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Journal articles on the topic 'Function approximation'

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Author: Grafiati

Published: 4 June 2021

Last updated: 31 January 2023

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1

Ludanov, Konstantin. "METHOD OF OBTAINING APPROXIMATE FORMULAS." EUREKA: Physics and Engineering 2 (March 30, 2018): 72–78. http://dx.doi.org/10.21303/2461-4262.2018.00589.

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The two-parameter method of approximating the sum of a power series in terms of its first three terms of the expansion, which allows one to obtain analytic approximations of various functions, decomposes into a Maclaurin series. As an approximation function of this approximation, it is proposed to use elementary functions constructed in the Nth degree, but with a "compressed" or "stretched" variable x due to the introduction of the numerical factor M (x ≡ ε ∙ m, M ≠ 0) into it. The use of this method makes it possible to significantly increase the range of very accurate approximation of the obtained approximate function with respect to a similar range of the output fragment of a series of three terms. Expressions for both the approximation parameters (M and N) are obtained in a general form and are determined by the coefficients of the second and third terms of the Maclaurin series. Also expressions of both approximation parameters are found for the case if the basis function and the approximant function decompose into the Maclaurin series in even powers of the argument. A number of examples of approximation of functions on the basis of the analysis of power series into which they decompose are given.

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2

Howard, Roy M. "Dual Taylor Series, Spline Based Function and Integral Approximation and Applications." Mathematical and Computational Applications 24, no. 2 (April 1, 2019): 35. http://dx.doi.org/10.3390/mca24020035.

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In this paper, function approximation is utilized to establish functional series approximations to integrals. The starting point is the definition of a dual Taylor series, which is a natural extension of a Taylor series, and spline based series approximation. It is shown that a spline based series approximation to an integral yields, in general, a higher accuracy for a set order of approximation than a dual Taylor series, a Taylor series and an antiderivative series. A spline based series for an integral has many applications and indicative examples are detailed. These include a series for the exponential function, which coincides with a Padé series, new series for the logarithm function as well as new series for integral defined functions such as the Fresnel Sine integral function. It is shown that these series are more accurate and have larger regions of convergence than corresponding Taylor series. The spline based series for an integral can be used to define algorithms for highly accurate approximations for the logarithm function, the exponential function, rational numbers to a fractional power and the inverse sine, inverse cosine and inverse tangent functions. These algorithms are used to establish highly accurate approximations for π and Catalan’s constant. The use of sub-intervals allows the region of convergence for an integral approximation to be extended.

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3

Howard, Roy M. "Arbitrarily Accurate Analytical Approximations for the Error Function." Mathematical and Computational Applications 27, no. 1 (February 9, 2022): 14. http://dx.doi.org/10.3390/mca27010014.

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A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be improved by utilizing the approximation erf(x)≈1 for |x|>xo and with xo optimally chosen. Two generalizations are possible; the first is based on demarcating the integration interval into m equally spaced subintervals. The second, is based on utilizing a larger fixed subinterval, with a known integral, and a smaller subinterval whose integral is to be approximated. Both generalizations lead to significantly improved accuracy. Furthermore, the initial approximations, and those arising from the first generalization, can be utilized as inputs to a custom dynamic system to establish approximations with better convergence properties. Indicative results include those of a fourth-order approximation, based on four subintervals, which leads to a relative error bound of 1.43 × 10−7 over the interval [0, ∞]. The corresponding sixteenth-order approximation achieves a relative error bound of 2.01 × 10−19. Various approximations that achieve the set relative error bounds of 10−4, 10−6, 10−10, and 10−16, over [0, ∞], are specified. Applications include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for exp(−x2) that have significantly higher convergence properties than a Taylor series approximation. Fourth, the definition of a complementary demarcation function eC(x) that satisfies the constraint eC2(x)+erf2(x)=1. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to an error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal that is modeled by the error function.

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4

Patseika, Pavel G., Yauheni A. Rouba, and Kanstantin A. Smatrytski. "On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions." Journal of the Belarusian State University. Mathematics and Informatics, no. 2 (July 30, 2020): 6–27. http://dx.doi.org/10.33581/2520-6508-2020-2-6-27.

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The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a > 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

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5

Malachivskyy, Petro. "Chebyshev approximation of the multivariable functions by some nonlinear expressions." Physico-mathematical modelling and informational technologies, no. 33 (September 2, 2021): 18–22. http://dx.doi.org/10.15407/fmmit2021.33.018.

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A method for constructing a Chebyshev approximation of the multivariable functions by exponential, logarithmic and power expressions is proposed. It consists in reducing the problem of the Chebyshev approximation by a nonlinear expression to the construction of an intermediate Chebyshev approximation by a generalized polynomial. The intermediate Chebyshev approximation by a generalized polynomial is calculated for the values of a certain functional transformation of the function we are approximating. The construction of the Chebyshev approximation of the multivariable functions by a polynomial is realized by an iterative scheme based on the method of least squares with a variable weight function.

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6

Wang, Zi, Aws Albarghouthi, Gautam Prakriya, and Somesh Jha. "Interval universal approximation for neural networks." Proceedings of the ACM on Programming Languages 6, POPL (January 16, 2022): 1–29. http://dx.doi.org/10.1145/3498675.

