Quadratic Polynomial Approximation Research Articles

Conversion map from quantitative parameter mapping to myelin water fraction: comparison with R1·R2* and myelin water fraction in white matter.

To clarify the relationship between myelin water fraction (MWF) and R1⋅R2* and to develop a method to calculate MWF directly from parameters derived from QPM, i.e., MWF converted from QPM (MWFQPM). Subjects were 12 healthy volunteers. On a 3T MR scanner, dataset was acquired using spoiled gradient-echo sequence for QPM. MWF and R1⋅R2* maps were derived from the multi-gradient-echo (mGRE) dataset. Volume-of-interest (VOI) analysis using the JHU-white matter (WM) atlas was performed. All the data in the 48 WM regions measured VOI were plotted, and quadratic polynomial approximations of each region were derived from the relationship between R1·R2* and the two-pool model-MWF. The R1·R2* map was converted to MWFQPM map. MWF atlas template was generated using converted to MWF from R1·R2* per WM region. The mean MWF and R1·R2* values for the 48 WM regions were 11.96 ± 6.63%, and 19.94 ± 4.59s-2, respectively. A non-linear relationship in 48 regions of the WM between MWF and R1·R2* values was observed by quadratic polynomial approximation (R2 ≥ 0.963, P < 0.0001). MWFQPM map improved image quality compared to the mGRE-MWF map. Myelin water atlas template derived from MWFQPM may be generated with combined multiple WM regions.

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.

Analysis and Identification of Root Cause of 5G Radio Quality Deterioration Using Machine Learning

It has become increasingly important for industries to promote digital transformation by utilizing 5G and industrial internet of things (IIoT) to improve productivity. To protect IIoT application performance (work speed, productivity, etc.), it is often necessary to satisfy quality of service (QoS) requirements precisely. For this purpose, there is an increasing need to automatically identify the root causes of radio-quality deterioration in order to take prompt measures when the QoS deteriorates. In this paper, a method for identifying the root cause of 5G radio-quality deterioration is proposed that uses machine learning. This Random Forest based method detects the root cause, such as distance attenuation, shielding, fading, or their combination, by analyzing the coefficients of a quadratic polynomial approximation in addition to the mean values of time-series data of radio quality indicators. The detection accuracy of the proposed method was evaluated in a simulation using the MATLAB 5G Toolbox. The detection accuracy of the proposed method was found to be 98.30% when any of the root causes occurs independently, and 83.13% when the multiple root causes occur simultaneously. The proposed method was compared with deep-learning methods, including bidirectional long short-term memory (bidirectional-LSTM) or one-dimensional convolutional neural network (1D-CNN), that directly analyze the time-series data of the radio quality, and the proposed method was found to be more accurate than those methods.

Estimation of the critical transition probability using quadratic polynomial approximation with skewness-based reject option

Critical transitions and early warning signals are gaining attention in various fields such as ecology, climatology, and economics. However, quantitative estimation of the critical transition probability remains difficult. In this study, I propose a method to estimate the critical transition probability. It is based on a previous method using quadratic polynomial approximation, and skewness filtering is added as a reject option. The proposed method is applied to May model, a mathematical model of an ecosystem, as an example case. The results of numerical simulations show that the proposed method has much better precision than the previous method without skewness filtering, achieving a relative error of approximately ±50% for the mean escape time.

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.

Viscoelastic wave propagation in transversely isotropic media based on constant-order fractional polynomial approximations

Accurately simulating viscoelastic wave propagation in the subsurface is essential to image subsurface reservoirs. One established method to investigate constant- Q effects is based on the variable-order fractional viscoelastic wave equation. However, the mixed-domain operators introduced in this approach are cumbersome to calculate and impede a high-efficiency numerical solver. To solve this issue, we approximate the mixed-domain operator by using a polynomial approximation and derive the corresponding viscoelastic wave equation in anisotropic attenuative media. The constant-order fractional polynomial method is further developed to improve the approximation accuracy. Based on these polynomial approximations, only several constant-order Laplacians remain in the new form of the viscoelastic wave equations. Therefore, these equations can be easily solved using Fourier transform and finite-difference methods. In addition, we also perform an accuracy analysis among the conventional quadratic, cubic, and constant-order fractional polynomial approximations. It indicates that the relative error of the constant-order fractional polynomial method is much smaller than that generated by the conventional quadratic and cubic polynomial approximations. Compared to the variable-order fractional viscoelastic wave equation, our constant-order viscoelastic wave equation can simulate seismic wave propagations with comparable accuracy but with high efficiency. Numerical results based on our approach can well match reference solutions in homogeneous and heterogeneous models, demonstrating the accuracy and effectiveness of the new method.

