Approximation Of Bivariate Functions Research Articles

Approximation of Bivariate Functions by Generalized Wendland Radial Basis Functions

Approximation of Bivariate Functions by Generalized Wendland Radial Basis Functions

On approximation of bivariate functions by Abel–Poisson and conjugate Abel–Poisson means

On approximation of bivariate functions by Abel–Poisson and conjugate Abel–Poisson means

Precise non‐linear flux linkage model for permanent magnet synchronous motors based on current injection and bivariate function approximation

Precise non‐linear flux linkage model for permanent magnet synchronous motors based on current injection and bivariate function approximation

Approximation of Bivariate Functions by Fourier-Tchebychev ``Circular'' Sums in $L_{2,\rho}$

В работе вычислены точные верхние грани приближения функций двух переменных круговыми частичными суммами двойного ряда Фурье - Чебышева на классе функций $L_{2,\rho}^{(r)} (D)$, $r\in \mathbb{N},$ в пространстве $L_{2,\rho}:=L_{2,\rho}(Q)$, где $\rho:=\rho(x,y)=1/\sqrt{(1-x^{2})(1-y^{2})}$, $Q:=\{(x,y):-1\leq x,y\leq1\}$, $D$ - оператор Чебышева - Эрмита второго порядка. Получены точные неравенства, в которых величины наилучших полиномиальных приближений оцениваются сверху посредством усредненных с весом значений обобщенных модулей непрерывности $m$-го порядка производной $D^r f$ $(r\in \mathbb{Z}_+)$ в метрике пространства $L_{2,\rho}$. Даны точные оценки наилучших приближений двойного ряда Фурье по ортогональным системам Фурье - Чебышева на классах функций многих переменных, характеризующихся обобщенным модулем непрерывности. Так как, в отличие от одномерного случая для двойных рядов, нет естественного способа построения частичных сумм, то мы строим некоторые классы функций, а затем соответствующий метод приближения - "круговые" частичные суммы двойного ряда Фуре - Чебышева. В вопросах, связанных с разложениями функций в ряд Фурье по тригонометрической системе и оценки их наилучших приближений, большую роль играют операторы сдвига. В работе, указывая на некоторые ранее известные результаты, построен оператор обобщенного сдвига, который позволяет определить класс функций, характеризующийся обобщенным модулем непрерывности. На этих классах вычислена верхняя грань значений, наилучшее среднеквадратическое приближение некоторых классов функций "круговыми" частичными суммами двойных рядов Фурье - Чебышева.

Saturation rates of filtered back projection approximations

This paper concerns the approximation of bivariate functions by using the well-known filtered back projection (FBP) formula from computerized tomography. We prove error estimates and convergence rates for the FBP approximation of target functions from Sobolev spaces mathrm H^alpha ({mathbb {R}}^2) of fractional order alpha >0, where we bound the FBP approximation error, which is incurred by the application of a low-pass filter, with respect to the weaker norms of the rougher Sobolev spaces mathrm H^sigma ({mathbb {R}}^2), for {0 le sigma le alpha }. In particular, we generalize our previous results to non band-limited filter functions and show that the decay rate of the error saturates at fractional order depending on smoothness properties of the filter’s window function at the origin. The theoretical results are supported by numerical simulations.

Singular value decomposition versus sparse grids: refined complexity estimates

Abstract We compare the cost complexities of two approximation schemes for functions that live on the product domain $\varOmega _1\times \varOmega _2$ of sufficiently smooth domains $\varOmega _1\subset \mathbb{R}^{n_1}$ and $\varOmega _2\subset \mathbb{R}^{n_2}$, namely the singular value / Karhunen–Lòeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders $r_1$ and $r_2$ of accuracy are given on the domains $\varOmega _1$ and $\varOmega _2$, respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation, which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from the study by Griebel & Harbrecht (2014, Approximation of bi-variate functions. Singular value decomposition versus sparse grids. IMA J. Numer. Anal., 34, 28–54). Especially, we consider the approximation of functions from generalized isotropic and anisotropic Sobolev spaces.

