Hermite Interpolation Research Articles

Numerical evaluation and analysis of highly oscillatory singular Bessel transforms with a particular oscillator

This paper proposes and analyzes two affordable and efficient quadrature rules for the numerical approximation of the oscillatory Bessel transform ∫0bxα(b−x)βf(x)Jν(ωxγ)dx with algebraic singularities, where b,α,β,ν,ω,γ denote the given constants. Firstly, we derive the explicit formula and asymptotic estimation of the generalized moments ∫0bxα′(b−x)β′Jν(ωxγ)dx with α′,β′>−1 by means of the Meijer G function. Furthermore, we design a modified Filon-type method based on a two-endpoint Taylor interpolation polynomial. In particular, we also give a more efficient Clenshaw–Curtis–Filon-type method in view of a special Hermite interpolation polynomial at the Clenshaw–Curtis points. Moreover, this method is easily implemented by the fast Fourier transform and fast computation of the modified moments. The useful homogeneous recurrence relation of the required modified moments is derived by the Bessel equation and the properties of the Chebyshev polynomial. Importantly, the rigorous error analyses in inverse powers of ω for the proposed numerical methods are carried out in details. Some primary numerical experiments can confirm our theoretical analysis, and verify the accuracy and efficiency of the proposed numerical methods.

Shishkin mesh based septic Hermite interpolation algorithm for time-dependent singularly perturbed convection–diffusion models

Shishkin mesh based septic Hermite interpolation algorithm for time-dependent singularly perturbed convection–diffusion models

Approximate Hermite Interpolations for Compactly Supported Linear Canonical Transforms

There has been several Lagrange and Hermite type interpolations of entire functions whose linear canonical transforms have compact supports inℝ. There interpolation representations are called sampling theorems for band-limited signals in signal analysis. The truncation, amplitude, and jitter errors associated with the Lagrange type interpolations are investigated rigorously. Nevertheless, the amplitude and jitter errors arising from perturbing samples and nodes are not studied before. The aim of this work is to establish rigorous analysis of their types of perturbation errors, which is important from both practical and theoretical points of view. We derive precise estimates for both types of errors and expose various numerical examples.

Free vibration analysis of rectangular thin plates with corner and inner cutouts using C1 Chebyshev spectral element method

This paper presents a novel higher-order C1 continuous Chebyshev spectral element method for the free vibration analysis of rectangular thin plates with corner and inner cutouts. To achieve C1 continuity at the interfaces of adjacent elements, the Hermite interpolation functions through nonuniformly distributed Chebyshev nodes are employed to construct the element displacement field. A Gauss–Lobatto type quadrature is used to efficiently evaluate the element matrices. Comparison with other well-established methods show that the proposed element has good convergence and accuracy. The influence of cutout size, location and eccentric ratio on the natural vibration characteristics of plates are investigated.

On the intersection of left and right ideals in the ring of quaternion polynomials

Given a left ideal [Formula: see text] and a right ideal [Formula: see text] in the ring of univariate quaternion polynomials, we present an explicit formula parametrizing the intersection [Formula: see text]. The motivation comes from interpolation theory where any such intersection appears to be the solution set of a two-sided homogeneous Lagrange–Hermite interpolation problem.

Hermite type radial basis function-based differential quadrature approach allows for free vibration beams for higher order equations

This paper's goal is to build a meshless radial basis function based on the differential quadrature approach that makes use of Hermite interpolation. This method approximates by adding a linear weighted sum of the sought function values defined at scattered nodes, one can calculate derivatives in a governing equation. On other hand the finite element analysis of mesh size is critical because the size of the mesh is closely related to the accuracy and number of mesh required for the meshing of the beam element. The stability of two created algorithms is discussed. The effectiveness of the algorithms is tested numerically, and cantilever beam of various reinforcing behaviours are provided. The proposed methods are discovered to be precise, straightforward, and quick, in order to extract free vibration with various boundary conditions and compare the natural frequency results, the method is applied to a cantilever beam with carbon glass fibre with varied bombyx mori reinforcement (10%, 20%, and 30%) as a numerical test.

Accurate total-body Ki parametric imaging with shortened dynamic 18 F-FDG PET scan durations via effective data processing.

