Quadratic Polynomial Interpolation Research Articles

Improved Algorithms Based on Trust Region Framework for Solving Unconstrained Derivative Free Optimization Problems

This paper is devoted to developing new derivative-free optimization (DFO) methods for solving optimization problems where derivative information is not available or cannot be calculated numerically. To overcome the computational difficulties arising from time-consuming estimations of the objective function, we propose two algorithms. One algorithm is a variant of the surrogate model-based DFO algorithm under the trust region method, where the surrogate model is formulated through sparse low-rank interpolation and quadratic polynomial interpolation. This algorithm serves as a comparative baseline. The second algorithm leverages the characteristics of the sparse regression model, which can handle sparsity and noise issues, to construct the surrogate model. The coefficients of the sparse surrogate model are then estimated using the alternating direction method of multipliers and refined through a correction strategy based on the R-square. Finally, numerical results, evaluated in terms of performance and data profiles, demonstrate the effectiveness and competitiveness of the proposed algorithms.

PSP-Equivalent Model for Double-Gate and Surrounding-Gate Field Effect Transistors for Circuit Simulation

We introduce a compact core model for double-gate (DGFET) and surrounding-gate (SGFET) MOSFETs designed for circuit simulations. Despite its high precision, the model is crafted to retain the same analytic formulation of the industry standard Pennsylvania State and Philips (PSP). Instead of linearizing the drain current as in the PSP model, we employ a quadratic symmetric polynomial interpolation of the charge in the channel. This eliminates the need for cumbersome derivatives, simplifications, and intricate coding when integrating into a circuit simulator, thereby preventing singularities during numerical iterations. Moreover, thanks to its mathematical formulation equivalent to PSP, this model simplifies the coding of terminal charges, capacitances, potentials, and electric fields in the channel within circuit simulators. We validate the accuracy of the model through comparisons with numerical solutions and experiments from the literature.

A unified core model of double-gate and surrounding-gate MOSFETs for circuit simulation

This paper presents a new core compact model of double-gate (DGFET) and surrounding-gate (SGFET) MOSFETs for circuit simulations. The current and the terminal charges are continuous with high computation efficiency and accuracy. Despite its accuracy, it retains the same simplicity of the industry standard transistors models. The drain current is worked out without invoking the charge-sheet approximation exploiting a quadratic symmetric polynomial interpolation of the charge in the channel. Apart this clear approximation, no other simplification is used to work out the drain current, the terminal charges, the potential, and electric field in the channel. The accuracy of the model is shown by comparison with the exact numerical solution and experimental data of the literature.

High-order schemes based on extrapolation for semilinear fractional differential equation

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order α∈(1,2).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,2).$$\\end{document} The error has the asymptotic expansion (d3τ3-α+d4τ4-α+d5τ5-α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\big ( d_{3} \ au ^{3- \\alpha } + d_{4} \ au ^{4-\\alpha } + d_{5} \ au ^{5-\\alpha } + \\cdots \\big ) + \\big ( d_{2}^{*} \ au ^{4} + d_{3}^{*} \ au ^{6} + d_{4}^{*} \ au ^{8} + \\cdots \\big ) $$\\end{document} at any fixed time tN=T,N∈Z+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_{N}= T, N \\in {\\mathbb {Z}}^{+}$$\\end{document}, where di,i=3,4,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{i}, i=3, 4,\\ldots $$\\end{document} and di∗,i=2,3,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{i}^{*}, i=2, 3,\\ldots $$\\end{document} denote some suitable constants and τ=T/N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au = T/N$$\\end{document} denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order α∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,2)$$\\end{document} is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.

A Fast High-Order Predictor–Corrector Method on Graded Meshes for Solving Fractional Differential Equations

In this paper, we focus on the computation of Caputo-type fractional differential equations. A high-order predictor–corrector method is derived by applying the quadratic interpolation polynomial approximation for the integral function. In order to deal with the weak singularity of the solution near the initial time of the fractional differential equations caused by the fractional derivative, graded meshes were used for time discretization. The error analysis of the predictor–corrector method is carefully investigated under suitable conditions on the data. Moreover, an efficient sum-of-exponentials (SOE) approximation to the kernel function was designed to reduce the computational cost. Lastly, several numerical examples are presented to support our theoretical analysis.