Hardware-Efficient QR-Decomposition Using Bivariate Numeric Function Approximation

Bivariate function approximation has proven its feasibility in terms of hardware-efficient arithmetic signal processing. However, its impact on high performance QR decomposition (QRD) has only been roughly studied so far. In this paper, a novel hardware architecture for Givens-Rotation-based QRD is proposed targeting hardware efficient signal processing. To this end, an ingenious triangular systolic array structure is considered. Complex-valued matrices are efficiently processed by means of a sophisticated bivariate numeric function approximation methodology. In order to get a comprehensive insight in the performance, exhaustive evaluation is carried out with a modern multi-antenna wireless communication system. In detail, the proposed QRD hardware architecture is used in a suitable channel pre-coding setup. For a meaningful proof-of-concept, our work is evaluated on several levels of the computing stack. In addition, our design is implemented and physically synthesized in a state-of-the-art 65-nm Taiwan Semiconductor Manufacturing Company technology and compared with other publications. The results indicate our approach to be a powerful solution for hardware-based QRD, especially in terms of energy and area requirements.

Approximation of Bivariate Functions via Smooth Extensions

For a smooth bivariate function defined on a general domain with arbitrary shape, it is difficult to do Fourier approximation or wavelet approximation. In order to solve these problems, in this paper, we give an extension of the bivariate function on a general domain with arbitrary shape to a smooth, periodic function in the whole space or to a smooth, compactly supported function in the whole space. These smooth extensions have simple and clear representations which are determined by this bivariate function and some polynomials. After that, we expand the smooth, periodic function into a Fourier series or a periodic wavelet series or we expand the smooth, compactly supported function into a wavelet series. Since our extensions are smooth, the obtained Fourier coefficients or wavelet coefficients decay very fast. Since our extension tools are polynomials, the moment theorem shows that a lot of wavelet coefficients vanish. From this, with the help of well-known approximation theorems, using our extension methods, the Fourier approximation and the wavelet approximation of the bivariate function on the general domain with small error are obtained.

Approximation of bi-variate functions: singular value decomposition versus sparse grids

We compare the cost complexities of two approximation schemes for functions $fin H^p(Omega_1times Omega_2)$ which live on the product domain $Omega_1timesOmega_2$ of sufficiently smooth domains $Omega_1subsetmathbb{R}^{n_1}$ and $Omega_2subsetmathbb{R}^{n_2}$, namely the singular value / Karhunen-L`oeve decomposition and the sparse grid representation. Here we assume that suitable finite element methods with associated {em fixed} order $r$ of accuracy are given on the domains $Omega_1$ and $Omega_2$. Then, the sparse grid approximation essentially needs only $mathcal{O}(varepsilon^{-q})$ with $q=frac{max{n_1,n_2}}{r}$ unknowns to reach a prescribed accuracy $varepsilon$ provided that the smoothness of $f$ satisfies $pge rfrac{n_1+n_2}{max{n_1,n_2}}$, which is an almost optimal rate. The singular value decomposition produces this rate only if $f$ is analytical since otherwise the decay of the singular values is not fast enough. If $p>rfrac{n_1+n_2}{max{n_1,n_2}}$, then the sparse grid approach gives essentially the rate $mathcal{O}(varepsilon^{-q})$ with $q=frac{n_1+n_2}{p}$ while, for the singular value decomposition, we can only prove the rate $mathcal{O}(varepsilon^{-q})$ with $q=frac{2min{r,p}min{n_1,n_2}+2pmax{n_1,n_2}}{(2p-min{n_1,n_2})min{r,p}}$. We derive the resulting complexities, compare the two approaches and present numerical results which demonstrate that these rates are also achieved in numerical practice. $p frac{n_1+n_2}{max{n_1,n_2}}$, then the sparse grid approach gives essentially the rate $mathcal{O} (varepsilon^{-q})$ with $q=frac{n_1+n_2}{p}$ while, for the singular value decomposition, we can only prove the rate $mathcal{O}(varepsilon^{-q})$ with $q=frac{2min{r,p}min{n_1,n_2} +2pmax{n_1,n_2}}{(2p-min{n_1,n_2})min{r,p}}$. We derive the resulting complexities, compare the two approaches and present numerical results which demonstrate that these rates are also achieved in numerical practi

