Search Results (38)

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 $\tau$ 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

Synthetic Aperture Radar (SAR) is a remote sensing technology that can realize all-weather and all-day monitoring, and it is widely used in ocean ship monitoring tasks. Recently, many oriented detectors were used for ship detection in SAR images. However, these methods often found it difficult to balance the detection accuracy and speed, and the noise around the target in the inshore scene of SAR images led to a poor detection network performance. In addition, the rotation representation still has the problem of boundary discontinuity. To address these issues, we propose S4Det, a Sinusoidal Single-Stage SAR image detection method that enables real-time oriented ship target detection. Two key mechanisms were designed to address inshore scene processing and angle regression challenges. Specifically, a Breadth Search Compensation Module (BSCM) resolved the limited detection capability issue observed within inshore scenarios. Neural Discrete Codebook Learning was strategically integrated with Multi-scale Large Kernel Attention, capturing context information around the target and mitigating the information loss inherent in dilated convolutions. To tackle boundary discontinuity arising from the periodic nature of the target regression angle, we developed a Sine Fourier Transform Coding (SFTC) technique. The angle is represented using diverse sine components, and the discrete Fourier transform is applied to convert these periodic components to the frequency domain for processing. Finally, the experimental results of our S4Det on the RSSDD dataset achieved 92.2% mAP and 31+ FPS on an RTXA5000 GPU, which outperformed the prevalent mainstream of the oriented detection network. The robustness of the proposed S4Det was also verified on another public RSDD dataset. Full article

Intelligent logistical systems are crucial for adapting to technological advancements and global supply chains, particularly at seaports. Automation can maximize port efficiency and adapt to changing circumstances, but port digitalisation is challenging due to the various parties and information flows involved. The port of Sines in Portugal is undergoing a digital transformation, specifically about the Smart Gate concept. The port administration and partners have developed a pre-gate, which is being examined for operations, technologies, and information models. This work uses simulation to analyse the pre-gate model dynamically. The discrete-event simulation model, using Anylogic software (version 8.9.0), forecasts possible problems and predicts pre-gate behaviour, facilitating ongoing enhancement of pre-gate procedures. The considered scenarios vary in two factors: the processing time at the bottleneck process and the number of active lanes at the same point. Four of the twenty tested alternatives were identified as balanced. Results allow drawing conclusions on the number of lanes to be open to prevent congestion, particularly when processing times increase. The study highlights the benefits of simulating complex systems to improve operations. Future work could involve adjusting parameters, incorporating advanced optimisation techniques, and expanding evaluated metrics. The ultimate goal is to develop a reliable digital twin for the port. Full article

The subject of this paper is the development of rationalized algorithms of discrete sinusoidal transform of type I for short sequences of length N = 2, 3, 4, 5, 6, 7, and 8. Here, by the word “rationalization”, we mean the reduction of the number of arithmetic operations required to implement the algorithms. The arithmetic complexity of the developed algorithms is presented in the final table. For each algorithm, we also provide data flow graphs demonstrating the space–time structure of the computational processes. The algorithms were tested to verify their validity using MATLAB software (version R2023). Full article

This paper presents an innovative numerical technique for specific classes of stochastic heat equations. Our approach uniquely combines a sixth-order compact finite difference algorithm with fast discrete Fourier transforms. While traditional discrete sine transforms are effective for approximating second-order derivatives, they are inadequate for first-order derivatives. To address this limitation, we introduce an innovative variant based on exponential transforms. This method is rigorously evaluated on two forms of stochastic heat equations, and the solutions are compared with those obtained using the established stochastic ten non-polynomial cubic-spline method. The results confirm the accuracy and applicability of our proposed method, highlighting its potential to enhance the numerical treatment of stochastic heat equations. Full article

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media. Full article

In this study, the energy control and asymptotic stability of the 1D sine-Gordon equation were investigated from the viewpoint of numerical approximation. An order reduction method was employed to transform the closed-loop system into an equivalent system, and an average-central finite difference scheme was constructed. This scheme is not only energy-preserving but also possesses uniform stability. The discrete multiplier method was utilized to obtain the uniformly asymptotic stability of the discrete systems. Moreover, to cope with the nonlinear term of the model, a discrete Wirtinger inequality suitable for our approximating scheme was established. Finally, several numerical experiments based on the eigenvalue distribution of the linearized approximation systems were conducted to demonstrate the effectiveness of the numerical approximating algorithm. Full article

