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26 pages, 15575 KB

Open AccessArticle

A Scalable and Consistent Method for Multi-Component Gravity-Gradient Data Processing

by Larissa Silva Piauilino, Vanderlei Coelho Oliveira Junior and Valeria Cristina Ferreira Barbosa

Appl. Sci. 2025, 15(15), 8396; https://doi.org/10.3390/app15158396 - 29 Jul 2025

Cited by 2 | Viewed by 858

We demonstrate the potential of using the convolutional equivalent layer to jointly process large gravity-gradient datasets. Based on the equivalent-layer principle, we assume a single fictitious physical property distribution on a planar layer can approximate all components of the gravity-gradient tensor. Estimating this distribution using the classical technique ensures physical consistency among components. However, the classical approach becomes computationally prohibitive for large datasets due to the need to solve a large-scale inversion with a massive sensitivity matrix. To overcome this limitation, we exploit the block-Toeplitz Toeplitz-block structure of the sensitivity matrix for data on a regular horizontal grid. This structure significantly reduces computational cost—by orders of magnitude—compared to the classical method. Applications to synthetic and real datasets show that our method offers a computationally efficient alternative for processing large gravity-gradient data from various acquisition systems (AGG and FTG), even when data are irregularly spaced or flight lines are misaligned. On a standard laptop configuration, our method processed over 290,000 AGG data points in a few tens of seconds. It also handled between 726,000 FTG and 1,250,000 AGG data points within seconds to a couple of minutes, demonstrating practical computational efficiency for large-scale datasets. Full article

(This article belongs to the Special Issue Advances in Geophysical Exploration)

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17 pages, 5008 KB

Open AccessArticle

Structure Approximation-Based Preconditioning for Solving Tempered Fractional Diffusion Equations

by Xuan Zhang and Chaojie Wang

Algorithms 2025, 18(6), 307; https://doi.org/10.3390/a18060307 - 23 May 2025

Cited by 1 | Viewed by 652

Abstract

Tempered fractional diffusion equations constitute a critical class of partial differential equations with broad applications across multiple physical domains. In this paper, the Crank–Nicolson method and the tempered weighted and shifted Grünwald formula are used to discretize the tempered fractional diffusion equations. The discretized system has the structure of the sum of the identity matrix and a diagonal matrix multiplied by a symmetric positive definite (SPD) Toeplitz matrix. For the discretized system, we propose a structure approximation-based preconditioning method. The structure approximation lies in two aspects: the inverse approximation based on the row-by-row strategy and the SPD Toeplitz approximation by the

τ

matrix. The proposed preconditioning method can be efficiently implemented using the discrete sine transform (DST). In spectral analysis, it is found that the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Numerical experiments demonstrate the effectiveness of the proposed preconditioner. Full article

(This article belongs to the Special Issue Numerical Optimization and Algorithms: 3rd Edition)

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24 pages, 6901 KB

Open AccessArticle

A Suitable Algorithm to Solve a Nonlinear Fractional Integro-Differential Equation with Extended Singular Kernel in (2+1) Dimensions

by Sameeha Ali Raad and Mohamed Abdella Abdou

Fractal Fract. 2025, 9(4), 239; https://doi.org/10.3390/fractalfract9040239 - 10 Apr 2025

Cited by 2 | Viewed by 894

Abstract

In this paper, the authors consider a problem with comprehensive properties in terms of form and content in the space

L_{2} [(a, b) \times (c, d)] \times C [0, T], T < 1

. In terms of time [...] Read more.

In this paper, the authors consider a problem with comprehensive properties in terms of form and content in the space

L_{2} [(a, b) \times (c, d)] \times C [0, T], T < 1

. In terms of time form, we assume that the time phase delay is implicitly contained in a nonlinear differential integral equation. The positional part is considered in two dimensions, and the position’s kernel is a general singular kernel, many different forms of which will be derived. In terms of content, all of the previously established numerical techniques are only appropriate for studying special cases of the kernel separately but are not suitable for studying the general kernel. This led to the use of the Toeplitz matrix method, which deals with the kernel in its extended nonlinear form and the special kernels will be studied as applications of the method. Moreover, this method has the advantage of converting all single integrals into regular integrals that can be easily solved. Additionally, the researchers examine the solution’s existence, uniqueness, and convergence in this paper. The error and its stability are also studied. At the end of the research, the authors studied some numerical applications of some of the singular kernels derived from the general kernel, examining the approximation error in each application separately. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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14 pages, 7993 KB