A second-order L2-1σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients

Based on quadratic and linear polynomial interpolation approximations and the Crank-Nicolson technique, a new second-order difference approximation method of the Caputo fractional derivative of order α∈(1,2) is proposed. This method is different from the known second-order methods, which is called the L2-1σ Crank-Nicolson method in this paper. Using the L2-1σ Crank-Nicolson method of the Caputo fractional derivative, a fully discrete L2-1σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients is developed. The unconditional stability and convergence of the method are rigorously proved. The optimal error estimates in the discrete L2-norm and H1-norm are obtained under a relatively weak regularity condition. The error estimates show that the method has the second-order convergence in both time and space for all α∈(1,2). In order to overcome the loss of the temporal convergence order caused by the stronger singularity of the exact solution at the initial time, a nonuniform L2-1σ Crank-Nicolson difference method is also developed on a class of general nonuniform time meshes. Numerical results confirm the theoretical convergence result. The effectiveness of the nonuniform method for non-smooth solutions with the stronger singularity at the initial time is tested on a class of initially graded time meshes.

Novel bi-layer beam elements for elastic fracture analysis of delaminated composite beams

A general finite element method is proposed for linear elastic fracture analysis of delaminated bilayer composite beams. An interface deformable bilayer beam model is adopted in modeling the virgin portions of bilayer beams, and it is capable of guaranteeing the continuity condition of internal forces at the tip of delamination because the bilayers are modeled as two separate shear deformable sub-beams and both the interface normal and shear compliances are introduced to account for interface deformation. The stiffness matrices of bilayer beams using the linear and quadratic interpolation polynomials are both derived using the principle of minimum potential energy. Then, a finite element program is developed by assembling the element stiffness matrices of virgin and delaminated portions of bilayer beams and applying the corresponding equivalent nodal forces. Two typical fracture specimens (i.e., single leg bending and end-notched flexure bilayer beams) are analyzed using the proposed bilayer beam elements. The computed displacement and energy release rate are compared well with those of analytical solutions, and the mode components of energy release rate for single leg bending specimen are also obtained using the virtual crack closure technique. It demonstrates that the proposed novel bilayer beam elements are capable of effectively and efficiently predicting the displacement and energy release rate (including its mode components) of delaminated bilayer beams. The developed bilayer beam element is general in nature and accurate and efficient in particular, because it can achieve the same accuracy as the high order beam theory and be easily implemented in the existing commercial finite element software.

Neuro‐controller for broadcast lighting light‐emitting diode to express white light with high color rendering index

AbstractTechnical characteristics analysis related to correlated color temperature (CCT), color rendering, and illuminance is required to use light‐emitting diode (LED) as broadcast lighting. In general, to realize a white light source with a high color rendering index (CRI), we selected the appropriate emission intensity of RGBW LED through trial and error. However, the characteristics of the LED light source and environmental conditions make it difficult to perform the procedure several times. The objective of this study was to design a system that could control illuminance, CCT, and ∆uv while having high CRI, as an LED control method for broadcasting lighting. The controller implements using a feed‐forward neural network with excellent nonlinear function approximation capability. We measure data directly from the red green blue white (RGBW) LED system for neural network training. We then select data with high CRI from the measured raw data and choose data for neural network learning by removing measurement noise using the quadratic polynomial interpolation method. The performance evaluation confirms that the proposed neural network controller shows excellent results as an RGBW LED controller for broadcast lighting in the Planckian locus and all regions of white light.

Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations

<p style='text-indent:20px;'>In this work, two fully novel finite difference schemes for two-dimensional time-fractional mixed diffusion and diffusion-wave equation (TFMDDWEs) are presented. Firstly, a Hermite and Newton quadratic interpolation polynomial have been used for time discretization and central quotient has used in spatial direction. The H2N2 finite difference is constructed. Secondly, in order to increase computational efficiency, the sum-of-exponential is used to approximate the kernel function in the fractional-order operator. The fast H2N2 finite difference is obtained. Thirdly, the stability and convergence of two schemes are studied by energy method. When the tolerance error <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> of fast algorithm is sufficiently small, it proves that both of difference schemes are of <inline-formula><tex-math id="M2">\begin{document}$ 3-\beta\; (1<\beta<2) $\end{document}</tex-math></inline-formula> order convergence in time and of second order convergence in space. Finally, numerical results demonstrate the theoretical convergence and effectiveness of the fast algorithm.</p>

A Second-Order Scheme with Nonuniform Time Steps for a Linear Reaction-Subdiffusion Problem

Stability and convergence of a time-weighted discrete scheme with nonuniform time steps are established for linear reaction-subdiffusion equations. The Caupto derivative is approximated at an offset point by using linear and quadratic polynomial interpolation. Our analysis relies on two tools: a discrete fractional Gr\"{o}nwall inequality and the global consistency analysis. The new consistency analysis makes use of an interpolation error formula for quadratic polynomials, which leads to a convolution-type bound for the local truncation error. To exploit these two tools, some theoretical properties of the discrete kernels in the numerical Caputo formula are crucial and we investigate them intensively in the nonuniform setting. Taking the initial singularity of the solution into account, we obtain a sharp error estimate on nonuniform time meshes. The fully discrete scheme generates a second-order accurate solution on the graded mesh provided a proper grading parameter is employed. An example is presented to show the sharpness of our analysis.

