Bernoulli-based random undersampling schemes for 2D seismic data regularization

Seismic data regularization is an important preprocessing step in seismic signal processing. Traditional seismic acquisition methods follow the Shannon–Nyquist sampling theorem, whereas compressive sensing（CS） provides a fundamentally new paradigm to overcome limitations in data acquisition. Besides the sparse representation of seismic signal in some transform domain and the 1-norm reconstruction algorithm, the seismic data regularization quality of CS-based techniques strongly depends on random undersampling schemes. For 2D seismic data, discrete uniform-based methods have been investigated, where some seismic traces are randomly sampled with an equal probability. However, in theory and practice, some seismic traces with different probability are required to be sampled for satisfying the assumptions in CS. Therefore, designing new undersampling schemes is imperative. We propose a Bernoulli-based random undersampling scheme and its jittered version to determine the regular traces that are randomly sampled with different probability, while both schemes comply with the Bernoulli process distribution. We performed experiments using the Fourier and curvelet transforms and the spectral projected gradient reconstruction algorithm for 1-norm（SPGL1）, and ten different random seeds. According to the signal-to-noise ratio（SNR） between the original and reconstructed seismic data, the detailed experimental results from 2D numerical and physical simulation data show that the proposed novel schemes perform overall better than the discrete uniform schemes.

Machine learning identification of impurities in the STM images

We train a neural network to identify impurities in the experimental images obtained by the scanning tunneling microscope(STM)measurements.The neural network is first trained with a large number of simulated data and then the trained neural network is applied to identify a set of experimental images taken at different voltages.We use the convolutional neural network to extract features from the images and also implement the attention mechanism to capture the correlations between images taken at different voltages.We note that the simulated data can capture the universal Friedel oscillation but cannot properly describe the non-universal physics short-range physics nearby an impurity,as well as noises in the experimental data.And we emphasize that the key of this approach is to properly deal with these differences between simulated data and experimental data.Here we show that even by including uncorrelated white noises in the simulated data,the performance of the neural network on experimental data can be significantly improved.To prevent the neural network from learning unphysical short-range physics,we also develop another method to evaluate the confidence of the neural network prediction on experimental data and to add this confidence measure into the loss function.We show that adding such an extra loss function can also improve the performance on experimental data.Our research can inspire future similar applications of machine learning on experimental data analysis.

The Efficient Finite Element Methods for Time-Fractional Oldroyd-B Fluid Model Involving Two Caputo Derivatives

In this paper,we consider the numerical schemes for a timefractionalOldroyd-B fluidmodel involving the Caputo derivative.We propose two efficient finite element methods by applying the convolution quadrature in time generated by the backward Euler and the second-order backward difference methods.Error estimates in terms of data regularity are established for both the semidiscrete and fully discrete schemes.Numerical examples for two-dimensional problems further confirmthe robustness of the schemes with first-and second-order accurate in time.

AN IMPROVEMENT OF A RESULT OF IVOCHKINA AND LADYZHENSKAYA ON A TYPE OF PARABOLIC MONGE-AMPèRE EQUATION

For the initial boundary value problem about a type of parabolicMonge Ampe re equation of the form (IBVP):{-D tu+( det D^(2)_(x)u) 1/n =f(x,t),(x,t)∈Q= Ω ×(0,T],u(x,t)=(x,t)(x,t)∈ pQ},where Ω is a bounded convex domain in R n ,the result in by Ivochkina and Ladyzheskaya is improved in the sense that, under assumptions that the data of the problem possess lower regularity and satisfy lower order compatibility conditions than those in , the existence of classical solution to (IBVP) is still established (see Theorem 1.1 below). This can not be realized by only using the method in . The main additional effort the authors have done is a kind of nonlinear perturbation.