Search Results (89)

We develop a high-order compact numerical scheme for solving a nonlinear Black–Scholes equation arising in option pricing under transaction costs. By leveraging a Hermite-enhanced Radial Basis Function-Finite Difference (RBF-HFD) method with three-point stencils, we achieve fourth-order spatial accuracy. The fully nonlinear PDE, driven by Gamma-dependent volatility models, is discretized via RBF-HFD in space and integrated using an explicit sixth-order Runge–Kutta scheme. Numerical results confirm the proposed method’s accuracy, stability, and its capability to capture sharp gradient behavior near strike prices. Full article

Smoothing and interpolation of zero-field (Z) and infield (I) heating steps in Thellier-type paleointensity determinations have been tested. Paleomagnetic samples of different materials were artificially magnetized with an applied field of 50 µT. Six samples were measured following the standard double-heating Coe-variation experimental protocol, and the obtained results were used to test several mathematical functions to smooth the experimental data. The best smoothed results were obtained using a Five Parameters Logistic (5PL) function that resulted in field estimates of good quality, although not better than those obtained experimentally. Therefore, the smoothing of de- and remagnetization data appears unnecessary. In addition to smoothing, the tested functions can be used to interpolate additional Z and, indirectly, also I steps. Interpolation using cubic Hermite splines (without any smoothing) displays a better performance than interpolation (and smoothing) using the 5PL function. A new single-step heating method is presented, combining experimental and interpolated de- and remagnetization steps. The new method would not be applicable for retrieving reliable ancient field intensities on its own, but it could save measuring time under some circumstances. Full article

Data from earth observation satellites provide unique and valuable information about water quality conditions in freshwater lakes but require significant processing before they can be used, even with the use of tools like Google Earth Engine. We use imagery from Sentinel 2 and MODIS and in situ data from the State of Utah Ambient Water Quality Management System (AQWMS) database to develop models and to generate a highly accessible, easy-to-use CSV file of chlorophyll-a (which is an indicator of algal biomass), turbidity, and water temperature measurements on Utah Lake. From a collection of 937 Sentinel 2 images spanning the period from January 2019 to May 2025, we generated 262,081 estimates each of chlorophyll-a and turbidity, with an additional 1,140,777 data points interpolated from those estimates to provide a dataset with a consistent time step. From a collection of 2333 MODIS images spanning the same time period, we extracted 1,390,800 measurements each of daytime water surface temperature and nighttime water surface temperature and interpolated or imputed an additional 12,058 data points from those estimates. We interpolated the data using piecewise cubic Hermite interpolation polynomials to preserve the original distribution of the data and provide the most accurate estimates of measurements between observations. We demonstrate the processing steps required to extract usable, accurate estimates of these three water quality parameters from satellite imagery and format them for analysis. We include summary statistics and charts for the resulting dataset, which show the usefulness of this data for informing Utah Lake management issues. We include the Jupyter Notebook with the implemented processing steps and the formatted CSV file of data as ^{supplemental materials}. The Jupyter Notebook can be used to update the Utah Lake data or can be easily modified to generate similar data for other waterbodies. We provide this method, tool set, and data to make remotely sensed water quality data more accessible to researchers, water managers, and others interested in Utah Lake and to facilitate the use of satellite data for those interested in applying remote sensing techniques to other waterbodies. Full article

The cislunar space navigation satellite system is essential infrastructure for lunar exploration in the next phase. It relies on high-precision orbit determination to provide the reference of time and space. This paper focuses on constructing a navigation constellation using special orbital locations such as Earth–Moon libration points and distant retrograde orbits (DRO), and it discusses the simplification of planetary perturbation models for their autonomous orbit determination on board. The gravitational perturbations exerted by major solar system bodies on spacecraft are first analyzed. The minimum perturbation required to maintain a precision of 10 m during a 30-day orbit extrapolation is calculated, followed by a simulation analysis. The results indicate that considering only gravitational perturbations from the Moon, Sun, Venus, Saturn, and Jupiter is sufficient to maintain orbital prediction accuracy within 10 m over 30 days. Based on these findings, a method for simplifying the ephemeris is proposed, which employs Hermite interpolation for the positions of the Sun and Moon at fixed time intervals, replacing the traditional Chebyshev polynomial fitting used in the JPL DE ephemeris. Several simplified schemes with varying time intervals and orders are designed. The simulation results of the inter-satellite links show that, with a 6-day orbit arc length, a 1-day lunar interpolation interval, and a 5-day solar interpolation interval, the accuracy loss for cislunar space navigation satellites remains within the meter level, while memory usage is reduced by approximately 60%. Full article