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To verify safety and robustness of neural networks, researchers have successfully applied abstract interpretation , primarily using the interval abstract domain. In this paper, we study the theoretical power and limits of the interval domain for neural-network verification. First, we introduce the interval universal approximation (IUA) theorem. IUA shows that neural networks not only can approximate any continuous function f (universal approximation) as we have known for decades, but we can find a neural network, using any well-behaved activation function, whose interval bounds are an arbitrarily close approximation of the set semantics of f (the result of applying f to a set of inputs). We call this notion of approximation interval approximation . Our theorem generalizes the recent result of Baader et al. from ReLUs to a rich class of activation functions that we call squashable functions . Additionally, the IUA theorem implies that we can always construct provably robust neural networks under ℓ ∞ -norm using almost any practical activation function. Second, we study the computational complexity of constructing neural networks that are amenable to precise interval analysis. This is a crucial question, as our constructive proof of IUA is exponential in the size of the approximation domain. We boil this question down to the problem of approximating the range of a neural network with squashable activation functions. We show that the range approximation problem (RA) is a Δ 2 -intermediate problem, which is strictly harder than NP -complete problems, assuming coNP ⊄ NP . As a result, IUA is an inherently hard problem : No matter what abstract domain or computational tools we consider to achieve interval approximation, there is no efficient construction of such a universal approximator. This implies that it is hard to construct a provably robust network, even if we have a robust network to start with.

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7

Patseika, Pavel G., and Yauheni A. Rouba. "Fejer means of rational Fourier – Chebyshev series and approximation of function |x|s." Journal of the Belarusian State University. Mathematics and Informatics, no. 3 (November 29, 2019): 18–34. http://dx.doi.org/10.33581/2520-6508-2019-3-18-34.

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Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function |x|s, 0 < s < 2, on the interval [−1,1], are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function |x|s, 0 < s < 2, on the interval [−1,1], the asymptotic expressions under n→∞ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found.

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8

Zakharchenko, S. M., N. A. Shydlovska, and I. L. Mazurenko. "DISCREPANCY PARAMETERS OF APPROXIMATIONS OF DISCRETELY SPECIFIED DEPENDENCIES BY ANALYTICAL FUNCTIONS AND SEARCH CRITERIA FOR OPTIMAL VALUES OF THEIR COEFFICIENTS." Praci Institutu elektrodinamiki Nacionalanoi akademii nauk Ukraini 2021, no. 59 (September 20, 2021): 11–19. http://dx.doi.org/10.15407/publishing2021.59.011.

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Universal discrepancy parameters of approximations of discretely specified dependencies by analytical functions and search criteria for optimal values of their coefficients, as well as analysis of features of their application are described. Discrepancy parameters of approximations, which do not depend on the ranges of variation of the values of functions and the number of points of a discretely specified dependence, are proposed. They can be effective for objectively comparing the quality of approximations of any dependencies by any functions. Approximations of a discretely specified dependence of the mathematical expectation of the equivalent electrical resistance of a layer of aluminum granules during spark-erosion dispersion in water on the instantaneous values of the discharge current are carried out. As approximating functions, we chose a power function with an exponent factor –1 and a function based on exponential. Using the criteria of the least approximation error, the optimal values of the coefficients of both approximating functions are founded. It is shown in which cases it is advisable to use the combined search criteria for the optimal values of the coefficients of the approximating functions, and in which are enough simple one-component ones. Ref. 27, fig. 2, tables 2.

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9

GREPL, MARTIN A. "CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS." Mathematical Models and Methods in Applied Sciences 22, no. 03 (March 2012): 1150015. http://dx.doi.org/10.1142/s0218202511500151.

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We present reduced basis approximations and associated a posteriori error bounds for parabolic partial differential equations involving (i) a nonaffine dependence on the parameter and (ii ) a nonlinear dependence on the field variable. The method employs the Empirical Interpolation Method in order to construct "affine" coefficient-function approximations of the "nonaffine" (or nonlinear) parametrized functions. We consider linear time-invariant as well as linear time-varying nonaffine functions and introduce a new sampling approach to generate the function approximation space for the latter case. Our a posteriori error bounds take both error contributions explicitly into account — the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. We show that these bounds are rigorous upper bounds for the approximation error under certain conditions on the function interpolation, thus addressing the demand for certainty of the approximation. As regards efficiency, we develop an offline–online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for the many-query or real-time contexts. Numerical results are presented to confirm and test our approach.

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10

M.Nasir, Haniffa, and Kamel Nafa. "A New Second Order Approximation for Fractional Derivatives with Applications." Sultan Qaboos University Journal for Science [SQUJS] 23, no. 1 (May 6, 2018): 43. http://dx.doi.org/10.24200/squjs.vol23iss1pp43-55.

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We propose a generalized theory to construct higher order Grünwald type approximations for fractional derivatives. We use this generalization to simplify the proofs of orders for existing approximation forms for the fractional derivative. We also construct a set of higher order Grünwald type approximations for fractional derivatives in terms of a general real sequence and its generating function. From this, a second order approximation with shift is shown to be useful in approximating steady state problems and time dependent fractional diffusion problems. Stability and convergence for a Crank-Nicolson type scheme for this second order approximation are analyzed and are supported by numerical results.

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11

Bagul, Yogesh J., and Christophe Chesneau. "Sigmoid functions for the smooth approximation to the absolute value function." Moroccan Journal of Pure and Applied Analysis 7, no. 1 (January 1, 2021): 12–19. http://dx.doi.org/10.2478/mjpaa-2021-0002.

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AbstractWe present smooth approximations to the absolute value function |x| using sigmoid functions. In particular, x erf(x/μ) is proved to be a better smooth approximation for |x| than x tanh(x/μ) and \sqrt {{x^2} + \mu } with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for the error function.

