the acoustic wave equation. These operators are applied within each block independently. The coupling between blocks is achieved through simultaneous approximation terms (SATs), which impose the interface condition weakly, i.e., by penalty. Ratio of the grid spacing of neighboring blocks is allowed to be rational number, for which specially designed interpolation formulas are presented. These interpolation formulas constitute key pieces of the simultaneous approximation terms. The overall discretization is shown to be energy-conserving and examined on test cases of both theoretical and practical interests, delivering accurate and stable simulation results.

High Efficiency Video Coding (HEVC) is the next generation video compression standard providing significant coding performance. It adopts 35 intra prediction modes with larger CU size to improve the intra encoding efficiency, so that cause a high computational complexity. In this paper, two fast intra-prediction algorithms are proposed to reduce the number of candidate modes for rate-distortion (RD) optimization. We obtain an optimal adjacent modes (OAM) list consisting of dominant directions through the analysis of costs of several general direction modes. Furthermore, we improve the most probable mode (MPM) algorithm to make full use of the spatial correlation between neighbour prediction blocks instead of simply merging the prediction modes of neighbour prediction blocks into the candidate list. Experimental results show that the proposed algorithms can reduce about 27.3% of the encoding time compared to the HEVC test model 14.0, while the decrease of coding quality is negligible.

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We consider numerical simulation of the isotropic elastic wave equations arising from seismic applications with non-trivial land topography. The more flexible finite element method is applied to the shallow region of the simulation domain to account for the topography, and combined with the more efficient finite difference method that is applied to the deep region of the simulation domain. We demonstrate that these two discretization methods, albeit starting from different formulations of the elastic wave equation, can be joined together smoothly via weakly imposed interface conditions. Discrete energy analysis is employed to derive the proper interface treatment, leading to an overall discretization that is energy-conserving. Numerical examples are presented to demonstrate the efficacy of the proposed interface treatment.