Rational polynomial image magnification based on edge and distance constraints

Surface fitting is one of the most effective methods in image magnification. The key is to construct the surface fitting the image.Image details and edges play a key role in the visual effect of the image. Therefore, one of the keys to constructing the fitting surface is to preserve the image details and edges.The surface fitting methods based on spline and polynomial cannot effectively preserve the edge information of the image, so that the magnified image has obvious jagged edges. In this paper, a new image magnification algorithm based on edge and distance constraints is proposed. The algorithm constructs a quadratic polynomial surface patch on the adjacent region of each pixel, which fits the pixels in the neighborhood, and a quadratic polynomial median surface patch is constructed on each quadrilateral mesh. The rational polynomial patch is generated from the weighted average of five patches on each quadrilateral mesh. The patch has the accuracy of the quadratic polynomial approximation, and the resulting image has a better visual effect. The accuracy and visual effects of the magnified image are improved by constructing error patches. The new algorithm calculates the constant term, the first term and the second term of the quadratic polynomial by different methods, which provides a new technique for constructing the polynomial function with constraints. The experimental results show that compared with other algorithms, the proposed algorithm not only has higher approximation accuracy but also has better visual effect of the magnified image.

The influence of thermophysical parameters on the prediction accuracy of the spindle thermal error model

The model based on the velocity and acceleration of temperature variation (MBOTV) was established for spindle thermal growth error. However, the thermophysical parameters of MBOTV, such as the thermal expansion coefficient related to the velocity of temperature variation, Hv, the thermal expansion coefficient related to the acceleration of temperature variation, Ha, and the thermal inertia coefficient of the spindle, τ, may change during actual working. In order to analyze the influence of thermophysical parameters on the prediction accuracy of MBOTV, it is necessary to calculate the reliability of MBOTV. Actually, it is difficult to obtain the function of MBOTV. Although the improved response surface method can be used to calculate the reliability of the model with implicit function, the accuracy is not enough obviously when the implicit function is replaced by linear polynomials. Therefore, an improved response surface method based on quadratic polynomial approximation was proposed in this paper to calculate the reliability of MBOTV. In this method, the linear polynomial was replaced by a quadratic polynomial without cross product terms to replace the implicit function, which can improve the accuracy of reliability calculation. The predicted residual errors of MBOTV with single-parameter fluctuation and multi-parameter fluctuation were given in this paper. Studies reveal that MBOTV has strong robustness when the thermophysical parameters change.

Medical Image Segmentation Based on Combination Method

This paper studies the data segmentation of medical images based on compressed sensing. We do not use image data as interpolation data, but as constraints to construct surface patches with quadratic polynomial approximation accuracy, study the properties and correlation of pixels, and take these edge points as constraints, and map the discrete image data into continuous mathematical functions, so that the continuous function or point data not only has the shape recommended by the image data, but also has a higher segmentation accuracy, and then realize the fine segmentation of images, which provides a new research idea to solve the difficult problems of spatial data representation, and has good theoretical significance and application value.

Modified Stencil Approximations for Fifth-Order Weighted Essentially Non-oscillatory Schemes

In this paper, a modified fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme is presented. The quadratic polynomial approximation of numerical flux on each candidate stencil of the traditional WENO-JS scheme is modified by adding a form of cubic terms such that the resulting stencil approximation achieves fourth-order accuracy. And the corresponding smoothness indicators are calculated. The modified candidate fluxes and local smoothness indicators, when used in the WENO-JS scheme, can make the resulting new scheme (called WENO-MS) achieve fifth-order convergence in smooth regions including first-order critical points. A series of one- and two-dimensional numerical examples are presented to demonstrate the performance of the new scheme. The numerical results show that the proposed WENO-MS scheme provides a comparable or higher resolution of fine structures compared with the WENO-M, WENO-Z and P-WENO schemes, while it increases only 7% of CPU time compared with the traditional WENO-JS scheme.