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12

DeVore, Ronald, Boris Hanin, and Guergana Petrova. "Neural network approximation." Acta Numerica 30 (May 2021): 327–444. http://dx.doi.org/10.1017/s0962492921000052.

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Neural networks (NNs) are the method of choice for building learning algorithms. They are now being investigated for other numerical tasks such as solving high-dimensional partial differential equations. Their popularity stems from their empirical success on several challenging learning problems (computer chess/Go, autonomous navigation, face recognition). However, most scholars agree that a convincing theoretical explanation for this success is still lacking. Since these applications revolve around approximating an unknown function from data observations, part of the answer must involve the ability of NNs to produce accurate approximations.This article surveys the known approximation properties of the outputs of NNs with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis, such as approximations using polynomials, wavelets, rational functions and splines. Comparisons are made with traditional approximation methods from the viewpoint of rate distortion, i.e. error versus the number of parameters used to create the approximant. Another major component in the analysis of numerical approximation is the computational time needed to construct the approximation, and this in turn is intimately connected with the stability of the approximation algorithm. So the stability of numerical approximation using NNs is a large part of the analysis put forward.The survey, for the most part, is concerned with NNs using the popular ReLU activation function. In this case the outputs of the NNs are piecewise linear functions on rather complicated partitions of the domain of f into cells that are convex polytopes. When the architecture of the NN is fixed and the parameters are allowed to vary, the set of output functions of the NN is a parametrized nonlinear manifold. It is shown that this manifold has certain space-filling properties leading to an increased ability to approximate (better rate distortion) but at the expense of numerical stability. The space filling creates the challenge to the numerical method of finding best or good parameter choices when trying to approximate.

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13

N. Ivanshin, Pyotr, and . "Continued Fractions and Conformal Mappings for Domains with Angel Points." International Journal of Engineering & Technology 7, no. 4.7 (September 27, 2018): 409. http://dx.doi.org/10.14419/ijet.v7i4.7.23039.

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Here we construct the conformal mappings with the help of the continued fraction approximations. We first show that the method of [19] works for conformal mappings of the unit disk onto domains with acute external angles at the boundary. We give certain illustrative examples of these constructions. Next we outline the problem with domains which boudary possesses acute internal angles. Then we construct the method of rational root approximation in the right complex half-plane. First we construct the square root approximation and consider approximative properties of the mapping sequence in Theorem 1. Then we turn to the general case, namely, the continued fraction approximation of the rational root function in the complex right half-plane. These approximations converge to the algebraic root functions , , , . This is proved in Theorem 2 of the aricle. Thus we prove convergence of this method and construct conformal approximate mappings of the unit disk onto domains with angles and thin domains. We estimate the convergence rate of the approximation sequences. Note that the closer the point is to zero or infinity and the lower is the ratio k/N the worse is the approximation. Also we give the examples that illustrate the conformal mapping construction.

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14

Eldracher, Martin, Alexander Staller, and René Pompl. "Adaptive Encoding Strongly Improves Function Approximation with CMAC." Neural Computation 9, no. 2 (February 1, 1997): 403–17. http://dx.doi.org/10.1162/neco.1997.9.2.403.

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The Cerebellar Model Arithmetic Computer (CMAC) (Albus 1981) is well known as a good function approximator with local generalization abilities. Depending on the smoothness of the function to be approximated, the resolution as the smallest distinguishable part of the input domain plays a crucial role. If the binary quantizing functions in CMAC are dropped in favor of more general, continuous-valued functions, much better results in function approximation for smooth functions are obtained in shorter training time with less memory consumption. For functions with discontinuities, we obtain a further improvement by adapting the continuous encoding proposed in Eldracher and Geiger (1994) for difficult-to-approximate areas. Based on the already far better function approximation capability on continuous functions with a fixed topologically distributed encoding scheme in CMAC (Eldracher et al. 1994), we present the better results in learning a two-valued function with discontinuity using this adaptive topologically distributed encoding scheme in CMAC.

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15

Shvai, Olga, and Kateryna Pozharska. "On some approximation properties of Gauss-Weierstrass singular operators." Ukrainian Mathematical Bulletin 18, no. 4 (November 12, 2021): 560–68. http://dx.doi.org/10.37069/1810-3200-2021-18-4-7.

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Approximation theorems were formulated for function continuous in the neighborhood of some point $x$, $-\infty <x<\infty $. Namely, the upper bounds were obtained for the function approximations by their Gauss-Weierstrass singular operators in terms of a majorant function for the first- and second-order continuity moduli of the relevant functions.

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16

Peric, Zoran H., Aleksandar V. Markovic, Natasa Z. Kontrec, Stefan R. Panic, and Petar C. Spalević. "Novel Composite Approximation for the Gaussian Q-Function." Elektronika ir Elektrotechnika 26, no. 5 (October 27, 2020): 33–38. http://dx.doi.org/10.5755/j01.eie.26.5.26012.

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This paper, by using Borjesson’s and Benitez’s approximation of Q-function, presents a novel and improved composite approximation of Q-function with wide applicability. The presented approach is very general and can be implemented on any observed interval. Based on the proposed approximation of Q-function, the average bit error rate is assessed by observing the transfer over Nakagami-m fading channel. The simplicity of the proposed approximation form in conjunction with yet another feature - utmost accurateness - appeared to be a better choice than the suggested approximations of similar complexity in terms of analyticity. The paper emphasizes the wide implementation possibilities in numerous tasks of communication theory and functional analysis that include Q-function.

