Interpolatory estimates in monotone piecewise polynomial approximation

Corrigendum to “Interpolatory estimates in monotone piecewise polynomial approximation” [J. Approx. Theory 223 (2017) 1–8

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

Let I be a finite interval and r,s∈N. Given a set M, of functions defined on I, denote by Δ+sM the subset of all functions y∈M such that the s-difference Δτsy(⋅) is nonnegative on I, ∀τ>0. Further, denote by Δ+sWpr, the class of functions x on I with the seminorm ‖x(r)‖Lp≤1, such that Δτsx≥0, τ>0. Let Mn(hk):={∑i=1ncihk(wit−θi)∣ci,wi, θi∈R, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions hk(t)=t+k, t∈R, k∈N0. We give two-sided estimates both of the best unconstrained approximation E(Δ+sWpr,Mn(hk))Lq, k=r−1,r, s=0,1,...,r+1, and of the best s-monotonicity preserving approximation E(Δ+sWpr,Δ+sMn(hk))Lq, k=r−1,r, s=0,1,...,r+1. The most significant results are contained in theorem 2.2.

SynopsisIf Ω is a bounded domain in Rn satisfying certain conditions, Ωk denotes its intersection with a k-dimensional hyperplane, 1 ≦ k ≦ n, it is shown that the embedding of the Sobolev space Ws,p(Ω), s>0, into Lq(Ωk) is of type lm if for q<p<∞. The same result is obtained for the space of Bessel potentials Ls,p(Ω). Piecewise polynomial and Fourier approximations of functions and interpolation theorems areused.

Over the last decades the finite element method, which was introduced by engineers in the 1960s, has become the perhaps most important numerical method for partial differential equations, particularly for equations of elliptic and parabolic types. This method is based on the variational form of the boundary value problem and approximates the exact solution by a piecewise polynomial function. It is more easily adapted to the geometry of the underlying domain than the finite difference method, and for symmetric positive definite elliptic problems it reduces to a finite linear system with a symmetric positive definite matrix. We first introduce this method in Sect. 5.1 for the case of a two-point boundary value problem and show a number of error estimates. In Sect. 5.2 we then formulate the method for a two-dimensional model problem. Here the piecewise polynomial approximations are defined on triangulations of the spatial domain, and in the following Sect. 5.3 we study such approximation in more detail. In Sect. 5.4 we show basic error estimates for the finite element method for the model problem, using piecewise linear approximating functions. All error bounds derived up to this point contain a norm of the unknown exact solution and are therefore often referred to as a priori error estimates. In Sect. 5.5 we show a so-called a posteriori error estimate in which the error bound is expressed in terms of the data of the problem and the computed solution. In Sect. 5.6 we analyze the effect of numerical integration, which is often used when the finite element equations are assembled in a computer program. In Sect. 5.7 we briefly describe a so-called mixed finite element method.KeywordsFinite Element MethodElliptic EquationQuadrature FormulaInterpolation ErrorPosteriori Error EstimateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In practice, we are interested in approximating a function f(x) near x = a by a polynomial of degree M. Other issues involved in approximation are how good the approximation is and over what interval can we expect the approximation to be good. In this paper we present approximation that is realized as piecewise polynomial function where in each interval ƒ(x) is approximated by a Taylor polynomial of low order. The piecewise polynomial approximation can be effectively implemented using Farrow structure.

The Brain Computer Interface (BCI) is a technology that provides non-muscular communication channels to utilize the external devices. The signal from the brain is recorded, then classified. To classify, time-frequency features are frequently extracted using Continuous Wavelet Transform (CWT). However, the CWT is computationally heavy. This becomes the bottleneck of the real-time processing of BCI systems. We introduce a new method to solve the CWT using piecewise polynomials approximation over mother wavelets. The implementation shows that our method, piecewise polynomials based CWT (PCWT), is clearly better than the state of the art computation using the Fast Fourier Transform in the smaller buffer, which is preferable in the real-time systems. The proposed method can apply to other applications and should provide an effective approach to the real-time signal processing.

