Hermite Basis Functions Research Articles

A quadrilateral inverse plate element for real-time shape-sensing and structural health monitoring of thin plate structures

The inverse finite element method (iFEM) emerged as a powerful tool in shape-sensing and structural health monitoring (SHM) applications with distinct advantages over existing methodologies. In this study, a quadrilateral inverse-plate element is formulated via a sub-parametric approach using bi-linear and non-conforming cubic Hermite basis functions for engineering structures, which can be modeled as thin plates. Numerical validation involves dense and assumed sparse sensor arrangements for in-plane, out-of-plane, and mixed general loading conditions. iFEM analysis reveals efficient monotonic convergence to analytical and high-fidelity finite element reference solutions. After successful numerical validation, defect detection analysis is performed considering minute geometric discontinuities and structural stiffness reduction because of latent subsurface defects under tensile and transverse loading conditions. The inverse formulation successfully resolves the presence of simulated defects under a sparse sensor arrangement. The proposed inverse-plate element is accurate in the full-field reconstruction of shape-sensing profiles and reliable in defect identification and quantification in thin plate structures.

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

Weighted Hermite Variable Projection Networks for Classifying Visually Evoked Potentials.

The occipital cortex responds to visual stimuli regardless of a patient's level of consciousness or attention, offering a noninvasive diagnostic tool for both ophthalmologists and neurologists. This response signal manifests as a unique waveform referred to as the visually evoked potential (VEP), which can be extracted from the electroencephalogram (EEG) activity of a human being. We propose a trainable VEP representation to disentangle the underlying explanatory factors of the data. To enhance the learning process with domain knowledge, we present an innovative parameterization of classical Hermite functions that effectively captures VEP pattern variations arising from patient-specific factors, disorders, and measurement setup influences. Then, we introduce a differentiable variable projection (VP) layer to fuse Hermite basis function expansions (BFEs) of VEP signals with machine learning (ML) approaches. We prove the existence of an optimal set of parameters in the least-squares sense, assess the representation power of such layers, and calculate their analytical derivatives, which allows us to utilize backpropagation for training. Finally, we evaluate the effectiveness of the proposed learning framework in VEP-based color classification. To achieve this, we have designed a novel measurement system dedicated to intraoperative clinical use cases, which presents new ways for patient monitoring during neurosurgical procedures.

A high frequency stock price prediction model based on Hermite basis expansion and LSTM neural network

Based on Hermite basis function expansion and ensemble learning, an improved LSTM neural network method is proposed in this paper. The proposed method can be used for high frequency stock price prediction. Considering the characteristics of high frequency time series data such as high dimensionality, large noise and instability, etc. In this paper, the function information extracted from the Hermite basis function expansion is used to predict the residual sequence predicted by the LSTM neural network. Since the components of the function feature vector are unknown to the underlying model structure of the response variable, the proposed method is processed by Bagging framework. It not only captures the structure of the latent model, but also balances the variance and bias of the model. In addition, the number of prediction periods of the LSTM neural network is a hyperparameter, and the model averaging method based on distance covariance weighting is considered in this paper for optimization. The results of actual data analysis show that the proposed method can effectively optimize the prediction accuracy of the LSTM neural network, and has certain robustness. Finally, on the one hand, this optimization framework can be used to improve other time series prediction models. On the other hand, the proposed method can play an important role in forecasting problems such as daily average temperature prediction and real-time monitoring of atmospheric environmental quality.

Extraction of stress intensity factors of biharmonic equations with corner singularities corresponding to mixed boundary conditions of clamped, simply supported, and free (II)

In [13], we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ2u=f in a cracked domain whose crack faces have clamped boundary conditions. In this paper, we extend our investigation to the derivation of the SIF extraction formulas of the biharmonic equation in non-convex polygonal domains containing cracks or reentrant corners when various BC such as clamped (CC), simply supported (SS), free (FF), or mixed conditions (CS, FC, SF) are imposed on the boundary. We prove that the SIF is expressed as the integral of fΨsk⁎−uΔ2Ψsk⁎ for a cut-off function Ψ and a dual singular function sk⁎ on a small neighborhood of the singularity. The dual singular function is determined in this paper. For a numerical approximation of u, we proposed an iteration method as well as a direct finite element method. For a finite element solution of Δ2u=f, we have to use C1-continuous basis functions. For continuous differentiable basis functions, we use either C1-continuous B-spline basis functions or the conventional Hermite basis function. Because of several advantages of B-spline functions in imposing complex boundary conditions, we choose B-spline basis functions for the finite element approximation of u. Moreover, in the direct numerical method, we propose to use implicitly enriched basis functions that resemble the singularities.

