Linear Interpolation Approximation Research Articles

Two new approximations to Caputo–Fabrizio fractional equation on non-uniform meshes and its applications

We present two numerical approximations with non-uniform meshes to the Caputo–Fabrizio derivative of order α (0 < α < 1). First, the L1 formula is obtained by using the linear interpolation approximation for constructing the second-order approximation. Next, the quadratic interpolation approximation is used for improving the accuracy in the temporal direction. Besides, we discretize the spatial derivative using the compact finite difference scheme. The accuracy of the suggested schemes is not dependent on the fractional α. The coefficients and the truncation errors are carefully investigated for two schemes, separately. Three examples are carried out to support the convergence orders and show the efficiency of the suggested scheme.

Block-centered local refinement methods for the time-fractional equations

In this paper, we present and analyze two block-centered local refinement (BLR) methods for solving the time-fractional equations. The main difference between the two methods is that different approximation methods are used for the value of the pressure at the slave nodes. One adopts a piecewise constant interpolation approximation (PCIA) method, which is called simple block-centered local refinement (S-BLR) method. The other utilizes a piecewise linear interpolation approximation (PLIA) method, which called more accurate block-centered local refinement (MA-BLR) method. The stability analysis is proved carefully. It is estimated that the discrete L2 errors for the velocity and pressure are O(▵t2−α+h3/2) and O(▵t2−α+h3/2) in use of S-BLR and MA-BLR methods, respectively. Where Δt is the time step and h is the maximal mesh size. These error estimate results are all established on locally refinement composite grids. Finally, a numerical experiment is presented to show that the convergence rates are in agreement with the theoretical analysis.

A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability

In the present work, first of all, a new numerical fractional differentiation formula (called the CF2 formula) to approximate the Caputo-Fabrizio fractional derivative of order $alpha,$ $(0<alpha<1)$ is developed. It is established by means of the quadratic interpolation approximation using three points $ (t_{j-2},y(t_{j-2}))$, $(t_{j-1},y(t_{j-1})) $ and $ (t_{j},y(t_{j})) $ on each interval $[t_{j-1},t_{j}]$ for $ ( j geq 2 )$, while the linear interpolation approximation is applied on the first interval $[t_{0},t_{1}]$. As a result, the new formula can be formally viewed as a modification of the classical CF1 formula, which is obtained by the piecewise linear approximation for $y(t)$. Both the computational efficiency and numerical accuracy of the new formula is superior to that of the CF1 formula. The coefficients and truncation errors of this formula are discussed in detail. {Two test example show} the numerical accuracy of the CF2 formula. The CF1 formula demonstrates that the new CF2 is much more effective and more accurate than the CF1 when solving fractional differential equations. Detailed stability analysis and region stability of the CF2 are also carefully investigated.

A difference method for solving the [formula omitted]-fractional differential equations

In this article, we study the Caputo type q-fractional differential equation: cDqαx(t)=f(t,x(t)),0<α,q<1. By using the piecewise linear interpolation approximation, we present a difference formula for discretizing the Caputo type q-fractional derivative cDqαx(t) and derive the truncation error boundness. This difference formula is used to solve the initial value problem of q-fractional differential equation. Numerical examples are provided to verify the effectiveness of our difference method.

Block-centred finite difference methods on rectangular composite grids with refinement in space for parabolic equation

ABSTRACTTwo block-centred finite difference schemes with local refinement in space are introduced and analysed to solve parabolic equation. One scheme is simple approximation scheme, in which piecewise constant interpolation approximation is used for slave nodes. The other is more accurate approximation scheme, with piecewise linear interpolation approximation along the interface. Error estimates in the discrete norm are derived of order for simple approximation scheme and of order for more accurate approximation scheme. Numerical experiments using these two schemes show that the convergence rates are in agreement with the theoretical analysis.

Confidence intervals for quantiles from histograms and other grouped data

We introduce methods to obtain confidence intervals for quantiles when only grouped data are available. Our preferred method is to approximate the underlying density using the Generalized Lambda Distribution to estimate the quantiles and variance of the estimators. We compare this method with a frequency approximation approach and another method that uses a linear interpolation approximation of the density. Simulations show that intervals with excellent coverage can be found for a wide number of distributions, including those that are highly skewed distributions. We also apply our methods to real data sets highlighting how intervals can be computed from just histograms.

A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications

In the present work, first, a new fractional numerical differentiation formula (called the L1-2 formula) to approximate the Caputo fractional derivative of order α (0<α<1) is developed. It is established by means of the quadratic interpolation approximation using three points (tj−2,f(tj−2)),(tj−1,f(tj−1)) and (tj,f(tj)) for the integrand f(t) on each small interval [tj−1,tj] (j⩾2), while the linear interpolation approximation is applied on the first small interval [t0,t1]. As a result, the new formula can be formally viewed as a modification of the classical L1 formula, which is obtained by the piecewise linear approximation for f(t). Both the computational efficiency and numerical accuracy of the new formula are superior to that of the L1 formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of L1-2 formula. Second, by the new formula, two improved finite difference schemes with high order accuracy in time for solving the time-fractional sub-diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed, respectively. In addition, the application of the new formula into solving fractional ordinary differential equations is also presented. Several numerical examples are computed. The comparison with the corresponding results of finite difference methods by the L1 formula demonstrates that the new L1-2 formula is much more effective and more accurate than the L1 formula when solving time-fractional differential equations numerically.

The support of the solution for stochastic differential equations driven by G-Brownian motion

By a linear interpolation approximation method, we obtain a characterization of the support of the solution for stochastic differential equations driven by G-Brownian motion.

