Seismic wavefield modeling is an important tool for the seismic interpretation. We consider modeling the wavefield in the frequency domain. This requires to solve a sequence of Helmholtz equations of wave numbers gove...
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Seismic wavefield modeling is an important tool for the seismic interpretation. We consider modeling the wavefield in the frequency domain. This requires to solve a sequence of Helmholtz equations of wave numbers governed by the Nyquist sampling theorem. Inevitably, we have to solve Helmholtz equations of large wave numbers, which is a challenging task numerically. To address this issue, we develop two methods for modeling the wavefield in the frequency domain to obtain an alias-free result using lower frequencies of a number fewer than typically required by the Nyquist sampling theorem. Specifically, we introduce two l(1) regularization models to deal with incomplete Fourier transforms, which arise from seismic wavefield modeling in the frequency domain, and propose a new sampling technique to avoid solving the Helmholtz equations of large wave numbers. In terms of the fixed-point equation via the proximity operator of the l(1) norm, we characterize solutions of the two l(1) regularization models and develop fixed-point algorithms to solve these two models. Numerical experiments are conducted on seismic data to test the approximation accuracy and the computational efficiency of the proposed methods. Numerical results show that the proposed methods are accurate, robust and efficient in modeling seismic wavefield in the frequency domain with only a few low frequencies. (C) 2020 Elsevier B.V. All rights reserved.
Sparsity promoting functions (SPFs) are commonly used in optimization problems to find solutions which are sparse in some basis. For example, the l(1)-regularized wavelet model and the Rudin-Osher-Fatemi total variati...
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Sparsity promoting functions (SPFs) are commonly used in optimization problems to find solutions which are sparse in some basis. For example, the l(1)-regularized wavelet model and the Rudin-Osher-Fatemi total variation (ROF-TV) model are some of the most well-known models for signal and image denoising, respectively. However, recent work demonstrates that convexity is not always desirable in SPFs. In this paper, we replace convex SPFs with their induced nonconvex SPFs and develop algorithms for the resulting model by exploring the intrinsic structures of the nonconvex SPFs. These functions are defined as the difference of the convex SPF and its Moreau envelope. We also present simulations illustrating the performance of a special SPF and the developed algorithms in image denoising.
We propose a sparse regularization model for inversion of incomplete Fourier transforms and apply it to seismic wavefield modeling. The objective function of the proposed model employs the Moreau envelope of the l(0) ...
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We propose a sparse regularization model for inversion of incomplete Fourier transforms and apply it to seismic wavefield modeling. The objective function of the proposed model employs the Moreau envelope of the l(0) norm under a tight framelet system as a regularization to promote sparsity. This model leads to a non-smooth, non-convex optimization problem for which traditional iteration schemes are inefficient or even divergent. By exploiting special structures of the l(0) norm, we identify a local minimizer of the proposed non-convex optimization problem with a globalminimizer of a convex optimization problem, which provides us insights for the development of efficient and convergence guaranteed algorithms to solve it. We characterize the solution of the regularization model in terms of a fixed-point of a map defined by the proximity operator of the l(0) norm and develop a fixed-point iteration algorithm to solve it. By connecting the map with an a-averaged nonexpansive operator, we prove that the sequence generated by the proposed fixed-point proximity algorithm converges to a local minimizer of the proposed model. Our numerical examples confirm that the proposed model outperforms significantly the existing model based on the l(1)-norm. The seismic wavefield modeling in the frequency domain requires solving a series of the Helmholtz equation with large wave numbers, which is a computationally intensive task. Applying the proposed sparse regularization model to the seismic wavefield modeling requires data of only a few low frequencies, avoiding solving the Helmholtz equation with large wave numbers. This makes the proposed model particularly suitable for the seismic wavefield (SW) modeling. Numerical results show that the proposed method performs better than the existing method based on the l(1) norm in terms of the SNR values and visual quality of the restored synthetic seismograms.
作者：
Zhang, NaLi, QiaSouth China Agr Univ
Coll Math & Informat Dept Appl Math Guangzhou 510642 Peoples R China Sun Yat Sen Univ
Sch Comp Sci & Engn Guangdong Prov Key Lab Computat Sci Guangzhou 510275 Peoples R China
In this paper, we consider a class of single-ratio fractional minimization problems, in which the numerator of the objective is the sum of a nonsmooth nonconvex function and a smooth nonconvex function while the denom...
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In this paper, we consider a class of single-ratio fractional minimization problems, in which the numerator of the objective is the sum of a nonsmooth nonconvex function and a smooth nonconvex function while the denominator is a nonsmooth convex function. In this work, we first derive its first-order necessary optimality condition, by using the first-order operators of the three functions involved. Then we develop first-order algorithms, namely, the proximity-gradientsubgradient algorithm (PGSA), PGSA with monotone line search (PGSA ML), and PGSA with nonmonotone line search (PGSA NL). It is shown that any accumulation point of the sequence generated by them is a critical point of the problem under mild assumptions. Moreover, we establish global convergence of the sequence generated by PGSA or PGSA ML and analyze its convergence rate, by further assuming the local Lipschitz continuity of the nonsmooth function in the numerator, the smoothness of the denominator, and the Kurdyka--\Lojasiewicz (KL) property of the objective. The proposed algorithms are applied to the sparse generalized eigenvalue problem associated with a pair of symmetric positive semidefinite matrices, and the corresponding convergence results are obtained according to their general convergence theorems. We perform some preliminary numerical experiments to demonstrate the efficiency of the proposed algorithms.
Sparse learning models are popular in many application areas. Objective functions in sparse learning models are usually non-smooth, which makes it difficult to solve them numerically. We develop a fast and convergent ...
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Sparse learning models are popular in many application areas. Objective functions in sparse learning models are usually non-smooth, which makes it difficult to solve them numerically. We develop a fast and convergent two-step iteration scheme for solving a class of non-differentiable optimization models motivated from sparse learning. To overcome the difficulty of the non-differentiability of the models, we first present characterizations of their solutions as fixed-points of mappings involving the proximity operators of the functions appearing in the objective functions. We then introduce a two-step fixed-point algorithm to compute the solutions. We establish convergence results of the proposed two-step iteration scheme and compare it with the alternating direction method of multipliers (ADMM). In particular, we derive specific two-step iteration algorithms for three models in machine learning: l(1)-SVM classification, l(1)-SVM regression, and the SVM classification with the group LASSO regularizer. Numerical experiments with some synthetic datasets and some benchmark datasets show that the proposed algorithm outperforms ADMM and the linear programming method in computational time and memory storage costs.
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