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17

Zaidi, H. R. "Green's function equations of motion for driven two-level atoms." Canadian Journal of Physics 63, no. 3 (March 1, 1985): 314–26. http://dx.doi.org/10.1139/p85-049.

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Using a diagram technique, equations of motion for N coherently driven two-level atoms are obtained from Green's functions. The longitudinal (Coulomb) dipole–dipole interaction is neglected. Three approximations (Born approximation, extended Born approximation, and screened interaction approximation) are considered. The existing theories are shown to be equivalent to the extended Born approximation. For a small sample, the equations of motion exhibit S conservation in this case. The screened interaction model predicts a different behaviour; in particular, S-breaking terms appear in the equations of motion for a small sample.

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18

Beare, Brendan K. "Distributional Replication." Entropy 23, no. 8 (August 17, 2021): 1063. http://dx.doi.org/10.3390/e23081063.

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A function which transforms a continuous random variable such that it has a specified distribution is called a replicating function. We suppose that functions may be assigned a price, and study an optimization problem in which the cheapest approximation to a replicating function is sought. Under suitable regularity conditions, including a bound on the entropy of the set of candidate approximations, we show that the optimal approximation comes close to achieving distributional replication, and close to achieving the minimum cost among replicating functions. We discuss the relevance of our results to the financial literature on hedge fund replication; in this case, the optimal approximation corresponds to the cheapest portfolio of market index options which delivers the hedge fund return distribution.

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19

TRIPŞA, Florența Violeta. "THE APPROXIMATION OF A CONTINUOUS FUNCTION USING BERNSTEIN POLYNOMIALS." SCIENTIFIC RESEARCH AND EDUCATION IN THE AIR FORCE 20 (June 18, 2018): 299–306. http://dx.doi.org/10.19062/2247-3173.2018.20.39.

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20

Perić, Zoran, Aleksandar Marković, Nataša Kontrec, Jelena Nikolić, Marko D. Petković, and Aleksandra Jovanović. "Two Interval Upper-Bound Q-Function Approximations with Applications." Mathematics 10, no. 19 (October 1, 2022): 3590. http://dx.doi.org/10.3390/math10193590.

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The Gaussian Q-function has considerable applications in numerous areas of science and engineering. However, the fact that a closed-form expression for this function does not exist encourages finding approximations or bounds of the Q-function. In this paper, we determine analytically two novel interval upper bound Q-function approximations and show that they could be used efficiently not only for the symbol error probability (SEP) estimation of transmission over Nakagami-m fading channels, but also for the average symbol error probability (ASEP) evaluation for two modulation formats. Specifically, we determine analytically the composition of the upper bound Q-function approximations specified at disjoint intervals of the input argument values so as to provide the highest accuracy within the intervals, by utilizing the selected one of two upper bound Q-function approximations. We show that a further increase of the accuracy, achieved in the case with two upper-bound approximations composing the interval approximation, can be obtained by forming a composite interval approximation of the Q-function that assumes another extra interval and by specifying the third form for the upper-bound Q-function approximation. The proposed analytical approach can be considered universal and widely applicable. The results presented in the paper indicate that the proposed Q-function approximations outperform in terms of accuracy other well-known approximations carefully chosen for comparison purposes. This approximation can be used in numerous theoretical communication problems based on the Q-function calculation. In this paper, we apply it to estimate the average bit error rate (ABER), when the transmission in a Nakagami-m fading channel is observed for the assumed binary phase-shift keying (BPSK) and differentially encoded quadrature phase-shift keying (DE-QPSK) modulation formats, as well as to design scalar quantization with equiprobable cells for variables from a Gaussian source.

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21

Huang, Changcun. "ReLU Networks Are Universal Approximators via Piecewise Linear or Constant Functions." Neural Computation 32, no. 11 (November 2020): 2249–78. http://dx.doi.org/10.1162/neco_a_01316.

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This letter proves that a ReLU network can approximate any continuous function with arbitrary precision by means of piecewise linear or constant approximations. For univariate function [Formula: see text], we use the composite of ReLUs to produce a line segment; all of the subnetworks of line segments comprise a ReLU network, which is a piecewise linear approximation to [Formula: see text]. For multivariate function [Formula: see text], ReLU networks are constructed to approximate a piecewise linear function derived from triangulation methods approximating [Formula: see text]. A neural unit called TRLU is designed by a ReLU network; the piecewise constant approximation, such as Haar wavelets, is implemented by rectifying the linear output of a ReLU network via TRLUs. New interpretations of deep layers, as well as some other results, are also presented.

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22

Sauer, A. "Meromorphic functions with prescribed asymptotic behaviour, zeros and poles and applications in complex approximation." Canadian Journal of Mathematics 51, no. 1 (February 1, 1999): 117–29. http://dx.doi.org/10.4153/cjm-1999-007-2.

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AbstractWe construct meromorphic functions with asymptotic power series expansion in z−1 at ∞ on an Arakelyan set A having prescribed zeros and poles outside A. We use our results to prove approximation theorems where the approximating function fulfills interpolation restrictions outside the set of approximation.

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23

Nikolić, Jelena, Zoran Perić, and Aleksandar Marković. "Proposal of Simple and Accurate Two-Parametric Approximation for the Q-Function." Mathematical Problems in Engineering 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/8140487.

