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In this article, we extend the fourth-order compact boundary scheme in Liao et al. (Numer Methods Partial Differential Equations 18 (2002), 340-354) to a 3D problem and then combine it with the fourth-order compact alternating direction implicit (ADI) method in Gu et al. (J Comput Appl Math 155 (2003), 1-17) to solve the 3D reaction-diffusion equation with Neumann boundary condition. First, the reaction-diffusion equation is solved with a compact fourth-order finite difference method based on the Padé approximation, which is then combined with the ADI method and a fourth-order compact scheme to approximate the Neumann boundary condition, to obtain fourth order accuracy in space. The accuracy in the temporal dimension is improved to fourth order by applying the Richardson extrapolation technique, although the unconditional stability of the numerical method is proved, and several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed new algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [ABSTRACT FROM AUTHOR]

A crucial part of successful wave propagation related inverse problems is an efficient and accurate numerical scheme for solving the seismic wave equations. In particular, the numerical solution to a multi-dimensional Helmholtz equation can be troublesome when the perfectly matched layer (PML) boundary condition is implemented. In this paper, we present a general approach for constructing fourth-order finite difference schemes for the Helmholtz equation with PML in the two-dimensional domain based on point-weighting strategy. Particularly, we develop two optimal fourth-order finite difference schemes, optimal point-weighting 25p and optimal point-weighting 17p. It is shown that the two schemes are consistent with the Helmholtz equation with PML. Moreover, an error analysis for the numerical approximation of the exact wavenumber is provided. Based on minimizing the numerical dispersion, we implement the refined choice strategy for selecting optimal parameters and present refined point-weighting 25p and refined point-weighting 17p finite difference schemes. Furthermore, three numerical examples are provided to illustrate the accuracy and effectiveness of the new methods in reducing numerical dispersion. [ABSTRACT FROM AUTHOR]

Conventional full waveform inversion (FWI) using least square distance (L 2 norm) between the observed and predicted seismograms suffers from local minima. Recently, the Wasserstein metric (W 1 metric) has been introduced to FWI to compute the misfit between two seismograms. Instead of comparisons bin by bin, the W 1 metric allows to compare signal intensities across different coordinates. This measure has great potential to account for time and space shifts of events within seismograms. However, there are two main challenges in application of the W 1 metric to FWI. The first one is that the compared signals need to satisfy nonnegativity and mass conservation assumptions. The second one is that the computation of W 1 metric between two seismograms is a computationally expensive problem. In this paper, a strategy is used to satisfy the two assumptions via decomposition and recombination of original seismic data. In addition, the computation of the W 1 metric based on dynamic formulation is formulated as a convex optimization problem. A primal-dual hybrid gradient method with linesearch has been developed to solve this large-scale optimization problem on GPU device. The advantages of the new method are that it is easy to implement and has high computational efficiency. Compared to the L 2 norm based FWI, the computation time of the proposed method will approximately increase by 11% in our case studies. A 1D time-shift signals case study has indicated that the W 1 metric is more effective in capturing time shift and makes the misfit function more convex. Two applications to synthetic data using transmissive and reflective recording geometries have demonstrated the effectiveness of the W 1 metric in mitigating cycle-skipping issues. We have also applied the proposed method to SEG 2014 benchmark data, which has further demonstrated that the W 1 metric can mitigate local minima and provide reliable velocity estimations without using low frequency information in the recorded data. • Apply Wasserstein metric to full waveform inversion to mitigate local minimum issue. • PDHG method is utilized to calculate the Wasserstein metric with dynamic formulation. • Adjoint-state method is applied to compute the gradient of seismic inverse problem. • The effectiveness and robustness of our method are verified by the benchmark data. [ABSTRACT FROM AUTHOR]

Many inverse problems can be formulated as constrained optimization problems in which the feasible set is a description of the a priori information. Different feasible sets represent different a priori information. In this work, we investigate the constrained optimization problem such that the feasible set is the intersection of several convex sets. A closed-form projection is necessary when using standard gradient projection methods to solve the constrained optimization problem since the projection has to be evaluated exactly. However, the numerical methods of projection onto the intersection of convex sets usually have an iterative structure which leads to an inexact projection result in practice. In this work, a feasible set expanding strategy is designed to overcome this issue. Based on this strategy, an inexact scaled gradient projection method is proposed. The convergence analysis is provided with proper assumptions. Numerical results for an inverse problem named full waveform inversion are provided. [ABSTRACT FROM AUTHOR]

Machine Learning models have been frequently used in transcriptome analyses. Particularly, Representation Learning (RL), e.g., autoencoders, are effective in learning critical representations in noisy data. However, learned representations, e.g., the "latent variables" in an autoencoder, are difficult to interpret, not to mention prioritizing essential genes for functional follow-up. In contrast, in traditional analyses, one may identify important genes such as Differentially Expressed (DiffEx), Differentially Co-Expressed (DiffCoEx), and Hub genes. Intuitively, the complex gene-gene interactions may be beyond the capture of marginal effects (DiffEx) or correlations (DiffCoEx and Hub), indicating the need of powerful RL models. However, the lack of interpretability and individual target genes is an obstacle for RL's broad use in practice. To facilitate interpretable analysis and gene-identification using RL, we propose "Critical genes", defined as genes that contribute highly to learned representations (e.g., latent variables in an autoencoder). As a proof-of-concept, supported by eXplainable Artificial Intelligence (XAI), we implemented eXplainable Autoencoder for Critical genes (XA4C) that quantifies each gene's contribution to latent variables, based on which Critical genes are prioritized. Applying XA4C to gene expression data in six cancers showed that Critical genes capture essential pathways underlying cancers. Remarkably, Critical genes has little overlap with Hub or DiffEx genes, however, has a higher enrichment in a comprehensive disease gene database (DisGeNET) and a cancer-specific database (COSMIC), evidencing its potential to disclose massive unknown biology. As an example, we discovered five Critical genes sitting in the center of Lysine degradation (hsa00310) pathway, displaying distinct interaction patterns in tumor and normal tissues. In conclusion, XA4C facilitates explainable analysis using RL and Critical genes discovered by explainable RL empowers the study of complex interactions. Author summary: We propose a gene expression data analysis tool, XA4C, which builds an eXplainable Autoencoder to reveal Critical genes. XA4C disentangles the black box of the neural network of an autoencoder by providing each gene's contribution to the latent variables in the autoencoder. Next, a gene's ability to contribute to the latent variables is used to define the importance of this gene, based on which XA4C prioritizes "Critical genes". Notably, we discovered that Critical genes enjoy two properties: (1) Their overlap with traditional differentially expressed genes and hub genes are poor, suggesting that they indeed brought novel insights into transcriptome data that cannot be captured by traditional analysis. (2) The enrichment of Critical genes in a comprehensive disease gene database (DisGeNET) and cancer-specific database (COSMIC) are higher than differentially expressed or hub genes, evidencing their strong relevance to disease pathology. Therefore, we conclude that XA4C can reveal an additional landscape of gene expression data. [ABSTRACT FROM AUTHOR]