A WSN Layer-Cluster Key Management Scheme Based on Quadratic Polynomial and Lagrange Interpolation Polynomial.

Since current key management schemes are mainly designed for static and planar networks, they are not very suitable for the layer-cluster wireless sensor networks (WSNs), a WSN layer-cluster key management scheme based on quadratic polynomial and Lagrange interpolation polynomial is proposed, in which the main idea of this scheme along the research line of broadcast identity authentication, session key, group key, network key and personal key. Specifically, authentication key can be established on the basis of Fourier series for identity authentication; session key is established by a multiple asymmetric quadratic polynomial, in which session key information is encrypted by the authentication key to ensure the security of intermediate interactive information; based on the former two keys, group key is established on the basis of Lagrange interpolation polynomial, in which the nodes of the cluster are not directly involved; the generation and management of network key is similar to the group key, in which the establishment idea is to regard the BS and all cluster heads as a group; the generation and management of personal key is also similar to the group key, the difference is that the personal key can be obtained by cluster nodes through getting the Lagrange interpolation polynomial coefficients based on their own random key information. It is analyzed that the proposed layer-cluster key management scheme can guarantee the identity of network nodes firstly through forward authentication and reverse authentication, and session key, group key and network key will guarantee the independence of the keys’ management and avoids the problem of single point failure compared with LEAP protocol, and personal key will guarantee the privacy of network.

Maximum Norm a Posteriori Error Estimation for a System of Singularly Perturbed Semilinear Reaction-Diffusion Equations

In this paper, a system of singularly perturbed semilinear reaction-diffusion equations is discretized on an arbitrary nonuniform mesh using an upwind finite difference scheme. Based on the quadratic polynomial interpolation function, a second-order maximum norm a posteriori error estimation is derived. Then, this bound is used to design an adaptive grid generation algorithm. Numerical results are given to verify the effectiveness of the presented adaptive grid method.<br />

An H2N2 Interpolation for Caputo Derivative with Order in (1, 2) and Its Application to Time-Fractional Wave Equations in More Than One Space Dimension

In this paper, a new derived method is developed for a known numerical differential formula of the Caputo fractional derivative of order $$\gamma \in (1,2)$$ (Li and Zeng in Numerical methods for fractional calculus. Chapman & Hall/CRC numerical analysis and scientific computing, CRC Press, Boca Raton, 2015) by means of the quadratic interpolation polynomials, and a concise expression of the truncation error is given. This new method will be called as the H2N2 method because of the application of the quadratic Hermite and Newton interpolation polynomials. A finite difference scheme with a second order accuracy in space and a $$(3-\gamma )$$-th order accuracy in time based on the H2N2 method is constructed for the initial boundary value problem of time-fractional wave equations. The stability and convergence of the difference scheme are proved. Furthermore, in order to increase computational efficiency, using the sum-of-exponentials to approximate the kernel $$t^{1-\gamma }$$, a fast difference scheme is presented. The problem with weak regularity at the initial time is also discussed with the help of the graded meshes. At each time level, the difference scheme is solved with a fast Poisson solver. Numerical results show the effectiveness of the two difference schemes and confirm our theoretical analysis.

A versatile algorithm for estimating natural frequencies with high accuracy

Detection of damage requires an accurate estimation of the natural frequencies of the monitored structure. This paper introduces an algorithm implemented in Python which improves the frequency readability by increasing the number of spectral lines without requiring a signal extension in the time domain. We achieve this by overlapping several spectra calculated from the acquired signal repeatedly shortened. In this way, the overlapped spectrum gets an increased number of spectral lines. The dense mesh of spectral lines permits us to obtain a fine frequency resolution without being necessary an extension of the signal in the time domain. The high density of the spectral lines ensures a sufficient number of points on the main lobes that permits performing an efficient quadratic polynomial interpolation to find the maximizer. It represents the amplitude of the real frequency and is typically located on an inter-line position, thus cannot be found by standard frequency estimation. We implemented the algorithm in Python and tested it successfully for generated signals, containing one or more harmonics, with known frequencies.