Open AccessArticle

An Improved Toeplitz Approximation Method for Coherent DOA Estimation in Impulsive Noise Environments

by Jiang’an Dai, Tianshuang Qiu, Shengyang Luan, Quan Tian and Jiacheng Zhang

Entropy 2023, 25(6), 960; https://doi.org/10.3390/e25060960 - 20 Jun 2023

Cited by 6 | Viewed by 2839

Abstract

Direction of arrival (DOA) estimation is an important research topic in array signal processing and widely applied in practical engineering. However, when signal sources are highly correlated or coherent, conventional subspace-based DOA estimation algorithms will perform poorly due to the rank deficiency in the received data covariance matrix. Moreover, conventional DOA estimation algorithms are usually developed under Gaussian-distributed background noise, which will deteriorate significantly in impulsive noise environments. In this paper, a novel method is presented to estimate the DOA of coherent signals in impulsive noise environments. A novel correntropy-based generalized covariance (CEGC) operator is defined and proof of boundedness is given to ensure the effectiveness of the proposed method in impulsive noise environments. Furthermore, an improved Toeplitz approximation method combined CEGC operator is proposed to estimate the DOA of coherent sources. Compared to other existing algorithms, the proposed method can avoid array aperture loss and perform more effectively, even in cases of intense impulsive noise and low snapshot numbers. Finally, comprehensive Monte-Carlo simulations are performed to verify the superiority of the proposed method under various impulsive noise conditions. Full article

(This article belongs to the Special Issue Entropy and Information Theory in Machine Learning: Theoretical Insights and Applications)

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28 pages, 2220 KB

Open AccessFeature PaperArticle

A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions

by Antonio Boccuto, Ivan Gerace and Valentina Giorgetti

Appl. Sci. 2023, 13(10), 5861; https://doi.org/10.3390/app13105861 - 9 May 2023

Cited by 2 | Viewed by 1988

Abstract

This paper focuses on reducing the computational cost of a GNC Algorithm for deblurring images when dealing with full symmetric Toeplitz block matrices composed of Toeplitz blocks. Such a case is widespread in real cases when the PSF has a vast range. The analysis in this paper centers around the class of gamma matrices, which can perform vector multiplications quickly. The paper presents a theoretical and experimental analysis of how

γ

-matrices can accurately approximate symmetric Toeplitz matrices. The proposed approach involves adding a minimization step for a new approximation of the energy function to the GNC technique. Specifically, we replace the Toeplitz matrices found in the blocks of the blur operator with

γ

-matrices in this approximation. The experimental results demonstrate that the new GNC algorithm proposed in this paper reduces computation time by over

20 %

compared with its previous version. The image reconstruction quality, however, remains unchanged. Full article

(This article belongs to the Special Issue Signal and Image Processing: From Theory to Applications)

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25 pages, 893 KB

Open AccessArticle

A Critical Evaluation and Modification of the Padé–Laplace Method for Deconvolution of Viscoelastic Spectra

by Siamak Shams Es-haghi and Douglas J. Gardner

Molecules 2021, 26(16), 4838; https://doi.org/10.3390/molecules26164838 - 10 Aug 2021

Cited by 3 | Viewed by 3268

Abstract

This paper shows that using the Padé–Laplace (PL) method for deconvolution of multi-exponential functions (stress relaxation of polymers) can produce ill-conditioned systems of equations. Analysis of different sets of generated data points from known multi-exponential functions indicates that by increasing the level of Padé approximants, the condition number of a matrix whose entries are coefficients of a Taylor series in the Laplace space grows rapidly. When higher levels of Padé approximants need to be computed to achieve stable modes for separation of exponentials, the problem of generating matrices with large condition numbers becomes more pronounced. The analysis in this paper discusses the origin of ill-posedness of the PL method and it was shown that ill-posedness may be regularized by reconstructing the system of equations and using singular value decomposition (SVD) for computation of the Padé table. Moreover, it is shown that after regularization, the PL method can deconvolute the exponential decays even when the input parameter of the method is out of its optimal range. Full article

(This article belongs to the Special Issue Thermal and Rheological Characterization of Polymeric Materials)

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12 pages, 395 KB

Open AccessArticle

An Improved Direction Finding Algorithm Based on Toeplitz Approximation

by Qing Wang, Hua Chen, Guohuang Zhao, Bin Chen and Pichao Wang

Sensors 2013, 13(1), 746-757; https://doi.org/10.3390/s130100746 - 7 Jan 2013

Cited by 11 | Viewed by 6773

Abstract

In this paper, a novel direction of arrival (DOA) estimation algorithm called the Toeplitz fourth order cumulants multiple signal classification method (TFOC-MUSIC) algorithm is proposed through combining a fast MUSIC-like algorithm termed the modified fourth order cumulants MUSIC (MFOC-MUSIC) algorithm and Toeplitz approximation. In the proposed algorithm, the redundant information in the cumulants is removed. Besides, the computational complexity is reduced due to the decreased dimension of the fourth-order cumulants matrix, which is equal to the number of the virtual array elements. That is, the effective array aperture of a physical array remains unchanged. However, due to finite sampling snapshots, there exists an estimation error of the reduced-rank FOC matrix and thus the capacity of DOA estimation degrades. In order to improve the estimation performance, Toeplitz approximation is introduced to recover the Toeplitz structure of the reduced-dimension FOC matrix just like the ideal one which has the Toeplitz structure possessing optimal estimated results. The theoretical formulas of the proposed algorithm are derived, and the simulations results are presented. From the simulations, in comparison with the MFOC-MUSIC algorithm, it is concluded that the TFOC-MUSIC algorithm yields an excellent performance in both spatially-white noise and in spatially-color noise environments. Full article

(This article belongs to the Section Physical Sensors)

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