Total-body dynamic positron emission tomography (dPET) imaging using 18 F-fluorodeoxyglucose(18 F-FDG) has received widespread attention in clinical oncology. However, the conventionally required scan duration of approximately 1 h seriously limits the application and promotion of this imaging technique. In this study, we investigated the possibility and feasibility of shortening the total-body dynamic scan duration to 30min post-injection (PI) with the help of a novel Patlak data processing algorithm for accurate Ki estimations of tumor lesions. Total-body dPET images acquired by uEXPLORER (United Imaging Healthcare Inc.) using 18 F-FDG of 15 patients with different tumor types were analyzed in this study. Dynamic images were reconstructed into 25 frames with a specific temporal dividing protocol for the scan data acquired 1 h PI. Patlak analysis-based Ki parametric imaging was conducted based on the imaging data corresponding to the first 30min PI, during which a Patlak data processing method based on cubic Hermite interpolation was applied. The resultant Ki images acquired by 30min dynamic PET data and the standard 1 h Ki images were compared in terms of visual imaging effect, region signal-to-noise ratio, and Ki estimation accuracy to evaluate the performance of the proposed Ki imaging method with a shortened scan duration. With the help of Patlak data processing, acceptable Ki parametric images were obtained from dynamic PET data acquired with a scan duration of 30min PI. Compared with Ki images obtained from unprocessed Patlak data, the resulting images from the proposed method performed better in terms of noise reduction. Moreover, Bland-Altman plot and Pearson correlation coefficient analysis showed that that 30min Ki images obtained from the processed Patlak data had higher accuracy for tumor lesions. Satisfactory Ki parametric images with high tumor accuracy can be acquired from dynamic imaging data corresponding to the first 30min PI. Patlak data processing can help achieve higher Ki imaging quality and higher accuracy regarding tumor lesion Ki values. Clinically, it is possible to shorten the dynamic scan duration of 18 F-FDG PET to 30min to acquire an accurate tumor Ki and further effective tumor detection with uEXPLORER scanners.

An explicit representation of the three-point Hermite interpolant for the numerical solution of singular boundary value problems

This study considers three-point Hermite interpolation as an approximation method that utilizes the values of a function and its derivatives at the two endpoints and another point of the domain to reconstruct the function. Therefore, it can also be considered as a three-point Taylor expansion. In the process of conducting the research, a novel and explicit form of the three-point Hermite interpolant is presented. Then, this interpolant is applied as the basis of a numerical method to approximate the solution of a general form of the second order singular boundary value problems. The needed data for the approximation are extracted by using the existed boundary conditions and the structure of the differential equation under consideration. The convergence analysis of the proposed scheme is also presented in detail. Furthermore, the efficiency and accuracy of the method are investigated and verified through the numerical illustration.

Optimizing the Layout of Run-of-River Powerplants Using Cubic Hermite Splines and Genetic Algorithms

Despite the clear advantages of mini hydropower technology to provide energy access in remote areas of developing countries, the lack of resources and technical training in these contexts usually lead to suboptimal installations that do not exploit the full potential of the environment. To address this drawback, the present work proposes a novel method to optimize the design of mini-hydropower plants with a robust and efficient formulation. The approach does not involve typical 2D simplifications of the terrain penstock layout. On the contrary, the problem is formulated considering arbitrary three-dimensional terrain profiles and realistic penstock layouts taking into account the bending effect. To this end, the plant layout is modeled on a continuous basis through the cubic Hermite interpolation of a set of key points, and the optimization problem is addressed using a genetic algorithm with tailored generation, mutation and crossover operators, especially designed to improve both the exploration and intensification. The approach is successfully applied to a real-case scenario with real topographic data, demonstrating its capability of providing optimal solutions while dealing with arbitrary terrain topography. Finally, a comparison with a previous discrete approach demonstrated that this algorithm can lead to a noticeable cost reduction for the problem studied.

Symmetry in multivariate ideal interpolation

An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes, while Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Gröbner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly to compute fundamental invariants and equivariants of a reflection group.

Numerical solution of fractional dynamical systems with impulsive effects

This paper proposes an effective numerical scheme for solving impulsive fractional differential equations. For this purpose, Hermite interpolation is used to approximate fractional-order integrals. The proposed methods convergence analysis is studied in detail by bounding the approximation error. Finally, the application and performance of the presented method are illustrated in two practical examples, including the impulsive control of the family of Lorenz systems, and the obtained results are compared with an existing method.