A predictor–corrector scheme for the tempered fractional differential equations with uniform and non-uniform meshes

Tempered fractional derivatives and the corresponding tempered fractional differential equations have played a key role in physical science. In this paper, for solving the tempered fractional ordinary differential equation, the predictor–corrector (PC) methods with uniform and non-uniform meshes of Deng et al. (Numer Algorithms 74(3):717–754, 2017) are developed, by using the piecewise quadratic interpolation polynomial. The error bounds of proposed predictor–corrector schemes with uniform and equidistributing meshes are obtained. We proved that the presented numerical method has a higher-order convergence order $$O(h^3)$$. Also, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical methods. Finally, the results of PC schemes with uniform and non-uniform given in Deng et al. (2017) and presented schemes (improved PC with uniform and non-uniform meshes) are compared for different values of parameters.

A new high order ADI numerical difference formula for time-fractional convection-diffusion equation

Based on exponential transformation, quadratic interpolation polynomial and Padé approximation, a new high order finite difference scheme is proposed for solving the two-dimensional (2D) time-fractional convection-dominated diffusion equation (of order α ∈ (0, 1)). The resulting scheme is of (3−α)-order accuracy in time and fourth-order accuracy in space. In order to reduce the amount of computation, the alternating direction implicit (ADI) scheme is further developed. Numerical experiments are given to demonstrate the high accuracy and robustness of our new scheme.

Eliminating starting hesitation for reliable operation of switched reluctance motor without machine parameters for light electric vehicle applications

In this study, an initial rotor position estimation at standstill condition is presented for reliable operation of switched reluctance motor (SRM) without initialising machine parameters for light electric vehicle (LEV) applications. The prime objective of this method is to startup an SRM without using the mechanical position sensor, and eliminate the starting hesitation and starting jerk issues from standstill condition. In the proposed method, the initial rotor position at standstill is determined by injecting a single pulse dc voltage simultaneously into all stator phase windings for a very short period (microsecond). The obtained current envelope data is then used to calculate the initial rotor position with the quadratic polynomial interpolation technique without any prior knowledge of other motor parameters. The experimental results for a 3.5 kW, 6/4 SRM are provided to substantiate the overall good performance. The developed system can be considered as a strong candidate for high precision LEV applications.

An emulator for the Lyman-α forest

We present methods for interpolating between the 1-D flux power spectrum of the Lyman-α forest, as output by cosmological hydrodynamic simulations. Interpolation is necessary for cosmological parameter estimation due to the limited number of simulations possible. We construct an emulator for the \\Lya forest flux power spectrum from 21 small simulations using Latin hypercube sampling and Gaussian process interpolation. We show that this emulator has a typical accuracy of 1.5% and a worst-case accuracy of 4%, which compares well to the current statistical error of 3–5% at z < 3 from BOSS DR9. We compare to the previous state of the art, quadratic polynomial interpolation. The Latin hypercube samples the entire volume of parameter space, while quadratic polynomial emulation samples only lower-dimensional subspaces. The Gaussian process provides an estimate of the emulation error and we show using test simulations that this estimate is reasonable. We construct a likelihood function and use it to show that the posterior constraints generated using the emulator are unbiased. We show that our Gaussian process emulator has lower emulation error than quadratic polynomial interpolation and thus produces tighter posterior confidence intervals, which will be essential for future Lyman-α surveys such as DESI.

Two-Step Interpolation Algorithm for Measurement of Longitudinal Cracks on Pipe Strings Using Circumferential Current Field Testing System

Pipe strings are critical facilities for well drilling, production, and transportation in oil and gas industry. Due to the stress corrosion cracking, longitudinal cracks are the most common defects on pipe strings. This paper presents a simple two-step interpolation algorithm based on a circumferential current field testing method for the measurement of longitudinal cracks on pipe strings. The theory and finite element method model of the circumferential current field testing method are analyzed. The two-step interpolation algorithm fitted by characteristic signals obtained from simulations is proposed to size longitudinal cracks. The first-step is to measure and calibrate the crack length by a quadratic polynomial interpolation formula and the second-step is to measure the crack depth by a cubic polynomial interpolation formula. Experiments are conducted to verify the efficacy of the proposed two-step interpolation algorithm based on the circumferential current field testing system. The results suggest that the two-step interpolation algorithm can obtain the length and depth information of longitudinal cracks effectively on pipe strings using the circumferential current field testing system.

A higher order numerical method for time fractional partial differential equations with nonsmooth data

Gao et al. [11] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O(k3−α), 0<α<1 by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu [20] (2016), where k is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu [20] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O(k3−α), 0<α<1 uniformly with respect to the time variable t. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate O(k3−α), 0<α<1 uniformly with respect to the time variable t. In this paper, we first obtain a similar approximation scheme to the Riemann–Liouville fractional derivative with the convergence rate O(k3−α), 0<α<1 as in Gao et al. [11] (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O(k3−α), 0<α<1 for any fixed tn>0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.