Pupillometry is commonly used to evaluate cognitive effort, attention, and facial expression response, offering valuable insights into human performance. The combination of eye tracking and facial expression data under the iMotions platform provides great opportunities for multimodal research. However, there is a lack of standardized pipelines for managing pupillometry data on a multimodal platform. Preprocessing pupil data in multimodal platforms poses challenges like timestamp misalignment, missing data, and inconsistencies across multiple data sources. To address these challenges, the authors introduced a systematic preprocessing pipeline for pupil diameter measurements collected using iMotions 10 (version 10.1.38911.4) during an endoscopy simulation task. The pipeline involves artifact removal, outlier detection using advanced methods such as the Median Absolute Deviation (MAD) and Moving Average (MA) algorithm filtering, interpolation of missing data using the Piecewise Cubic Hermite Interpolating Polynomial (PCHIP), and mean pupil diameter calculation through linear regression, as well as normalization of mean pupil diameter and integration of the pupil diameter dataset with facial expression data. By following these steps, the pipeline enhances data quality, reduces noise, and facilitates the seamless integration of pupillometry other multimodal datasets. In conclusion, this pipeline provides a detailed and organized preprocessing method that improves data reliability while preserving important information for further analysis. Full article

Transfinite interpolation, originally proposed in the early 1970s as a global interpolation method, was first implemented using Lagrange polynomials and cubic Hermite splines. While initially developed for computer-aided geometric design (CAGD), the method also found application in global finite element analysis. With the advent of isogeometric analysis (IGA), Bernstein–Bézier polynomials have increasingly replaced Lagrange polynomials, particularly in conjunction with tensor product B-splines and non-uniform rational B-splines (NURBSs). Despite its early promise, transfinite interpolation has seen limited adoption in modern CAD/CAE workflows, primarily due to its mathematical complexity—especially when blending polynomials of different degrees. In this context, the present study revisits transfinite interpolation and demonstrates that, in four broad classes, Lagrange polynomials can be systematically replaced by Bernstein polynomials in a one-to-one manner, thus giving the same accuracy. In a fifth class, this replacement yields a robust dual set of basis functions with improved numerical properties. A key advantage of Bernstein polynomials lies in their natural compatibility with weighted formulations, enabling the accurate representation of conic sections and quadrics—scenarios where IGA methods are particularly effective. The proposed methodology is validated through its application to a boundary-value problem governed by the Laplace equation, as well as to the eigenvalue analysis of an acoustic cavity, thereby confirming its feasibility and accuracy. Full article

High-quality data are foundational to reliable environmental monitoring and urban planning in smart cities, yet challenges like missing values and outliers in air pollution and meteorological time series data are critical barriers. This study developed and validated a dual-phase framework to improve data quality using a 60-month gas and weather dataset from Jubail Industrial City, Saudi Arabia, an industrial region. First, outliers were identified via statistical methods like Interquartile Range and Z-Score. Machine learning algorithms like Isolation Forest and Local Outlier Factor were also used, chosen for their robustness to non-normal data distributions, significantly improving subsequent imputation accuracy. Second, missing values in both single and sequential gaps were imputed using linear interpolation, Piecewise Cubic Hermite Interpolating Polynomial (PCHIP), and Akima interpolation. Linear interpolation excelled for short gaps (R² up to 0.97), and PCHIP and Akima minimized errors in sequential gaps (R² up to 0.95, lowest MSE). By aligning methods with gap characteristics, the framework handles real-world data complexities, significantly improving time series consistency and reliability. This work demonstrates a significant improvement in data reliability, offering a replicable model for smart cities worldwide. Full article