A fourth order orthogonal spline collocation method Interface boundary value problem

Objective: A higher order numerical scheme for two-point boundary interface problem with Dirichlet and Neumann boundary condition on two different sides is propounded. Methods: Orthogonal cubic spline collocation techniques have been used (OSC) for the two-point interface boundary value problem. To approximate the solution a piecewise Hermite cubic basis functions have been used. Findings: Remarkable features of the OSC are accounted for the numerous applications, theoretical clarity, and convenient execution. The stability and efficiency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against finite element methods, determining the approximate solution and the coefficients of stiffness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method. Novelty: As against the existing methodologies it becomes clear from our findings that OSC is dominantly computationally superior. A computational treatment has been implemented on the two-point interface boundary value problem with super-convergent results of derivative at the nodal points, being the noteworthy finding of the study. Keywords: Helmholtz problem; Orthogonal spline collocation techniques (OSC); Discontinuous data; Super Convergence; Piecewise cubic Hermite basis functions; Almost block diagonal (ABD) structure

Stability analysis of high order methods for the wave equation

In this paper, we investigate the stability of a numerical method for solving the wave equation. The method uses explicit leap-frog in time and high order continuous and discontinuous (DG) finite elements using the standard Lagrange and Hermite basis functions in space. Matrix eigenvalue analysis is used to calculate time-step restrictions. We show that the time-step restriction for continuous Lagrange elements is independent of the nodal distribution, such as equidistributed Lagrange nodes and Gauss–Lobatto nodes. We show that the time-step restriction for the symmetric interior penalty DG schemes with the usual penalty terms is tighter than for continuous Lagrange finite elements. Finally, we conclude that the best time-step restriction is obtained for continuous Hermite finite elements up to polynomial degrees p=13.

An automatic arrhythmia classification model based on improved Marine Predators Algorithm and Convolutions Neural Networks

Preparation of Convolutional Neural Networks (CNNs) for classification purposes depends heavily on the knowledge of hyper-parameters tuning. This study aims, in particular in task of automated electrocardiograms (ECG), to minimize the user variability in the CNN training by searching and optimizing the CNN parameters automatically. In the clinical practice, the task of ECG classification analysis is restricted by existing models’ configuration. The hyper-parameters of the CNN model should be adjusted for the ECG classification problem. The best configuration for hyper-parameters is selected to have an impact on the production of the model. Deep knowledge of deep learning algorithms and suitable optimization techniques are also needed. Although there are many strategies for automated optimization, different benefits and disadvantages occur as they are applied to ECG classification problem. Here we present a CNN model for classification of non-ectopic (N), ventricular ectopic (V), supraventricular ectopic (S), and fusion (F) ECG rhythmias by the hybrid models based on modified version of Marine Predators algorithm (MPA) and CNN, known as the IMPA-CNN. The proposed model summarizes the feature extraction techniques of major features and, thus, outperforms other current classification models through automatically select the best hyper-parameters configuration of the CNN model. To reduce the time and complication complexity, optimum characteristics have been extracted directly from the raw signal using 1D-local binary pattern, higher order statistics, central moment, Hermite basis function discrete wavelet transform, and R–R intervals. Then, a modified version of MPA algorithm is used to select appropriate hyper-parameters for the CNN model like initial learning rate for the CNN model that is one of the major hyper parameters effect output performance, optimizer type which can be set to stochastic gradient descent (SGD), adaptive moment estimation (Adam), root mean square propagation (RMSprop), the activation function form used for modeling non-linear functions, set to ‘Rectified Linear Unit (ReLU), or ‘sigmoid’ and some other hyper-parameters are related to the optimization and training process of CNN model. Many available optimization algorithms for hyper-parameters optimization problems are provided. In addition, experiments with well know data sets like MIT-BIH arrhythmia, European ST-T database, and St. Petersburg INCART database are carried out to compare the performance of various optimization approaches and to provide practical illustration of the optimization of hyper-parameters for the proposed CNN model.