Reducing the number of measuring points of the LED-based colorimetric probe

In this paper, reducing the number of necessary measuring points for estimating a reflected electromagnetic spectrum of a printed color patch is presented. In our previous work, a machine learning-based method was proven to be superior to Cubic Hermite interpolation in estimating spectrum based on six measured values provided by measuring reflection of six LED sources (400 nm, 457 nm, 517 nm, 572 nm, 632 nm, and 700 nm). Now, the new hypothesis is that the number of measuring points LEDs could be decreased without the significant loss of the spectrum estimation. The ECI2002 test chart was used to create the dataset, which was further divided into training and test subset. For all the colors on the test chart, the measurements were performed on printed patches with the device proposed in our previous work, as well as with the commercial spectrophotometer X-Rite i1 Publish Pro2, which were then used as the ground truth, or reference values. The Artificial Neural Networks were trained to estimate spectrums based on measurements acquired with our device. The results proved satisfactory even when the number of measuring points is reduced from six to three RGB LEDs (457 nm, 517 nm, and 632 nm).

Spatial interpolation method comparison for physico-chemical parameters of river water in Klang River using MATLAB

Water quality is one of the most highly debated issues worldwide at the moment. Inadequate water supplies affect human health, hinder food production, and degrade the environment. Using contemporary technology to analyze pollution statistics can help solve pollution issues. One option is to take advantage of advancements in intelligent data processing to conduct hydrological parameter analysis. To perform conclusive water quality studies, a lot of data is necessary. Unfilled data (information gaps) in the long-term hydrological data set may be due to equipment faults, collection schedule delays, or the data collection officer’s absence. The lack of hydrological data skews its interpretation. Therefore, interpolation is used to recreate and fill missing hydrological data. From 2012 to 2017, the Klang River’s biochemical oxygen demand (BOD) in Selangor, Malaysia, was sampled. This study examined three methods of interpolation for their effectiveness using the MATLAB software: piecewise cubic hermite interpolating polynomial (PCHIP), cubic Spline data interpolation (Spline), and modified Akima partitioned cubic hermite interpolation (Makima). The accuracy is assessed using root mean square error (RMSE). All interpolation algorithms offer excellent results with low RMSE. However, PCHIP delivers the best match between interpolated and original data.

Compact Finite Difference Scheme with Hermite Interpolation for Pricing American Put Options Based on Regime Switching Model

American put options with the regime-switching model is a system of coupled free boundary problems. In this study, we present an accurate finite difference method coupled with the Hermite interpolation for solving this system. To this end, we first employ the logarithmic transformation to map the free boundary for each regime to a fixed interval and then eliminate the first-order derivatives in the transformed model by taking derivatives to obtain a system of partial differential equations which we call the asset-delta-gamma-speed equations. We then discretize the system using the fourth-order compact scheme coupled with the Crank–Nicholson method. At the same time, the influence of other asset options and option sensitivities are estimated based on the third-order Hermite interpolation. As such, the overall scheme consists of four tridiagonal linear systems, which can be easily solved using the Thomas algorithm and the Gauss–Seidel iteration. The obtained scheme is then applied for the model with two, four, and sixteen regimes, respectively. Our results show that the scheme provides an accurate solution that is fast in computation as compared with other existing numerical methods.

G1 Hermite interpolation method for spatial PH curves with PH planar projections

The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or PHoPH curve. We investigate the algebraic structure of PHoPH curves and show that their hodographs are obtained by applying two squaring maps successively to quaternion generator polynomials. The simplest nontrivial PHoPH curves generated from linear quaternion generators are quintic curves, which have adequate degrees of freedom to solve the G1 Hermite interpolation problem. From the algebraic structure, we can derive a system of nonlinear equation for G1 interpolation, which is addressable by numerical methods. We also suggest the choice of initial values for the numerical method. The solvability is not guaranteed for arbitrary G1 data in general, however, we show the feasibility of the system for the G1 data taken from a small segment of reference curves without inflection points using extensive Monte-Carlo simulation. We also present a few illustrative examples of PHoPH spline curves that approximate the given reference curves.

Fully Automatic Classification of Cardiotocographic Signals with 1D-CNN and Bi-directional GRU.