To address the challenges of automatic detection caused by the variation of surface normal vectors in automotive steering knuckles, an automatic detection method based on ultrasonic phased array technology is herein proposed. First, a point cloud model of the workpiece was constructed using ultrasonic distance measurement, and Gaussian-weighted principal component analysis was used to estimate the normal vectors of the point cloud. By utilizing the normal vectors, water layer thickness during detection, and the incident angle of the sound beam, the probe pose information corresponding to the detection point was precisely calculated, ensuring the stability of the sound beam incident angle during the detection process. At the same time, in the trajectory planning process, piecewise cubic Hermite interpolation was used to optimize the detection trajectory, ensuring continuity during probe movement. Finally, an automatic detection system was set up to test a steering knuckle specimen with surface circumferential cracks. The results show that the point cloud data of the steering knuckle specimen, obtained using phased array ultrasound, had a relative measurement error controlled within 1.4%, and the error between the calculated probe angle and the theoretical angle did not exceed 0.5°. The probe trajectory derived from these data effectively improved the B-scan image quality during the automatic detection of the steering knuckle and increased the defect signal amplitude by 5.6 dB, demonstrating the effectiveness of this method in the automatic detection of automotive steering knuckles. Full article

The Finite-Difference Method (FDM) plays a pivotal role in the field of stability prediction, particularly in the modeling and stability analysis of cutting process dynamics. However, traditional approaches to optimizing the FDM often treat system state terms and time-delay terms as a monolithic entity, failing to explicitly distinguish between them, which leads to a lack of specificity in selecting optimization targets. In this study, an innovative approach is introduced by incorporating the third-order Newton interpolation method and the fourth-order Hermite interpolation method. By comparing the computational accuracy and convergence speed, it is found that the 3N-FDM (third-order Newton Finite-Difference Method) exhibits superior overall performance, and it is clearly pointed out that increasing the order does not always result in better outcomes. Additionally, this study selects different discretization numbers, denoted as m, for comparative analysis to thoroughly evaluate their impact on computational accuracy. Experimental validation demonstrates the high accuracy of the 3N-FDM. Through a one-way ANOVA (analysis of variance) of tool wear and workpiece surface roughness, it is revealed that changes in system state terms have the most significant impact on the feed rate $f$, followed by the cutting depth $a_p$, and finally the spindle speed $n$. Based on the experimental results and analysis mentioned above, this study concludes that optimizing system state terms can more effectively explore the combined influences of processing parameters on processing quality, production efficiency, and tool wear. Full article

This paper is devoted to numerical algorithms based on harmonic transformations with two goals: (1) face boundary formulation by blending techniques based on the known characteristic nodes and (2) some challenging examples of face resembling. The formulation of the face boundary is imperative for face recognition, transformation, and combination. Mapping between the source and target face boundaries with constituent pixels is explored by two approaches: cubic spline interpolation and ordinary differential equation (ODE) using Hermite interpolation. The ODE approach is more flexible and suitable for handling different boundary conditions, such as the clamped and simple support conditions. The intrinsic relations between the cubic spline and ODE methods are explored for different face boundaries, and their combinations are developed. Face combination and resembling are performed by employing blending curves for generating the face boundary, and face images are converted by numerical methods for harmonic models, such as the finite difference method (FDM), the finite element method (FEM) and the finite volume method (FVM) for harmonic models, and the splitting–integrating method (SIM) for the resampling of constituent pixels. For the second goal, the age effects of facial appearance are explored to discover that different ages of face images can be produced by integrating the photos and images of the old and the young. Then, the following challenging task is targeted. Based on the photos and images of parents and their children, can we obtain an integrated image to resemble his/her current image as closely as possible? Amazing examples of face combination and resembling are reported in this paper to give a positive answer. Furthermore, an optimal combination of face images of parents and their children in the least-squares sense is introduced to greatly facilitate face resembling. Face combination and resembling may also be used for plastic surgery, finding missing children, and identifying criminals. The boundary and numerical techniques of face images in this paper can be used not only for pattern recognition but also for face morphing, morphing attack detection (MAD), and computer animation as Sora to greatly enhance further developments in AI. Full article

A kind of finite difference Hermite WENO (HWENO) method is presented in this paper to deal with convection-dominated convection-diffusion equations in uniform grids. The benefit of the HWENO method is its compactness, allowing great accuracy to be attained in the solution’s smooth regions and maintaining the essential nonoscillation in the solution’s discontinuities. We discretize the convection term using the HWENO method and the diffusion term using the Hermite central interpolation schemes. However, it is difficult to deal with mixed derivative terms when solving two-dimensional problems using the HWENO method mentioned. To address this problem, we also employ the Hermite interpolation approach, which can keep the compactness. Lastly, we apply this method to two-dimensional Navier-Stokes problems that are incompressible. The efficiency and stability of the presented method are illustrated through numerous numerical experiments. Full article