An Empirical Study of Moment Estimators for Quantile Approximation

We empirically evaluate lightweight moment estimators for the single-pass quantile approximation problem, including maximum entropy methods and orthogonal series with Fourier, Cosine, Legendre, Chebyshev and Hermite basis functions. We show how to apply stable summation formulas to offset numerical precision issues for higher-order moments, leading to reliable single-pass moment estimators up to order 15. Additionally, we provide an algorithm for GPU-accelerated quantile approximation based on parallel tree reduction. Experiments evaluate the accuracy and runtime of moment estimators against the state-of-the-art KLL quantile estimator on 14,072 real-world datasets drawn from the OpenML database. Our analysis highlights the effectiveness of variants of moment-based quantile approximation for highly space efficient summaries: their average performance using as few as five sample moments can approach the performance of a KLL sketch containing 500 elements. Experiments also illustrate the difficulty of applying the method reliably and showcases which moment-based approximations can be expected to fail or perform poorly.

Unit Heads Amidst COVID 19 Pandemic: Work-Life Balance, Talent Management Skills and Institutional Performance

Purpose: The study aims to determine the level of work-life balance, talent management skills, and institutional performance of Northern Iloilo Polytechnic State College (NIPSC) key officials during the COVID 19 pandemic. Methods: This study is descriptive-correlational research using purposive samplings. The 97 key officials from the 7 campuses were selected as respondents of this study. The researcher adopted a questionnaire focused on work balance, talent management skills, and institutional performance. To analyze the data, statistical tools like frequency count, percentage, mean, standard deviation for descriptive, and for inferential were t-test, one-way analysis of variance, and Pearson’s r was utilized. Results: The results revealed that the work-life balance of different units’ heads was ”Highly Balanced”. While the talent management skills were ”Highly Managed”. And, the level of key officials’ institutional performance was ”Outstanding.” Furthermore, the findings also indicate that there are no significant differences in the level of work-life balance, talent management skills, and institutional performance of school officials. The school key officials’ highly balanced lifestyle was able them to cater to the needs of their work and personal life in this catastrophe. Their experiences as key officials improved their management skills and impacted and influenced them to cope with the crisis. Novelty: Higher Education Institutions (HEIs) unit heads should know work-life balance for teachers and management skills to inspire colleagues for institutional performance in this virus crisis. This pandemic teaches key officials to become dynamic and visionary to manage the new educational policies by implementing different modalities and training for teachers. The four mandates such as instruction, research, extension services, and production of the college are prospering despite health protocols restrictions. Keywords: Northern Iloilo Polytechnic State College; key Officials; Pandemic; WorkLife Balance; Management Skills; Institutional Performance 1 Introductionbjective: A higher order numerical scheme for two-point boundary interface problem with Dirichlet and Neumann boundary condition on two different sides is propounded. Methods: Orthogonal cubic spline collocation techniques have been used (OSC) for the two-point interface boundary value problem. To approximate the solution a piecewise Hermite cubic basis functions have been used. Findings: Remarkable features of the OSC are accounted for the numerous applications, theoretical clarity, and convenient execution. The stability and efficiency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against finite element methods, determining the approximate solution and the coefficients of stiffness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method. Novelty: As against the existing methodologies it becomes clear from our findings that OSC is dominantly computationally superior. A computational treatment has been implemented on the two-point interface boundary value problem with super-convergent results of derivative at the nodal points, being the noteworthy finding of the study. Keywords: Helmholtz problem; Orthogonal spline collocation techniques (OSC); Discontinuous data; Super Convergence; Piecewise cubic Hermite basis functions; Almost block diagonal (ABD) structure

Evaluation of numerical integration by using Hermite wavelets

In this research, we present a numerical scheme to evaluate the numerical integration of a given function using Hermite wavelets. The proposed technique is based on the expansion of the given function into a series of Hermite wavelets basis functions. Some numerical experiments have been performed to illustrate the accuracy of the proposed method.