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The approximations for the Q-function reported in the literature so far have mainly been developed to overcome not only the difficulties, but also the limitations, caused in different research areas, by the nonexistence of the closed form expression for the Q-function. Unlike the previous papers, we propose the novel approximation for the Q-function not for solving some particular problem. Instead, we analyze this problem in one general manner and we provide one general solution, which has wide applicability. Specifically, in this paper, we set two goals, which are somewhat contrary to each other. The one is the simplicity of the analytical form of Q-function approximation and the other is the relatively high accuracy of the approximation for a wide range of arguments. Since we propose a two-parametric approximation for the Q-function, by examining the effect of the parameters choice on the accuracy of the approximation, we manage to determine the most suitable parameters of approximation and to achieve these goals simultaneously. The simplicity of the analytical form of our approximation along with its relatively high accuracy, which is comparable to or even better than that of the previously proposed approximations of similar analytical form complexity, indicates its wide applicability.

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24

Malachivskyy, P., L. Melnychok, and Ya Pizyur. "Chebyshev approximation of multivariable functions with the interpolation." Mathematical Modeling and Computing 9, no. 3 (2022): 757–66. http://dx.doi.org/10.23939/mmc2022.03.757.

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A method of constructing a Chebyshev approximation of multivariable functions by a generalized polynomial with the exact reproduction of its values at a given points is proposed. It is based on the sequential construction of mean-power approximations, taking into account the interpolation condition. The mean-power approximation is calculated using an iterative scheme based on the method of least squares with the variable weight function. An algorithm for calculating the Chebyshev approximation parameters with the interpolation condition for absolute and relative error is described. The presented results of solving test examples confirm the rapid convergence of the method when calculating the parameters of the Chebyshev approximation of tabular continuous functions of one, two and three variables with the reproduction of the values of the function at given points.

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25

NEWMAN, RHYS. "A FUNCTION APPROXIMATION ALGORITHM USING SEQUENTIAL COMPOSITION." International Journal of Neural Systems 04, no. 02 (June 1993): 187–99. http://dx.doi.org/10.1142/s012906579300016x.

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A new method for approximating one dimensional functions is developed based on structural capabilities of multilayer feedforward neural networks. It possesses notable but unproven convergence properties which are examined in a series of examples. It is shown that it outperforms conventional networks for complicated one dimensional problems. An adaptive version of the algorithm whose approximation changes to best use the data available is also presented. Experiments indicate that this method is very stable in the presence of noise. For straightforward function approximation however, other conventional routines generally perform better. Nevertheless, noise stability, adaptability and other properties make the new method useful in context.

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26

Pal, Mahendra Kumar, M. L. L. Wijerathne, and Muneo Hori. "Numerical Modeling of Brittle Cracks Using Higher Order Particle Discretization Scheme–FEM." International Journal of Computational Methods 16, no. 04 (May 13, 2019): 1843006. http://dx.doi.org/10.1142/s0219876218430065.

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Higher order extension of Particle Discretization Scheme (HO-PDS), its implementation in FEM framework (HO-PDS-FEM) and applications in efficiently simulating cracks are presented in this paper. PDS is an approximation scheme which uses a conjugate domain tessellation pair like Voronoi and Delaunay in approximating a function and its derivatives. In approximating a function (or derivatives), HO-PDS first produces local polynomial approximations for the target function (or derivatives) within each element of respective tessellation. The approximations over the whole domain are then obtained by taking the union of those respective local approximations. These approximations are inherently discontinuous along the boundaries of the respective tessellation elements since the support of the local approximations is confined to the domain of respective tessellation elements and no continuity conditions are enforced. HO-PDS-FEM utilizes these inherent discontinuities in function approximation to efficiently model discontinuities such as cracks. Higher order PDS is implemented in FEM framework to solve boundary value problem of elastic solids, including mode-I crack problems. With several benchmark problems, it is shown that HO-PDS-FEM has higher expected accuracy and convergence rate. J-integral around a mode-I crack tip is calculated to demonstrate the improvement in the accuracy of the crack tip stress field. Further, it is shown that HO-PDS-FEM significantly improves the traction along the crack surfaces, compared to the zeroth-order PDS-FEM [Hori, M., Oguni, K. and Sakaguchi, H. [2005] “Proposal of FEM implemented with particle discretization scheme for analysis of failure phenomena,” J. Mech. Phys. Solids 53, 681–703].

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Rosado, Alberto López, Federico Prieto Muñoz, and Roberto Alvarez Fernández. "An Analytic Expression for the Inverse Involute." Mathematical Problems in Engineering 2019 (September 29, 2019): 1–9. http://dx.doi.org/10.1155/2019/3586012.

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This article introduces new types of rational approximations of the inverse involute function, widely used in gear engineering, allowing the processing of this function with a very low error. This approximated function is appropriate for engineering applications, with a much reduced number of operations than previous formulae in the existing literature, and a very efficient computation. The proposed expressions avoid the use of iterative methods. The theoretical foundations of the approximation theory of rational functions, the Chebyshev and Jacobi polynomials that allow these approximations to be obtained, are presented in this work, and an adaptation of the Remez algorithm is also provided, which gets a null error at the origin. This way, approximations in ranges or degrees different from those presented here can be obtained. A rational approximation of the direct involute function is computed, which avoids the computation of the tangent function. Finally, the direct polar equation of the circle involute curve is approximated with some application examples.

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Thonet, Thibaut, Yagmur Gizem Cinar, Eric Gaussier, Minghan Li, and Jean-Michel Renders. "Listwise Learning to Rank Based on Approximate Rank Indicators." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 8 (June 28, 2022): 8494–502. http://dx.doi.org/10.1609/aaai.v36i8.20826.

