A training sample reduction strategy for data-driven surrogate modeling of wind-induced structural responses

In this paper, we developed and constructed the Haar wavelet (HW) for the two-dimensional (2D) time-fractional neuronal dynamics model (TFNDM) with the dynamical electro-diffusion behaviour of ions in nerve cells. The Haar wavelet method is considered in space and the difference method in time for the time-fractional Riemann–Liouville (TFRL) derivative. The calculation CPU time of this proposed method is very short because the Haar matrix and Haar integral matrix are stored only once and used for each iteration. Moreover, the results show that the solution of the Haar wavelet method is good even when there are fewer grid points.

Myocardial infarction (MI) is a life-threatening cardiovascular disorder caused by a partial or complete interruption of oxygenated blood flow to the myocardium, leading to high mortality rates if not diagnosed promptly. Although electrocardiogram (ECG) signals are widely used due to their non-invasive and low-cost nature, MI-specific abnormalities may be subtle and subject to inter-observer variability. Therefore, reliable artificial intelligence-based decision support systems are essential to enhance diagnostic classification accuracy. In this study, only the Lead II derivation from 12-lead ECG recordings of 52 healthy individuals and 148 MI patients was analyzed. To effectively characterize the non-stationary nature of ECG signals, a hybrid time-frequency feature extraction framework was employed. Five-level intrinsic mode functions and wavelet detail and approximation coefficients were obtained using Empirical Mode Decomposition and Discrete Wavelet Transform with a Daubechies-6 wavelet. From these components, 390 times, nonlinear and complexity-based features were extracted using 23 entropy-driven measures. Particle Swarm Optimization was applied to select the most discriminative feature subset, significantly enhancing classification performance. The optimized features were evaluated using Support Vector Machines, Artificial Neural Networks, k-Nearest Neighbors, and Bagged Tree classifiers. The Bagged Trees classifier achieved the best classification performance with an overall correct classification rate of 97.6%. The results demonstrate that the proposed hybrid feature representation combined with PSO-based selection provides a robust and reliable framework for MI detection, offering strong potential for clinical decision support applications.

Abstract With the rapid development of offshore photovoltaic (PV) systems towards higher voltages and larger capacities, the fire risk induced by DC arc faults has become a critical factor threatening platform safety. Compared to parallel arcs, series DC arc faults are considerably more challenging to detect in complex marine environments due to their extremely subtle characteristics. To address the unique safety challenges of offshore platforms, this paper proposes a DC series arc fault detection method based on wavelet transform. A Cassie arc model was established, a MATLAB/Simulink simulation model was constructed, and an experimental platform compliant with the UL1699B standard was built to collect authentic arc data. The proposed method innovatively employs the Bior4.4 wavelet base to perform multi-scale decomposition of the current signal. By combining this with a statistical analysis of the wavelet approximation coefficients, it ultimately achieves accurate and reliable detection of series DC arcs under complex marine operating conditions.

Multimodal Image Fusion (MMIF) aims to integrate complementary information from different imaging modalities to overcome the limitations of individual sensors. It enhances image quality and facilitates downstream applications such as remote sensing, medical diagnostics, and robotics. Despite significant advancements, current MMIF methods still face challenges such as modality misalignment, high-frequency detail destruction, and task-specific limitations. To address these challenges, we propose AdaSFFuse, a novel framework for task-generalized MMIF through adaptive cross-domain co-fusion learning. AdaSFFuse introduces two key innovations: the Adaptive Approximate Wavelet Transform (AdaWAT) for frequency decoupling, and the Spatial-Frequency Mamba Blocks for efficient multimodal fusion. AdaWAT adaptively separates the high- and low-frequency components of multimodal images from different scenes, enabling fine-grained extraction and alignment of distinct frequency characteristics for each modality. The Spatial-Frequency Mamba Blocks facilitate cross-domain fusion in both spatial and frequency domains, enhancing this process. These blocks dynamically adjust through learnable mappings to ensure robust fusion across diverse modalities. By combining these components, AdaSFFuse improves the alignment and integration of multimodal features, reduces frequency loss, and preserves critical details. Extensive experiments on four MMIF tasks-Infrared-Visible Image Fusion (IVF), Multi-Focus Image Fusion (MFF), Multi-Exposure Image Fusion (MEF), and Medical Image Fusion (MIF)-demonstrate AdaSFFuse's superior fusion performance, ensuring both low computational cost and a compact network, offering a strong balance between performance and efficiency. The code will be publicly available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/Zhen-yu-Liu/AdaSFFuse</uri>.