Wrinkle synthesis for cloth mesh with hermite radial basis functions

Designing virtual clothing has received much attention recently due to the increasing need for synthesizing realistically dressed digital humans for various applications. Wrinkles are an important appearance feature of the garment in virtual environments. Generating such wrinkles currently requires a computationally expensive simulation or specialized design skills. We present a new geometric method for adding believable wrinkles to existing virtual clothing. The key novelty of our work is to use Hermite Radial Based Functions (HRBF) to reconstruct an approximation of the clothing mesh. Our method takes advantage of angles between gradients of adjacent HRBF scalar fields to trace spatially and temporally coherent wrinkle curves on a cloth mesh. We generate plausible wrinkle geometry using implicit deformers following the wrinkle curves. Our method can be used as a post-processing step on any garment simulation system. The results obtained demonstrate that our approach produces believable wrinkles and it is very satisfactory in terms of performance. The method is fully automatic and provides a large set of parameters that can be modified by the user in order to control the appearance of the resulting wrinkles in real-time and thus obtain the desired result.

A Finite Element Versus Analytical Approach to the Solution of the Current Diffusion Equation in Tokamaks

This paper deals with two efficient approaches for solving the current diffusion equation (CDE), which governs current diffusion through the conductive plasma inside a tokamak and compares them to CRONOS tokamak simulation suite, as well. Namely, CDE is solved via the finite-element method (FEM) and an analytical technique, respectively, and the obtained results are subsequently compared with the solution obtained from the state-of-the-art CRONOS suite with finite-difference calculations. The FEM solution is carried out featuring the use of linear and Hermite type shape functions, respectively, while the analytical solution is obtained by applying certain approximations to the CDE. The tradeoff between different approaches has been undertaken. Thus, the results obtained via the FEM approach (with Hermite basis function, in particular) show very good agreement with the CRONOS results, while also providing the stability of the solution. On the other hand, the results obtained via the analytical solution clearly demonstrate a good agreement with the numerical results in the edge region, which makes it very useful for various applications, e.g., for benchmarking purposes.

ECG-derived respiration using Hermite expansion

Abstract This paper presents a new ECG-derived respiration (EDR) technique from single-lead ECG based on the decomposition of each QRS complex of an ECG using the Hermite basis functions. We hypothesize that the respiration process, which is a quasi-periodic process, affects the energy in the QRS complexes of ECG signal and the distribution of energy along each component in the orthogonal signal expansion of QRS complex using the Hermite basis functions also. Respiratory-induced beat-to-beat variations in the QRS complexes of ECG signal are monitored by computing the energy and the standard deviation of the Hermite coefficients for deriving the respiratory signal. The performance of the proposed EDR technique is assessed over the Fantasia data by computing the correlation coefficient and respiratory rate errors between the EDR and the recorded respiration. The performance results of the proposed technique are compared with some of the well-known EDR techniques based on the principal component analysis (PCA), R-peak amplitudes (RPA), respiratory sinus arrhythmia (RSA), slopes of the QRS complex, and R-wave angle. Results demonstrate that the overall performance of the proposed EDR technique is better than the existing methods. Upon investigating the young and elderly subjects separately, the proposed method provided effective performance for the young subjects but outperformed the existing methods in the elderly subjects. The performance results confirm that the proposed EDR technique can be utilized for continuous monitoring of the respiration using single-lead ECG.

Computation of Hermite interpolation in terms of B-spline basis using polar forms

The aim of this paper is to present an Hermite interpolation problem with B-splines of high degree of smoothness. More precisely, we use polar forms to find the B-spline control points for Hermite interpolation. The resulting formula is used to give Hermite basis functions. In particular, Quadratic C1 and cubic C2 interpolations with sharp parameters are analyzed.

An algorithm for sleep apnea detection from single-lead ECG using Hermite basis functions