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We study here a way to approximate information retrieval metrics through a softmax-based approximation of the rank indicator function. Indeed, this latter function is a key component in the design of information retrieval metrics, as well as in the design of the ranking and sorting functions. Obtaining a good approximation for it thus opens the door to differentiable approximations of many evaluation measures that can in turn be used in neural end-to-end approaches. We first prove theoretically that the approximations proposed are of good quality, prior to validate them experimentally on both learning to rank and text-based information retrieval tasks.

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Kivinukk, Andi, Anna Saksa, and Maria Zeltser. "On a cosine operator function framework of approximation processes in Banach space." Filomat 33, no. 13 (2019): 4213–28. http://dx.doi.org/10.2298/fil1913213k.

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We introduce the cosine-type approximation processes in abstract Banach space setting. The historical roots of these processes go back to W. W. Rogosinski in 1926. The given new definitions use a cosine operator functions concept. We proved that in presented setting the cosine-type operators possess the order of approximation, which coincide with results known in trigonometric approximation. Moreover, a general method for factorization of certain linear combinations of cosine operator functions is presented. The given method allows to find the order of approximation using the higher order modulus of continuity. Also applications for the different type of approximations are given.

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Patseika, P. G., and Y. A. Rovba. "On approximations of the function |x|^s by the Vallee Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 3 (October 7, 2019): 263–82. http://dx.doi.org/10.29235/1561-2430-2019-55-3-263-282.

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The approximative properties of the Valle Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions in the approximation of the function |x|s, 0 < s < 2 are investigated. The introduction presents the main results of the previously known works on the Vallee Poussin means in the polynomial and rational cases, as well as on the known literature data on the approximations of functions with power singularity. The Valle Poussin means on the interval [–1,1] as a method of summing the Fourier series by one system of the Chebyshev – Markov rational fractions are introduced. In the main section of the article, a integral representation for the error of approximations by the rational Valle Poussin means of the function |x|s, 0 < s < 2, on the segment [–1,1], an estimate of deviations of the Valle Poussin means from the function |x|s, 0 < s < 2, depending on the position of the point on the segment, a uniform estimate of deviations on the segment [–1,1] and its asymptotic expression are found. The optimal value of the parameter is obtained, at which the deviation error of the Valle Poussin means from the function |x|s, 0 < s <2, on the interval [–1,1] has the highest velocity of zero. As a consequence of the obtained results, the problem of approximation of the function |x|s, s > 0, by the Valle Poussin means of the Fourier series by the system of the Chebyshev first-kind polynomials is studied in detail. The pointwise estimation of approximation and asymptotic estimation are established.The work is both theoretical and applied. Its results can be used to read special courses at mathematical faculties and to solve specific problems of computational mathematics.

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Free, J. W., A. R. Parkinson, G. R. Bryce, and R. J. Balling. "Approximation of Computationally Expensive and Noisy Functions for Constrained Nonlinear Optimization." Journal of Mechanisms, Transmissions, and Automation in Design 109, no. 4 (December 1, 1987): 528–32. http://dx.doi.org/10.1115/1.3258832.

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The use of statistical experimental designs is explored as a method of approximating computationally expensive and noisy functions. The advantages of experimental designs and function approximation for use in optimization are discussed. Several test problems are reported showing the approximation method to be competitive with the most efficient optimization algorithms when no noise is present. When noise is introduced, the approximation method is more efficient and solves more problems than conventional nonlinear programming algorithms.

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Smila, T. H., and L. L. Pecherytsia. "Adaptation of gas-dynamic characteristic arrays to automated ballistics support of spacecraft flight." Technical mechanics 2021, no. 4 (December 7, 2021): 89–103. http://dx.doi.org/10.15407/itm2021.04.089.

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The current level of the design and use of new-generation spacecraft calls for a maximally automated ballistics support of engineering developments. An integral part of the solution of this problem is the development of an effective tool to adapt discrete functions of gas-dynamic characteristics to the solution of various problems that arise in the development and use of space complexes. Simplifying the use of bulky information arrays together with improving the accuracy of approximation of key coefficients will significantly improve the ballistics support quality. The aim of this work is to choose an optimum method for the approximation of a discrete function of two variable spacecraft aerodynamic characteristics. Based on the analysis of the advantages and drawbacks of basic methods of approximation by two fitting criteria: the maximum error and the root-mean-square deviation, recommendations on this choice were made. The methods were assessed by the example of the aerodynamic coefficients of the Sich-2M spacecraft’s simplified geometrical model tabulated as a function of the spacecraft orientation angles relative to the incident flow velocity. Multiparameter numerical studies were conducted for different approximation methods with varying the parameters of the approximation types under consideration and the approximation grid density. It was found that increasing the number of nodes of an input array does not always improve the accuracy of approximation. The node arrangement exerts a greater effect on the approximation quality. It was established that the most easily implementable method among those considered is a step interpolation, whose advantages are simplicity, quickness, and limitless possibilities in accuracy improvement, while its significant drawbacks are the lack of an analytical description and the dependence of the accuracy on the grid density. It was shown that spline functions feature the best approximating properties in comparison with other mathematical models. A polynomial approximation or any approximation by a general form function provide an analytical description with a single approximating function, but their accuracy of approximation is not so high as that provided by splines. It was found that there exists no approximation method that would be best by all criteria taken together: each method has some advantages, but at the same time, it has significant drawbacks too. An optimum approximation method is chosen according to the features of the problem, the priorities in approximation requirements, the required degree of approximation, and the initial data organization method.