The approximation of bivariate functions by modified bivariate operators and GBS operators associated

In this paper we demonstrate a Voronovskaja-type theorem and approximation theorem for a class of modified operators and Generalized Boolean Sum (GBS) associated operators obtained (see (3)) from given operators.

The approximation of bivariate functions by bivariate operators and GBS operators

In this paper we demonstrate a general approximation theorem for the bivariate functions by bivariate operators and GBS (Generalized Boolean Sum) operators.

On error formulas for approximation by sums of univariate functions

The purpose of the paper is to develop a new method for obtaining explicit formulas for the error of approximation of bivariate functions by sums of univariate functions. It should be remarked that formulas of this type have been known only for functions defined on a rectangle with sides parallel to coordinate axes. Our method, based on a maximization process over closed bolts, allows the consideration of functions defined on a hexagon or octagon with sides parallel to coordinate axes.

Near minimax polynomial approximation

In this paper a representation for linear polynomial projection is given. Then it is shown that the near minimax approximation is Equivalant to a problem of the best approximation of bivariate functions. And some properties for near minimax are obtained.

On the failure of proximinality of tensor-product subspaces

An important open problem concerning the approximation of bivariate functions by separable functions is whether thc tensor-product subspace, C( S) ⊗ H + G ⊗ C( T), is proximinal in C( S × T), when H and G are Haar subspaces of C( T) and C( S), respectively. In the present paper, we prove that, in general, this subspace is not proximinal.

A test function theorem and apporoximation by pseudopolynomials

We prove a Korovkin-type theorem on approximation of bivariate functions in the space ofB-continuous functions (introduced by K. Bögel in 1934). As consequences, some sequences of uniformly approximating pseudopolynomials are obtained.

The approximation of bivariate functions by sums of univariate ones using the L1-metric

We shall study a special case of the following abstract approximation problem: givena normed linear space E and two subspaces, M1 and M2, of E, we seek to approximate f ∈ E by elements in the sum of M1 and M2. In particular, we might ask whether closest points to f from M = M1 + M2 exist, and if so, how they are characterised. If we can define proximity maps p1 and p2 for M1 and M2, respectively, then an algorithm analogous to the one given by Diliberto and Straus [4] can be defined by the formulae

Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones

The problem of finding optimal cycles in a doubly weighted directed graph (Problem A) is closely related to the problem of approximating bivariate functions by the sum of two univariate functions with respect to the supremum norm (Problem B). The close relationship between Problem A and Problem B is detected by the characterization (7.4) of the distance dist (f, t) of Problem B. In Part 1 we construct an algorithm for Problem A where the essential role is played by the minimal lengthsy j(k) defined by (2.3). If weight functiont?1 then the minimum of Problem A is computed by equality (2.4). Ift?1 then the minimum is obtained by a binary search procedure, Algorithm 3. In Part 2 we construct our algorithms for solving Problem B by following exactly the ideas of Part 1. By Algorithm 4 we compute the minimal pseudolengthsh k(y, M) defined by (7.5). If weight functiont?1 then the infimum dist(f,t) of Problem B is obtained by equality (7.12) which is closely related to (2.4). Ift?1 we compute the infimum dist(f,t) by the binary search procedure Algorithm 5. Additionally, Algorithm 4 leads to a constructive proof of the existence of continuous optimal solutions of Problem B (see Theorem 7.1e) which is already known in caset?1 but unknown in caset?1. Interesting applications to the steady-state behaviour of industrial processes with interference (Sect. 3) and the solution of integral equations (Problem C) are included.