The subject of this work is the development of fast algorithms for the discrete sinusoidal transformation of the second type (DST-II) for sequences of input data of small length N = 2, 3, 4, 5, 6, 7, 8. The starting point for the development of algorithms is the well-known possibility of representing any discrete transformation in the form of a matrix–vector product. Due to the remarkable structural properties of the matrices of the DST-II transformation base, these matrices can be successfully factorized, which should lead to a reduction in the computational complexity of the procedure as a whole. You can factorize matrices in different ways. The art of designing fast algorithms is to find the factorization that produces the maximum effect. We justified the correctness of the obtained algorithmic solutions theoretically, using strict mathematical derivations of each of them. The developed algorithms were then further tested using MATLAB R2023b software to finally confirm their performance. Finally, we presented estimates of the computational complexity for each solution obtained and compared them with direct computational methods that rely on the direct calculation of matrix–vector products. Full article

Transformer has emerged as one of the modern neural networks that has been applied in numerous applications. However, transformers’ large and deep architecture makes them computationally and memory-intensive. In this paper, we propose the block g-circulant matrices to replace the dense weight matrices in the feedforward layers of the transformer and leverage the DCT-DST algorithm to multiply these matrices with the input vector. Our test using Portuguese-English datasets shows that the suggested method improves model memory efficiency compared to the dense transformer but at the cost of a slight drop in accuracy. We found that the model Dense-block 1-circulant DCT-DST of 128 dimensions achieved the highest model memory efficiency at 22.14%. We further show that the same model achieved a BLEU score of 26.47%. Full article

The classification of time series using machine learning (ML) analysis and entropy-based features is an urgent task for the study of nonlinear signals in the fields of finance, biology and medicine, including EEG analysis and Brain–Computer Interfacing. As several entropy measures exist, the problem is assessing the effectiveness of entropies used as features for the ML classification of nonlinear dynamics of time series. We propose a method, called global efficiency (GEFMCC), for assessing the effectiveness of entropy features using several chaotic mappings. GEFMCC is a fitness function for optimizing the type and parameters of entropies for time series classification problems. We analyze fuzzy entropy (FuzzyEn) and neural network entropy (NNetEn) for four discrete mappings, the logistic map, the sine map, the Planck map, and the two-memristor-based map, with a base length time series of 300 elements. FuzzyEn has greater GEFMCC in the classification task compared to NNetEn. However, NNetEn classification efficiency is higher than FuzzyEn for some local areas of the time series dynamics. The results of using horizontal visibility graphs (HVG) instead of the raw time series demonstrate the GEFMCC decrease after HVG time series transformation. However, the GEFMCC increases after applying the HVG for some local areas of time series dynamics. The scientific community can use the results to explore the efficiency of the entropy-based classification of time series in “The Entropy Universe”. An implementation of the algorithms in Python is presented. Full article

This paper investigates the distribution characteristics of Fourier, discrete cosine, and discrete sine transform coefficients in T1 MRI images. This paper reveals their adherence to Benford’s law, characterized by a logarithmic distribution of first digits. The impact of Rician noise on the first digit distribution is examined, which causes deviations from the ideal distribution. A novel methodology is proposed for noise level estimation, employing metrics such as the Bhattacharyya distance, Kullback–Leibler divergence, total variation distance, Hellinger distance, and Jensen–Shannon divergence. Supervised learning techniques utilize these metrics as regressors. Evaluations on MRI scans from several datasets coming from a wide range of different acquisition devices of 1.5 T and 3 T, comprising hundreds of patients, validate the adherence of noiseless T1 MRI frequency domain coefficients to Benford’s law. Through rigorous experimentation, our methodology has demonstrated competitiveness with established noise estimation techniques, even surpassing them in numerous conducted experiments. This research empirically supports the application of Benford’s law in transforms, offering a reliable approach for noise estimation in denoising algorithms and advancing image quality assessment. Full article