Adaptive placement method on pricing arithmetic average options

Since there is no analytic solution for arithmetic average options until present, developing an efficient numerical algorithm becomes a promising alternative. One of the most famous numerical algorithms is introduced by Hull and White (J Deriv 1:21–31, 1993). Motivated by the common idea of reducing the nonlinearity error in the adaptive mesh model in Figlewski and Gao (J Financ Econ 53:313–351, 1999) and the adaptive quadrature method, we propose an adaptive placement method to replace the logarithmically equally-spaced placement rule in the Hull and White’s model by placing more representative average prices in the highly nonlinear area of the option value as the function of the arithmetic average stock price. The basic idea of this method is to design a recursive algorithm to limit the error of the linear interpolation between each pair of adjacent representative average prices. Numerical experiments verify the superior performance of this method for reducing the interpolation error and hence improving the convergence rate. To show that the adaptive placement method can improve any numerical algorithm with the techniques of augmented state variables and the piece-wise linear interpolation approximation, we also demonstrate how to integrate the adaptive placement method into the GARCH option pricing algorithm in Ritchken and Trevor (J Finance 54:377–402, 1999). Similarly great improvement of the convergence rate suggests the potential applications of this novel method to a broad class of numerical pricing algorithms for exotic options and complex underlying processes.

Compositional dependence of longitudinal sound velocities of piezoelectric (111) InxGa(1−x)As measured by picosecond ultrasonics

Picosecond ultrasonic experiments were performed to generate and detect acoustic pulses in piezoelectric (111) epilayers of InxGa(1−x)As, and the longitudinal sound velocities in the [111] direction of InxGa(1−x)As were thus directly determined by controlling the indium content x. We found that the sound velocity exhibits an obvious bowing with different alloy compositions and can be described as a quadratic function of VL(x)=458x2−1451x+5414. This result supports a nonlinear compositional dependence of the elastic constants of InxGa(1−x)As and indicates that the linear-interpolation approximation is inapplicable for estimating sound velocity in this specific case.

Evaluation of boundary element methods for the EEG forward problem: effect of linear interpolation.

We implement the approach for solving the boundary integral equation for the electroencephalography (EEG) forward problem proposed by de Munck [1], in which the electric potential varies linearly across each plane triangle of the mesh. Previous solutions have assumed the potential is constant across an element. We calculate the electric potential and systematically investigate the effect of different mesh choices and dipole locations by using a three concentric sphere head model for which there is an analytic solution. Implementing the linear interpolation approximation results in errors that are approximately half those of the same mesh when the potential is assumed to be constant, and provides a reliable method for solving the problem.

Systematic consideration of the superiority of generalized linear interpolation approximation

AbstractThis paper considers the set of waveforms for which the weighted power‐of‐p (1 &lt; p &lt; +∞) integral of the Fourier spectrum is bounded. Since different weight functions can be used in different frequency bands, as long as they are positive, the restriction of the waveform by the power‐of‐p integral is sufficiently general. It is assumed that the waveform belonging to such a set is added to several given linear time‐invariant filters. A theory of the optimum approximation of the original waveform is established using the sampling sequences of the corresponding outputs. The measure of the approximation error is defined as the upper limit of the absolute value of the difference between the original and the approximating waveforms when the original waveforms are changed one by one over the whole set of waveforms.Considering the sampling points, which substantially affect the approximation at a fixed time t, a theorem is presented which indicates that there exists a waveform which has zero value at all such sampling points and has the absolute value of the approximation error which reaches the above upper limit. It is shown also that a similar result can be established for the multidimensional waveform.In this paper, it is shown that the proposed method of linear approximation is better than any other approximation using the same sample values, including the nonlinear signal processing, as long as the forementioned set of waveforms and the measure for the approximation error are considered.

A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control

Using the decomposition of solution of SDE, we consider the stochastic optimal control problem with anticipative controls as a family of deterministic control problems parametrized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier stochastic process (nonanticipativity equality constraint). It is shown that the value function of these problems is the unique global solution of a robust equation (random partial differential equation) associated to a linear backward Hamilton-Jacobi-Bellman stochastic partial differential equation (HJB SPDE). This appears as limiting SPDE for a sequence of random HJB PDE's when linear interpolation approximation of the Wiener process is used. Our approach extends the Wong-Zakai type results [20] from SDE to the stochastic dynamic programming equation by showing how this arises as average of the limit of a sequence of deterministic dynamic programming equations. The stochastic characteristics method of Kunita [13] is used to represent the ...

Study of the superiority of generalized linear interpolation approximation

Consider a set of waveforms whose Fourier spectra are band-limited within a certain frequency band. We assume that the weighted Lp integrals (1<p< +∞) of the Fourier spectra are bounded with respect to a given weight function. We present a method which optimally approximates the original waveform contained in the above set with respect to an appropriate measure of error. We use a series of sample values of the outputs obtained when the waveform passes through a given finite number of linear time-invariant filters. Here, we establish a theorem which shows that there exists a waveform among all those waveforms whose sample values at the sample points are all zero, whose error is a maximum at some arbitrary fixed time t. Further, under the condition that p = 2 and the sample values, which are in general not all zero, are given, we prove that the upper limit of approximation error is proportional to the maximum of absolute value of the approximation error described above. As a result, it is shown that as long as the set of waveforms and the measure of approximation error mentioned above are adopted, the proposed linear approximation method is superior to all other approximation methods even including, for example, non-linear signal processing using the same sample values. Further, at the end of this paper, we show that our argument can be applied to more general sets of waveforms.