Prenatal fetal monitoring, which can monitor the growth and health of the fetus, is vital for pregnant women before delivery. During pregnancy, it is essential to classify whether the fetus is abnormal, which helps physicians carry out early intervention to avoid fetal heart hypoxia and even death. Fetal heart rate and uterine contraction signals obtained by fetal heart monitoring equipment are essential to estimate fetal health status. In this paper, we pre-process the obtained data set and enhance them using Hermite interpolation on the abnormal classification in the samples. We use the 1D-CNN and GRU hybrid models to extract the abstract features of fetal heart rate and uterine contraction signals. Several evaluation metrics are used for evaluation, and the accuracy is 96 %, while the sensitivity is 95 %, and the specificity is 96 %. The experiments show the effectiveness of the proposed method, which can provide physicians and users with more stable, efficient, and convenient diagnosis and decision support.

Derivative Interpolating Subspace Frameworks for Nonlinear Eigenvalue Problems

We first consider the problem of approximating a few eigenvalues of a proper rational matrix-valued function closest to a prescribed target. It is assumed that the proper rational matrix-valued function is expressed in the transfer function form $H(s) = C (sI - A)^{-1} B$, where the middle factor is large, whereas the number of rows of $C$ and the number of columns of $B$ are equal and small. We propose a subspace framework that performs two-sided projections on the state-space representation of $H(\cdot)$, commonly employed in model reduction and giving rise to a reduced transfer function. At every iteration, the projection subspaces are expanded to attain Hermite interpolation conditions at the eigenvalues of the reduced transfer function closest to the target, which in turn leads to a new reduced transfer function. We prove in theory that, when a sequence of eigenvalues of the reduced transfer functions converges to an eigenvalue of the full problem, it converges at least at a quadratic rate. In the second part, we extend the proposed framework to locate the eigenvalues of a general square large-scale nonlinear meromorphic matrix-valued function $T(\cdot)$, where we exploit a representation ${\mathcal R}(s) = C(s) A(s)^{-1} B(s) - D(s)$ defined in terms of the block components of $T(\cdot)$. The numerical experiments illustrate that the proposed framework is reliable in locating a few eigenvalues closest to the target point, and that, with respect to runtime, it is competitive to established methods for nonlinear eigenvalue problems.

Reliability modelling of wheel wear deterioration using conditional bivariate gamma processes and Bayesian hierarchical models

Wear of wheel profiles is a common degradation process that significantly impacts the reliability of wheelsets. In this study, a conditional bivariate Gamma process is constructed to characterize wheel wear, and hierarchical Bayesian models are employed to identify the differences in wear between each vehicle within a train. Before computation, the field data are modified by isotonic regression and monotone cubic Hermite interpolants. Due to the failure modes involving flange thickness and wheel diameter difference, the distributions of first-passage times are calculated using an analytical approach and kernel density estimation, respectively. The proposed model is validated with a real-world case study of a train consisting of eight vehicles. The results indicate that the laws of wear vary throughout the train, besides left and right wheels. Based on the results, conclusions can be drawn that the first and last vehicles in a train have higher mean wear rates than others, and that different vehicles have different levels of reliability during different stages; the early failure rates are close to zero, but they rapidly increase in the late stage.

On speed of mean convergence of Lagrange and Hermite interpolation based on the roots of Laguerre polynomials

On speed of mean convergence of Lagrange and Hermite interpolation based on the roots of Laguerre polynomials

A method of generating car sounds based on granular synthesis algorithm

The active sound generation technique for automobiles, where car sounds can be synthesized by means of electronic sound production, is one of the effective methods to achieve the target sound. A method for active sound generation of automobile based on the granular synthesis algorithm is put forward here to avoid the broadband beating phenomena that occur due to the mismatch of parameters of sound granular signals. The comparison of the function expression of sound signal and Hilbert transformation is performed based on the principle of overlap and add; moreover, those parameters (phase, frequency and amplitude) of sound signals are interpolated by means of the Hermite interpolation algorithm which can ensure the continuity of the phase, frequency and amplitude curves. Thus, the transition audio is constructed by means of the sound signal function here to splice the adjacent sound granules. Our simulations show that our method can be applied to solve the current broadband beating issue for splicing sound granules and achieve natural continuity of synthesized car sounds. The subjective test results also indicate that our transition audio can produce high quality audio restitution.