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Anastassiou, George A. "Multivariate and abstract approximation theory for Banach space valued functions." Demonstratio Mathematica 50, no. 1 (August 28, 2017): 208–22. http://dx.doi.org/10.1515/dema-2017-0020.

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Abstract Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

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Yessenbayeva, Gulsim A., Gulmira A. Yessenbayeva, A. T. Kasimov, and N. K. Syzdykova. "On the boundedness of the partial sums operator for the Fourier series in the function classes families associated with harmonic intervals." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 103, no. 3 (September 30, 2021): 131–39. http://dx.doi.org/10.31489/2021m3/131-139.

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The article is devoted to the study of some data from the theory of functions approximation by trigonometric polynomials with a spectrum from special sets called harmonic intervals. Due to the limited perception range of devices, the perception range of the senses of the person himself, when studying a mathematical model it is often enough to find an approximation of the object so that the error (noise, interference, distortion) is outside the interval of perception. Harmonic intervals model problems of this kind to some extent. In the article the main components of the approximation theory of functions by trigonometric polynomials with a spectrum from harmonic intervals are presented, the theorem on estimating the best approximation of a function by trigonometric polynomials through the best approximations of a function by trigonometric polynomials with a spectrum from harmonic intervals is proved. Theorems on the boundedness of the partial sums operator for the Fourier series in the function classes families associated with harmonic intervals are considered; such a theorem for the Lorentz space is generalized and proved. The article is mainly aimed at scientific researchers dealing with practical applications of the approximation theory of functions by trigonometric polynomials with a spectrum from special sets.

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Hajjar, Amjad F., and Mohammad H. Awedh. "Efficient Logarithmic Function Approximation." International Journal of Scientific Engineering and Technology 4, no. 7 (July 1, 2015): 387–91. http://dx.doi.org/10.17950/ijset/v4s7/701.

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Eshghi, Nasim, Lothar Reichel, and Miodrag M. Spalević. "Enhanced matrix function approximation." ETNA - Electronic Transactions on Numerical Analysis 27 (2018): 197–206. http://dx.doi.org/10.1553/etna_vol47s197.

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KYU, Shon MIN, Junichi MURATA, and Kotaro HIRASAWA. "Function Approximation Using LVQ." Transactions of the Society of Instrument and Control Engineers 39, no. 5 (2003): 513–19. http://dx.doi.org/10.9746/sicetr1965.39.513.

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PARKER, PHILIP J., and BRIAN D. O. ANDERSON. "Unstable rational function approximation." International Journal of Control 46, no. 5 (November 1987): 1783–801. http://dx.doi.org/10.1080/00207178708934010.

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39

Hannan, E. J. "Rational Transfer Function Approximation." Statistical Science 2, no. 2 (May 1987): 135–51. http://dx.doi.org/10.1214/ss/1177013343.

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40

Sejeeni, Fowzi Ahmed. "Approximation in function modules." Bulletin of the Australian Mathematical Society 43, no. 2 (April 1991): 295–302. http://dx.doi.org/10.1017/s0004972700029087.

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We investigate the existence of best approximation of an element α in a function module from a subfunction module whose fibers satisfy the intersection property of balls. Also we investigate the lower semicontinuity of the metric projection associated with such a subfunction module.

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41

Ylostalo, J. "Function approximation using polynomials." IEEE Signal Processing Magazine 23, no. 5 (September 2006): 99–102. http://dx.doi.org/10.1109/msp.2006.1708417.

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42

Wang, Dong Dong, You Jun Chen, and Hai Jie Pang. "Rational Function Functional Networks Based on Function Approximation." Applied Mechanics and Materials 220-223 (November 2012): 2264–68. http://dx.doi.org/10.4028/www.scientific.net/amm.220-223.2264.

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In order to solve function approximation, a mathematic model of Rational Function Functional Networks (RFFN) based on approximation was proposed and the learning algorithm for function approximation was presented. This algorithm used the lease square method thought and constructed auxiliary function by Lagrange multiplier method, and the parameters of the rational function functional networks were determined by solving a system of linear equations. Results illustrate the effectiveness of the rational function functional networks in solving approximation problems of the function with a pole.

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43

Chukhrova, Nataliya, and Arne Johannssen. "Improved binomial and Poisson approximations to the Type-A operating characteristic function." International Journal of Quality & Reliability Management 36, no. 4 (April 1, 2019): 620–52. http://dx.doi.org/10.1108/ijqrm-10-2017-0203.

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PurposeIn acceptance sampling, the hypergeometric operating characteristic (OC) function (so called type-A OC) is used to be approximated by the binomial or Poisson OC function, which actually reduce computational effort, but do not provide suffcient approximation results. The purpose of this paper is to examine binomial- and Poisson-type approximations to the hypergeometric distribution, in order to find a simple but accurate approximation that can be successfully applied in acceptance sampling.Design/methodology/approachThe authors present a new binomial-type approximation for the type-A OC function, and derive its properties. Further, the authors compare this approximation via an extensive numerical study with other common approximations in terms of variation distance and relative efficiency under various conditions on the parameters including limiting cases.FindingsThe introduced approximation generates best numerical results over a wide range of parameter values, and ensures arithmetic simplicity of the binomial distribution and high accuracy to meet requirements regarding acceptance sampling problems. Additionally, it can considerably reduce the computational effort in relation to the type-A OC function and therefore is strongly recommended for calculating sampling plans.Originality/valueThe newly presented approximation provides a remarkably close fit to the type-A OC function, is discrete and needs no correction for continuity, and is skewed in the same direction by roughly the same amount as the exact OC. Due to less factorials, this OC in general involves lower powers than the type-A OC function. Moreover, the binomial-type approximation is easy to fit to the conventional statistical computing packages.