We prove the inequalities of the weighted Hermite–Hadamard type the and Hermite–Hadamard–Mercer type for an extremely rich class of geometrically arithmetically-h-convex functions (GA-h-CFs) via generalized Hadamard–Fractional integral operators (HFIOs). The two generalized fractional integral operators (FIOs) are Hadamard proportional fractional integral operators (HPFIOs) and Hadamard k-fractional integral operators (HKFIOs). Moreover, we also present the results for subclasses of GA-h-CFs and show that the inequalities proved in this paper unify the results from the recent related literature. Furthermore, we compare the two generalizations in view of the fractional operator parameters that contribute to the generalizations of the results and assess the better approximation via graphical tools. Finally, we present applications of the new inequalities via HPFIOs and HKFIOs by establishing interpolation relations between arithmetic mean and geometric mean and by proving the new upper bounds for the Tsallis relative operator entropy. Full article

The trajectory data mining and analysis of maritime targets are of great significance in furthering the construction of maritime traffic facilities, improving the ability of marine supervision and maintaining national marine security. However, due to factors such as detection means and environmental interference, a large number of trajectory data have problems such as large space-time span, uneven sampling, and poor continuity, which seriously restrict the effect of trajectory mining. Therefore, this paper proposes a method of trajectory reconstruction based on navigation state recognition and bidirectional kinematic interpolation. The method mainly includes three steps: (1) data preprocessing, (2) navigation state recognition, and (3) trajectory interpolation. The method can recognize the navigation state of the targets in different segments, and then adaptively select the interpolation method to reconstruct the trajectories, that is, linear interpolation in the straight segments and bidirectional kinematic interpolation in the turning segments. Among them, bidirectional kinematic interpolation uses the cubic Hermite function to nonlinearly fit the acceleration of the interpolation section, and then calculates the velocity and coordinates of the interpolation points by time weighting from the positive and negative directions. The proposed method is verified and analyzed on the contest dataset of “Intelligent classification and recognition of XX trajectories”. Compared with the existing methods, the reconstruction results of the proposed method are closer to the real trajectories, and it can effectively reconstruct the target trajectories with better accuracy and stability. At the same time, the effect of trajectories classification based on the Long Short-Term Memory (LSTM) model, which uses trajectories before and after reconstruction, is compared and analyzed. The results show that the model has a higher classification accuracy for reconstructed trajectory, which proves the necessity of trajectory reconstruction. Full article

In this article, we present a novel three-step with-memory iterative method for solving nonlinear equations. We have improved the convergence order of a well-known optimal eighth-order iterative method by converting it into a with-memory version. The Hermite interpolating polynomial is utilized to compute a self-accelerating parameter that improves the convergence order. The proposed uni-parametric with-memory iterative method improves its R-order of convergence from 8 to $8.8989$. Additionally, no more function evaluations are required to achieve this improvement in convergence order. Furthermore, the efficiency index has increased from $1.6818$ to $1.7272$. The proposed method is shown to be more effective than some well-known existing methods, as shown by extensive numerical testing on a variety of problems. Full article

This paper proposes a new fault diagnosis model for wind power systems called residual convolution nested long short-term memory network with an attention mechanism (rlaNet). The method first preprocesses the SCADA data through feature engineering, uses the Hermite interpolation method to handle missing data, and uses the mutual information-based dimensionality reduction technique to improve data quality and eliminate redundant information. rlaNet combines residual networks and nested long short-term memory networks to replace traditional convolutional neural networks and standard long short-term memory architectures, thereby improving feature extraction and ensuring the abstractness and depth of the extracted features. In addition, the model emphasizes the weighted learning of spatiotemporal features in the input data, enhances the focus on key features, and improves training efficiency. Experimental results show that rlaNet achieves an accuracy of more than 90% in wind turbine fault diagnosis, showing good robustness. Furthermore, noise simulation experiments verify the model’s resistance to interference, providing a reliable solution for wind turbine fault diagnosis under complex operating conditions. Full article