Expectation–Maximization-Based Passive Localization Relying on Asynchronous Receivers: Centralized Versus Distributed Implementations

This paper considers a passive localization scenario relying on a single transmitter, several receivers, and multiple moving targets to be located. The so-called “passive” targets equipped with RFID reflectors are capable of reflecting the signals from the transmitter to the receivers. Existing approaches assume that the transmitter and receivers are synchronous or quasi-synchronous, which is not always realistic in practical scenarios. Hence, an asynchronous wireless network is considered, where different clock offsets are assumed at different receivers. We propose a centralized expectation–maximization-based passive localization method for asynchronous receivers (EMpLaR) by treating the clock offsets as hidden variables. Thereby, the proposed algorithm makes use of Taylor expansions to arrive at a closed-form maximization. Furthermore, to improve the robustness to link failures and to reduce the energy consumption, we propose a distributed localization approach based on average consensus formulation to locate the target at each receiver. By applying a quadratic polynomial approximation of the function on which consensus has to be reached, both the computational complexity and the communications overhead are significantly reduced. The Cramer–Rao bound of the target location is derived as a benchmark of our proposed algorithms. Our simulation results show that the proposed centralized and distributed EMpLaR algorithms match the Cramer–Rao bound and significantly improve the localization performance compared with the conventional methods.

Static and dynamic FE analysis of piezolaminated composite shells considering electric field nonlinearity under thermo-electro-mechanical loads

The present article focuses on the static and dynamic finite element simulations of smart piezolaminated composite shell structures considering strong electric field nonlinearity under thermo-electro-mechanical loads. To model the electromechanical behaviour of piezoelectric patches or layers under large applied electric fields more efficiently, two-way coupled rotationally invariant second-order nonlinear constitutive relations are used in the variational principle approach. Furthermore, the nonlinear piezoelectric element formulations are further extended to capture the response under temperature gradients. Quadratic and cubic polynomial approximations are deemed to represent the electric potential and temperature fields, respectively. Validation of the present element formulation has been done in comparison to experimental and numerical investigations of those available in the literature. Moreover, numerical simulations are performed to study the large electric field nonlinearities of piezolaminated structures in static and dynamic as well as active vibration control problems under both mechanical and thermal loads. The numerical simulations have shown that using the piezoelectric nonlinearity, both the static shape control and vibration suppression either under mechanical or thermal loads can be accomplished at much lower actuation voltages than estimated by the linear model.

A coupled electro-thermal Discontinuous Galerkin method

This paper presents a Discontinuous Galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro-thermal problems.In this paper we discuss the fundamental equations for the transport of electricity and heat, in terms of macroscopic variables such as temperature and electric potential. A fully coupled nonlinear weak formulation for electro-thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients. The weak form can thus be formulated as a Discontinuous Galerkin method.The existence and uniqueness of the weak form solution are proved. The numerical properties of the nonlinear elliptic problems i.e., consistency and stability, are demonstrated under specific conditions, i.e. use of high enough stabilization parameter and at least quadratic polynomial approximations. Moreover the prior error estimates in the H1-norm and in the L2-norm are shown to be optimal in the mesh size with the polynomial approximation degree.

Fourier Type Super Convergence Study on DDGIC and Symmetric DDG Methods

In this paper, using Fourier analysis technique, we study the super convergence property of the DDGIC (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and the symmetric DDG (Vidden and Yan in J Comput Math 31(6):638–662, 2013) methods for diffusion equation. With $$k\hbox {th}$$ degree piecewise polynomials applied, the convergence to the solution’s spatial derivative is $$k\hbox {th}$$ order measured under regular norms. On the other hand when measuring the error in the weak sense or in its moment format, the error is super convergent with $$(k+2)\hbox {th}$$ and $$(k+3)\hbox {th}$$ orders for its first two moments with even order degree polynomial approximations. We carry out Fourier type (Von Neumann) error analysis and obtain the desired super convergent orders for the case of $$P^2$$ quadratic polynomial approximations. The theoretical predicted errors agree well with the numerical results.