A filtering approach for speech emotion recognition using wavelet approximation coefficient

ABSTRACT This paper introduces a novel numerical methodology employing wavelet‐based ‐ discretization and its higher order convergence analysis to address the time‐fractional Black‐Scholes model within a jump‐diffusion framework, specifically in the context of pricing European options. The Haar wavelet approximation is applied to solve the resulting semi‐discrete problem, and the fractional time operator is effectively approximated through ‐ discretization. The theoretical estimates of the error bounds are provided. A detailed analysis demonstrates that the method yields a second‐order accuracy over the space‐time domain, depending on the solution's regularity as required for theoretical convergence. Moreover, the study explores the impact of fractional operators on option pricing through various test examples and applications based on Merton's as well as Kou's jump‐diffusion model. We rigorously compared our results with the spline‐based finite difference method and an implicit finite difference method, explicitly highlighting the efficacy and accuracy of our proposed method. In addition, the present research contributes not only to a robust numerical solution for time‐fractional Black‐Scholes models but also valuable insights into the influence of fractional operators on option pricing dynamics under jump‐diffusion.

$$(\beta ,\gamma )$$-Pseudo-Chebyshev Wavelets Approximation and Wavelets Coapproximation

General Background: Wavelet approximations are fundamental in numerical analysis and signal processing, with classical orthogonal polynomials like Jacobi and Chebyshev serving as key tools due to their strong approximation properties. Specific Background: The use of Chebyshev wavelets has been extended through generalized polynomial frameworks, such as Koornwinder’s generalization of Jacobi polynomials, offering more flexibility for function approximation on finite intervals. Knowledge Gap: Despite existing wavelet frameworks, the integration of generalized Jacobi and Chebyshev structures into a unified wavelet approximation scheme remains underexplored. Aims: This study introduces the Generalized Jacobi Chebyshev Wavelet (GJCW) approximation, establishing its theoretical foundations and demonstrating convergence and approximation capabilities. Results: It is shown that for a uniformly bounded function expanded in the GJCW basis, the partial sums yield both convergent and best uniform polynomial approximations. Novelty: The formulation of a new wavelet approximation based on a hybrid of generalized Jacobi and Chebyshev polynomials constitutes a novel contribution, supported by rigorous recurrence relations and multiresolution analysis. Implications: This work enhances the theoretical landscape of wavelet-based function approximation, with potential applications in computational mathematics, signal analysis, and numerical solutions of differential equations. Highlight : Wavelet Construction: The paper defines and constructs generalized Jacobi Chebyshev wavelets using orthogonal polynomials. Approximation Theory: It proves that if the wavelet series converges, then a uniform best polynomial approximation exists. Multiresolution Framework: The approach is grounded in Mallat’s multiresolution analysis, enabling efficient function approximation. Keywords : Jacobi Polynomials, Chebyshev Wavelets, Multiresolution Analysis, Polynomial Approximation, Orthonormal Basis

In this article, we propose the Haar wavelet operational matrix method (HWOMM) for the numerical analysis of a free convection vertical flat plate embedded in a saturated porous medium with thermal radiation. Also, we justify the convergence of our method. The Haar wavelet operational matrix of integration is used with the collocation technique to introduce the Haar wavelet approximation and obtain the approximate solution. Also, we have approximate mathematical models to demonstrate our major consequences. The governing equation of the suggested problem with appropriate boundary conditions can be reduced to the nonlinear boundary value problems over the infinite domain through the similarity transformations. The HWOMM is used to address the nonlinear boundary value problems with the help of Matlab software. The obtained results are compared with the existing numerical results, and the results are found to be more accurate, which confirms and verifies the HWOMM. The accuracy of the result increases when the level of resolution J increases. The solutions for the temperature exponent λ=0, 1/3 are selected, and the effects of the parameters of thermal radiation Rd , injection or suction Rd velocity or temperature gradient and shear stress profiles are established and deliberated graphically.