This paper introduces a methodology for the detection of sleep apnea based on single-lead electrocardiogram (ECG) of the patient. In the proposed technique, each QRS complex of the ECG signal is approximated using a linear combination of the lower order Hermite basis functions. The coefficients of the Hermite expansion are then used to discriminate the apnea and normal segments along with three features based on R–R time series (mean of R–R intervals, the standard deviation of R–R intervals) and energy in the error of the QRS approximation. To perform classification between the apnea and normal segments, four different types of classifiers (K-nearest neighbor (KNN), multilayer perceptron neural network (MLPNN), support vector machine (SVM), and least-square support vector machine (LS-SVM)) are used in this work.In total, 70 ECG recordings from Apnea-ECG dataset are used in this study and the performance of the proposed algorithm is evaluated based on the minute-by-minute (per-segment) classification, and per-recording (where the entire ECG recording of a subject is discriminated as the apnea or normal one) classification. By considering the events of apnea and hypopnea together, an accuracy of about 84% is achieved on the minute-by-minute basis classification using the LS-SVM classifier with the Gaussian radial basis function (RBF) kernel.On the other hand, an accuracy of about 97.14% is achieved for per-recording classification using the SVM, and LS-SVM classifiers. From the results, it is observed that the proposed methodology provides comparable accuracy with the methods existing in the literature at reduced computational cost due to the lesser number of features selected for the classification.

Isogeometric configuration design sensitivity analysis of finite deformation curved beam structures using Jaumann strain formulation

Using an isogeometric approach, a continuum-based configuration design sensitivity analysis (DSA) method is developed for curved Kirchhoff beams with multi-patch junctions. Under the total Lagrangian formulation, large deformations considering the initial curvature of curved beams are described by geometrically exact beam theory (GEBT) and Jaumann strain formulation. In the isogeometric approach, the higher order continuity and the exact description of initial geometry are naturally embedded using NURBS basis functions. In multi-patch models, C0-continuity of physical displacement or C1-continuity of displacement component at junction is weakly imposed using the Lagrange multiplier method. The superior accuracy of isogeometric analysis (IGA) is verified through the comparison with the results of finite element analysis (FEA) using cubic Hermite interpolation. In the DSA, a material derivative is utilized and the kinematical description of GEBT is consistently employed to express orientation design variations. Contrary to the IGA-based DSA, the Hermite basis function explicitly depends on design in the FEA-based DSA due to its element length parameter. Moreover, since the design velocity field is approximated using the nodal velocity imposed at nodal tangential vector, the amount of design perturbations should be very small to obtain precise design sensitivity.

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

A spectral method for the numerical solution of the multi-dimensional Vlasov–Maxwell equations is presented. The plasma distribution function is expanded in Fourier (for the spatial part) and Hermite (for the velocity part) basis functions, leading to a truncated system of ordinary differential equations for the expansion coefficients (moments) that is discretized with an implicit, second order accurate Crank–Nicolson time discretization. The discrete non-linear system is solved with a preconditioned Jacobian–Free Newton–Krylov method.It is shown analytically that the Fourier–Hermite method features exact conservation laws for total mass, momentum and energy in discrete form. Standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically. The whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities. Some preconditioning strategies are presented, showing that the number of linear iterations of the Krylov solver can be drastically reduced and a significant gain in performance can be obtained.

Multiple neural network integration using a binary decision tree to improve the ECG signal recognition accuracy

Abstract The paper presents a new system for ECG (ElectroCardioGraphy) signal recognition using different neural classifiers and a binary decision tree to provide one more processing stage to give the final recognition result. As the base classifiers, the three classical neural models, i.e., the MLP (Multi Layer Perceptron), modified TSK (Takagi-Sugeno-Kang) and the SVM (Support Vector Machine), will be applied. The coefficients in ECG signal decomposition using Hermite basis functions and the peak-to-peak periods of the ECG signals will be used as features for the classifiers. Numerical experiments will be performed for the recognition of different types of arrhythmia in the ECG signals taken from the MIT-BIH (Massachusetts Institute of Technology and Boston’s Beth Israel Hospital) Arrhythmia Database. The results will be compared with individual base classifiers’ performances and with other integration methods to show the high quality of the proposed solution

A new spectral method for numerical solution of the unbounded rough surface scattering problem

A new spectral method is developed to solve the unbounded rough surface scattering problem. An unbounded rough surface is referred to as a non-local perturbation of an infinite plane surface such that the whole rough surface lies within a finite distance of the original plane. The method uses a transformed field expansion to reduce the boundary value problem with a complex scattering surface into a successive sequence of transmission problems of a planar surface. Hermite orthonormal basis functions are adopted to further simplify these problems to fully decoupled one-dimensional two-point boundary value problems, which are solved efficiently by the Legendre–Galerkin method. Numerical results indicate that the method is efficient, accurate, and well-suited for solving the scattering problem by unbounded rough surfaces.