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

The piecewise polynomial collocation method is discussed to solve linear Volterra-Basset integro-differential equations with weakly singular or other nonsmooth kernels. Using special graded grids, global convergence estimates are derived. The error analysis is based on certain regularity properties of the solution of the initial value problem.

This paper presents an event-driven simulation methodology for injection-locked oscillators. The proposed method adopts a phase-domain macromodel based on a perturbation projection vector (PPV), which expresses the oscillator's phase response using a nonlinear, time-varying differential equation. By expressing the PPV using a piecewise polynomial function, the PPV model can be simulated in an event-driven fashion on a digital logic simulator like SystemVerilog. While this piecewise polynomial approximation introduces a trade-off between the simulation accuracy and speed, our results show that the simulation accuracy quickly reaches its optimal value as the piecewise interval number increases even with first-order polynomials. The experimental results with an LC oscillator, ring oscillator, and burst-mode CDR models operating with both periodic and aperiodic inputs demonstrate that the proposed models achieve speeds up to 2 ~ 30× that of transistor-level Spectre simulations with the same accuracy.

Continuous numerical methods for ODEs with defect control

High frequency scattering by convex curvilinear polygons

The two-level local projection stabilisation on triangular meshes is based on the refinement of a macro cell into three child cells by connecting the barycentre with the vertices of the macro cell. This refinement technique leads to non-nested meshes with large inner angles and to non-nested finite element spaces. We show that also the red refinement where a triangle is divided into four child cells by connecting the midpoints of the edges can be used. This avoids the above mentioned disadvantages. For the red refinement a local inf-sup condition for the continuous, piecewise polynomial approximation spaces of order less than or equal to r???2 living on the refined mesh and discontinuous, piecewise polynomial projection spaces of order less than or equal to r???1 living on the coarser mesh is established. Numerical tests compare the local projection stabilisation resulting from both refinement rules in case of convection-diffusion problems.

The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F -spaces N 0 p , n {\mathbf {N}}_0^{p,n} (introduced for this purpose in [2]). We obtain two o -results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in N 0 p , n {\mathbf {N}}_0^{p,n} we define a derivative D n , σ f {D^{n,\sigma }}f in L σ , σ = ( n + p − 1 ) − 1 {L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}} , which agrees with D n f {D^n}f for smooth f , and prove several properties of D n , σ {D^{n,\sigma }} .

A given real continuous function f on [a, b] is approximated by polynomials P n {P_n} of degree n that are subject to certain restrictions. Let 1 ≦ k 1 > ⋯ > k p ≦ n 1 \leqq {k_1} > \cdots > {k_p} \leqq n be given integers, ε i = ± 1 {\varepsilon _i} = \pm 1 , given signs. It is assumed that P n ( k i ) ( x ) P_n^{({k_i})}(x) has the sign of ε i , i = 1 , … , p , a ≦ x ≦ b {\varepsilon _i},i = 1, \ldots ,p,a \leqq x \leqq b . Theorems are obtained which describe the polynomials of best approximation, and (for p = 1 p = 1 ) establish their uniqueness. Relations to Birkhoff interpolation problems are of importance. Another tool are the sets A, where | f ( x ) − P n ( x ) | |f(x) - {P_n}(x)| attains its maximum, and the sets B i {B_i} with P n ( k i ) ( x ) = 0 P_n^{({k_i})}(x) = 0 . Conditions are discussed which these sets must satisfy for a polynomial P n {P_n} of best approximation for f. Numbers of the points of sets A, B i {B_i} are studied, the possibility of certain extreme situations established. For example, if p = 1 , k 1 = 1 , n = 2 q + 1 p = 1,{k_1} = 1,n = 2q + 1 , it is possible that | A | = 3 , | B | = n |A| = 3,|B| = n .