Reducing error accumulation of optimized finite-difference scheme using the minimum norm

Abstract

The finite-difference scheme is popular in the field of seismic exploration for numerical simulation of wave propagation; however, its accuracy and computational efficiency are restricted by the numerical dispersion caused by numerical discretization of spatial partial derivatives using coarse grid. The constant-coefficient optimization method is widely used for suppressing the numerical dispersion by tuning the finite-difference weights. While gaining a wider effective bandwidth under a given error tolerance, this method undoubtedly encounters larger errors at low wavenumbers and accumulates significant errors. We proposed an approach to reduce the error accumulation. First, we construct an objective function based on the L1 norm, which can better constrain the total error than the L2 and L∞ norms. Second, we translated our objective function into a constrained L1 norm minimization model, which can be solved by the alternating direction method of multipliers. Finally, we perform theoretical analyses and numerical experiments to illustrate the accuracy improvement. The proposed method is shown to be superior to the existing constant-coefficient optimization methods at the low-wavenumber region; thus, we can obtain higher accuracy with less error accumulation, particularly at longer simulation times. The widely used objective functions, defined by the L2 and L∞ norms could handle a relatively wider range of accurate wavenumbers, compared with our objective function defined by the L1 norm, but their actual errors would be much larger than the given error tolerance at some azimuths rather than axis directions (e.g., about twice at 45°), which greatly degrade the overall numerical accuracy. In contrast, our scheme can obtain a relatively even 2D error distribution at various azimuths, with an apparently smaller error. The peak error of the proposed method is only 40~65% that of the L2 norm under the same error tolerance, or only 60~80% that of the L2 norm under the same effective bandwidth.

[Retracted] Automatic Classification Method of Music Genres Based on Deep Belief Network and Sparse Representation

“…However, formula ( 14) has an NP problem, which is difficult to solve. erefore, solving the NP problem by minimizing the problem [19,20] is expressed as follows:…”