Short-term power load forecasting refers to the use of load and weather information to forecast the Day-ahead load, which is very important for power dispatch and the establishment of the power spot market. In this manuscript, a comprehensive study on the frame of input data for electricity load forecasting is proposed based on the extreme gradient boosting algorithm. Periodicity was the first of the historical load data to be analyzed using discrete Fourier transform, autocorrelation function, and partial autocorrelation function to determine the key width of a sliding window for an optimization load feature. The mean absolute error (MAE) of the frame reached 52.04 using a boosting model with a 7-day width in the validation dataset. Second, the fusing of datetime variables and meteorological information factors was discussed in detail and determined how to best improve performance. The datetime variables were determined as a form of integer, sine–cosine pairs, and Boolean-type combinations, and the meteorological features were determined as a combination with 540 features from 15 sampled sites, which further decreased MAE to 44.32 in the validation dataset. Last, a training method for day-ahead forecasting was proposed to combine the Minkowski distance to determine the historical span. Under this framework, the performance has been significantly improved without any tuning for the boosting algorithm. The proposed method further decreased MAE to 37.84. Finally, the effectiveness of the proposed method is evaluated using a 200-day load dataset from the Estonian grid. The achieved MAE of 41.69 outperforms other baseline models, with MAE ranging from 65.03 to 104.05. This represents a significant improvement of 35.89% over the method currently employed by the European Network of Transmission System Operators for Electricity (ENTSO-E). The robustness of the proposal method can be also guaranteed with excellent performance in extreme weather and on special days. Full article

Discrete cosine and sine transforms closely approximate the Karhunen–Loeve transform for first-order Markov stationary signals with high and low correlation coefficients, respectively. Discrete sinusoidal transforms can be used in data compression, digital filtering, spectral analysis and pattern recognition. Short-time transforms based on discrete sinusoidal transforms are suitable for the adaptive processing and time–frequency analysis of quasi-stationary data. The generalized sliding discrete transform is a type of short-time transform, that is, a fixed-length windowed transform that slides over a signal with an arbitrary integer step. In this paper, eight fast algorithms for calculating various sliding sinusoidal transforms based on a generalized solution of a second-order linear nonhomogeneous difference equation and pruned discrete sine transforms are proposed. The performances of the algorithms in terms of computational complexity and execution time were compared with those of recursive sliding and fast discrete sinusoidal algorithms. The low complexity of the proposed algorithms resulted in significant time savings. Full article

In this paper, we present two systolic array algorithms for efficient Very-Large-Scale Integration (VLSI) implementations of the 1-D Modified Discrete Sine Transform (MDST) using the systolic array architectural paradigm. The new algorithms decompose the computation of the MDST into modular and regular computational structures called pseudo-circular correlation and pseudo-cycle convolution. The two computational structures for pseudo-circular correlation and pseudo-cycle convolution both have the same form. This feature can be exploited to significantly reduce the hardware complexity since the two computational structures can be computed on the same linear systolic array. Moreover, the second algorithm can be used to further reduce the hardware complexity by replacing the general multipliers from the first one with multipliers with a constant that have a significantly reduced complexity. The resulting VLSI architectures have all the advantages of a cycle convolution and circular correlation based systolic implementations, such as high-speed using concurrency, an efficient use of the VLSI technology due to its local and regular interconnection topology, and low I/O cost. Moreover, in both architectures, a cost-effective application of an obfuscation technique can be achieved with low overheads. Full article

This article addresses the telecommunications industry’s priority of ensuring information security during the transition to next-generation networks. It proposes an image encryption system that combines watermarking techniques and a discrete fractional sine chaotic map. The authors also incorporate the principles of blockchain to enhance the security of transmitted and received image data. The proposed system utilizes a newly developed sine chaotic map with a fractional difference operator, exhibiting long-term chaotic dynamics. The complexity of this map is demonstrated by comparing it with three other fractional chaotic maps from existing literature, using bifurcation diagrams and the largest Lyapunov exponent. The authors also show the map’s sensitivity to changes in initial conditions through time-series diagrams. To encrypt images, the authors suggest a method involving watermarking of two secret images and encryption based on blockchain technology. The cover image is watermarked with the two hidden images using discrete wavelet transformations. Then, the image pixels undergo diffusion using a chaotic matrix generated from the discrete fractional sine chaotic map. This encryption process aims to protect the image data and make it resistant to unauthorized access. To evaluate the algorithm, the authors perform statistical analysis and critical sensitivity analysis to examine its characteristics. They also analyse different attacks to assess the algorithm’s ability to resist such threats and maintain image quality after decryption. The results demonstrate that the proposed algorithm effectively defends against attacks and ensures image security. Full article