A high order finite element scheme for pricing options under regime switching jump diffusion processes

This paper considers the numerical pricing of European, American and Butterfly options whose asset price dynamics follow the regime switching jump diffusion process. In an incomplete market structure and using the no-arbitrage pricing principle, the option pricing problem under the jump modulated regime switching process is formulated as a set of coupled partial integro-differential equations describing different states of a Markov chain. We develop efficient numerical algorithms to approximate the spatial terms of the option pricing equations using linear and quadratic basis polynomial approximations and solve the resulting initial value problem using exponential time integration. Various numerical examples are given to demonstrate the superiority of our computational scheme with higher level of accuracy and faster convergence compared to existing methods for pricing options under the regime switching model.

A practical numerical scheme for the ternary Cahn–Hilliard system with a logarithmic free energy

We consider a practically stable finite difference method for the ternary Cahn–Hilliard system with a logarithmic free energy modeling the phase separation of a three-component mixture. The numerical scheme is based on a linear unconditionally gradient stable scheme by Eyre and is solved by an efficient and accurate multigrid method. The logarithmic function has a singularity at zero. To remove the singularity, we regularize the function near zero by using a quadratic polynomial approximation. We perform a convergence test, a linear stability analysis, and a robustness test of the ternary Cahn–Hilliard equation. We observe that our numerical solutions are convergent, consistent with the exact solutions of linear stability analysis, and stable with practically large enough time steps. Using the proposed numerical scheme, we also study the temporal evolution of morphology patterns during phase separation in one-, two-, and three-dimensional spaces.

Mixed Convective Heat and Mass Transfer Flow of Nanofluids in Concentric Annulus

This paper studies the non-Darcy natural convective heat transfer flow of nanofluid flow through a porous medium in a co-axial cylindrical duct where the boundaries are maintained at constant temperature and concentration. The flow in the porous material is given by a linear Brinkman-Forchheimer-extended Darcy equation. The Boussinesq approximation is invoked so that the effect of density variation is combined to the buoyancy forces. The equations of momentum, energy and diffusion are coupled and linear. Galerkin finite element analysis is employed with quadratic polynomial approximations. At different axial positions, analysis is conducted for the behaviour of velocity, temperature and concentration.

A planar quadratic clipping method for computing a root of a polynomial in an interval

This paper presents a new quadratic clipping method for computing a root of a polynomial f(t) of degree n within an interval. Different from the traditional one in R1 space, it derives three quadratic curves in R2 space for approximating (t,f(t)) instead, which leads to a higher approximation order. Two bounding polynomials are then computed in O(n2) for bounding the roots of f(t) within the interval. The new clipping method achieves a convergence rate of 4 for a single root, compared with that of 3 from traditional method using quadratic polynomial approximation in R1 space. When f(t) is convex within the interval, the two bounding polynomials are able to be directly constructed without error estimation, which leads to computational complexity O(n). Numerical examples show the approximation effect and efficiency of the new method. The method is particularly useful for the fly computation in many geometry processing and graphics rendering applications.

효율적인 에너지 절감을 위한 센서 LED 스탠드 개발

In this paper, we present a new lighting control method considering ambient light in addition to the required lighting illumination for efficient energy saving of a LED stand. We estimate accurate environmental illuminance using a cheap illuminance sensor by modeling measured- and actualilluminance using quadratic polynomial approximation. The relation between PWM(Pulse Width Modulation) duty ratio and illuminance intensity is modeled by a linear model. Illumination of the LED stand is controlled by estimating the difference of required illumination and the estimated ambient illumination. The developed LED stand has reduced electric energy consumption compared with a conventional manually controlled LED stand with the same lighting source. In addition, human subject evaluation shows that the LED stand, which is applied the proposed method, is more satisfactory than conventional ones since the proposed automatic controlled illumination produce more accurately required lighting and it is convenient.

J122042 LED放熱用ヒートシンクの最適設計

As we know that LED has a bright future for its long life time, good endurance of heavy impact, no mercury containing and most importantly, potentially high efficiency, energy conservation and low carbon dioxide emissions. This paper has concentrated on the optimum design of LED heat sink in order to increase the efficiency of LED external heat diffusion. At first, we have selected the height of heat sink, the thickness of fin, the width of fin, the radius and the fin number as design variables, and have assigned the design points via the orthogonal array based on the design of experiment method. Then, the simulation models of LED heat sink are created by Solidworks, and then the simulation was implemented by the COMSOL software. Next, we have constructed the second-order polynomial approximation of response surface for the maximum temperature by the multiple regression analysis. At last, the optimum design point was determined by the use of the quadratic polynomial approximation of response surface.