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44

Patseika, P. G., and Y. A. Rouba. "The Abel – Poisson means of conjugate Fourier – Chebyshev series and their approximation properties." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 57, no. 2 (July 16, 2021): 156–75. http://dx.doi.org/10.29235/1561-2430-2021-57-2-156-175.

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Herein, the approximation properties of the Abel – Poisson means of rational conjugate Fourier series on the system of the Chebyshev–Markov algebraic fractions are studied, and the approximations of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] by this method are investigated. In the introduction, the results related to the study of the polynomial and rational approximations of conjugate functions are presented. The conjugate Fourier series on one system of the Chebyshev – Markov algebraic fractions is constructed. In the main part of the article, the integral representation of the approximations of conjugate functions on the segment [–1,1] by the method under study is established, the asymptotically exact upper bounds of deviations of conjugate Abel – Poisson means on classes of conjugate functions when the function satisfies the Lipschitz condition on the segment [–1,1] are found, and the approximations of the conjugate Abel – Poisson means of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] are studied. Estimates of the approximations are obtained, and the asymptotic expression of the majorant of the approximations in the final part is found. The optimal value of the parameter at which the greatest rate of decreasing the majorant is provided is found. As a consequence of the obtained results, the problem of approximating the conjugate function with density | x |s , s ∈(1, 2), by the Abel – Poisson means of conjugate polynomial series on the system of Chebyshev polynomials of the first kind is studied in detail. Estimates of the approximations are established, as well as the asymptotic expression of the majorants of the approximations. This work is of both theoretical and applied nature. It can be used when reading special courses at mathematical faculties and for solving specific problems of computational mathematics.

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45

Maldonado, Wilfredo L., and B. F. Svaiter. "Hölder continuity of the policy function approximation in the value function approximation." Journal of Mathematical Economics 43, no. 5 (June 2007): 629–39. http://dx.doi.org/10.1016/j.jmateco.2007.01.004.

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Park, J., and I. W. Sandberg. "Universal Approximation Using Radial-Basis-Function Networks." Neural Computation 3, no. 2 (June 1991): 246–57. http://dx.doi.org/10.1162/neco.1991.3.2.246.

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There have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables. Some of these studies deal with cases in which the hidden-layer nonlinearity is not a sigmoid. This was motivated by successful applications of feedforward networks with nonsigmoidal hidden-layer units. This paper reports on a related study of radial-basis-function (RBF) networks, and it is proved that RBF networks having one hidden layer are capable of universal approximation. Here the emphasis is on the case of typical RBF networks, and the results show that a certain class of RBF networks with the same smoothing factor in each kernel node is broad enough for universal approximation.

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Opschoor, Joost A. A., Philipp C. Petersen, and Christoph Schwab. "Deep ReLU networks and high-order finite element methods." Analysis and Applications 18, no. 05 (February 21, 2020): 715–70. http://dx.doi.org/10.1142/s0219530519410136.

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Approximation rate bounds for emulations of real-valued functions on intervals by deep neural networks (DNNs) are established. The approximation results are given for DNNs based on ReLU activation functions. The approximation error is measured with respect to Sobolev norms. It is shown that ReLU DNNs allow for essentially the same approximation rates as nonlinear, variable-order, free-knot (or so-called “[Formula: see text]-adaptive”) spline approximations and spectral approximations, for a wide range of Sobolev and Besov spaces. In particular, exponential convergence rates in terms of the DNN size for univariate, piecewise Gevrey functions with point singularities are established. Combined with recent results on ReLU DNN approximation of rational, oscillatory, and high-dimensional functions, this corroborates that continuous, piecewise affine ReLU DNNs afford algebraic and exponential convergence rate bounds which are comparable to “best in class” schemes for several important function classes of high and infinite smoothness. Using composition of DNNs, we also prove that radial-like functions obtained as compositions of the above with the Euclidean norm and, possibly, anisotropic affine changes of co-ordinates can be emulated at exponential rate in terms of the DNN size and depth without the curse of dimensionality.

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Anandakrishnan, Ramu. "A Partition Function Approximation Using Elementary Symmetric Functions." PLoS ONE 7, no. 12 (December 12, 2012): e51352. http://dx.doi.org/10.1371/journal.pone.0051352.

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49

Freedman, M. A. "Riemann step function approximation of Bochner integrable functions." Proceedings of the American Mathematical Society 96, no. 4 (April 1, 1986): 605. http://dx.doi.org/10.1090/s0002-9939-1986-0826489-4.

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Leinonen, Leena, Marko Leinonen, and Tapani Matala-aho. "On approximation measures of q-exponential function." International Journal of Number Theory 12, no. 01 (February 2016): 287–303. http://dx.doi.org/10.1142/s1793042116500172.

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We shall present effective approximation measures for certain infinite products related to [Formula: see text]-exponential function [Formula: see text]. There are two main targets. First we shall prove an explicit irrationality measure result for the values [Formula: see text], where [Formula: see text], and [Formula: see text], [Formula: see text]. Then, if we restrict the approximations of [Formula: see text] to rational numbers of the shape [Formula: see text], where [Formula: see text], we may replace Bundschuh’s irrationality exponent [Formula: see text] by [Formula: see text]

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