A new approach to the generalized Touchard wavelet approximation of fractional integro-differential equations with weakly singular kernels: Moduli of continuity and convergence

An error estimation of absolutely continuous signals and solution of Abel’s integral equation using the first kind pseudo-Chebyshev wavelet technique

This paper presents a novel computational approach to tackle challenges in approximation theory. The proposed method leverages pseudo-Chebyshev wavelet approximations, a concept introduced by Lal et al. in 2022, based on pseudo-Chebyshev functions. The paper provides a detailed description of the method, followed by an error analysis for a given function. Key results are illustrated through an example, highlighting the accuracy and efficiency of the pseudo-Chebyshev wavelet approximation technique. Fur-thermore, the paper derives error estimates for functions of bounded variation using pseudo-Chebyshev wavelets via orthogonal projection operators, demonstrating that these estimators are exceptionally precise and optimal in the context of wavelet analysis.

We introduce the Hölder width, which measures the best error performance of some recent nonlinear approximation methods, such as deep neural network approximation. Then, we investigate the relationship between Hölder widths and other widths, showing that some Hölder widths are essentially smaller than n-Kolmogorov widths and linear widths. We also prove that, as the Hölder constants grow with n, the Hölder widths are much smaller than the entropy numbers. The fact that Hölder widths are smaller than the known widths implies that the nonlinear approximation represented by deep neural networks can provide a better approximation order than other existing approximation methods, such as adaptive finite elements and n-term wavelet approximation. In particular, we show that Hölder widths for Sobolev and Besov classes, induced by deep neural networks, are O(n−2s/d) and are much smaller than other known widths and entropy numbers, which are O(n−s/d).

ABSTRACT This paper presents wavelet and other numerical techniques for solving systems of tumour-immune-vitamin intervention (TIV) model with non-integer order and nonlinear. For the first time, fractional order TIV system has been solved by using Bernstein wavelet collocation method (BWCM). This process uses wavelet approximations depend on Bernstein wavelets and their non-integer integral to transform non-integer differential systems into algebraic equation. This fractional TIV model will represent the effects of memory on the system. Three elements make up this partial sequence TIV system: vitamin intervention, immune cells, and tumour cells. Boundedness and non-negativity, residual error and convergence analysis of solutions have been determined using the iterative approach and BWCM. This investigation shows that the technique is very much useful and efficient for TIV intervention. Also, we have discussed the existence and uniqueness of the non-integer order TIV model. According to the findings, the TIV system would benefit from the use of the BWCM and this numerical method will open more research. Our findings will be valuable to biologists and researcher in the treatment of TIV.

The quantification of credit portfolio losses using the wavelet approach offers an innovative methodology for assessing the financial risks associated with credit.This approach uses advanced mathematical techniques to analyse temporal fluctuations in credit data.In terms of quantifying losses, the wavelet approach allows the decomposition of loss time series into different time scales.This makes it possible to identify short-and long-term trends as well as irregular variations.By analysing these scales, analysts can better understand the dynamics of credit losses and identify the underlying factors that contribute to fluctuations.To quantify credit portfolio losses, the cumulative loss function is approximated by a finite combination of wavelet basis functions by computing the coefficients of the wavelet approximation (WA).Wavelet approximation is an accurate, robust and fast method that enables VaR to be estimated much more quickly than with other loss quantification methods, such as the Monte Carlo MC method.

Approximation of one and two dimensional nonlinear generalized Benjamin-Bona-Mahony Burgers' equation with local fractional derivative

Analysis of nonlinear Burgers equation with time fractional Atangana-Baleanu-Caputo derivative

In this paper, we introduce two novel wavelet approximations tailored for functions exhibiting a restricted second derivative and a bounded derivative. Employing the Hermite wavelet method, we derive these approximations to address the need for effective representations of such functions. Our findings reveal that these new wavelet approximations offer enhanced accuracy and efficiency in capturing the underlying structure of , making them valuable tools in various applications requiring precise function approximations with limited derivative constraints.

Brightness Aware Pixel Stretching for Perceptually Invisible Images Using Wavelet Approximation Balancing