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Journal articles on the topic 'Local approximation of a function'

Author: Grafiati

Published: 25 May 2024

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1

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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Fan, Jiang, Qinghao Yuan, Fulei Jing, Hongbin Xu, Hao Wang, and Qingze Meng. "Adaptive Local Maximum-Entropy Surrogate Model and Its Application to Turbine Disk Reliability Analysis." Aerospace 9, no. 7 (June 30, 2022): 353. http://dx.doi.org/10.3390/aerospace9070353.

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The emerging Local Maximum-Entropy (LME) approximation, which combines the advantages of global and local approximations, has an unsolved issue wherein it cannot adaptively change the morphology of the basis function according to the local characteristics of the sample, which greatly limits its highly nonlinear approximation ability. In this research, a novel Adaptive Local Maximum-Entropy Surrogate Model (ALMESM) is proposed by constructing an algorithm that adaptively changes the LME basis function and introduces Particle Swarm Optimization to ensure the optimality of the adaptively changed basis function. The performance of the ALMESM is systematically investigated by comparison with the LME approximation, a Radial basis function, and the Kriging model in two explicit highly nonlinear mathematical functions. The results show that the ALMESM has the highest accuracy and stability of all the compared models. The ALMESM is further validated by a highly nonlinear engineering case, consisting of a turbine disk reliability analysis under geometrical uncertainty, and achieves a desirable result. Compared with the direct Monte Carlo method, the relative error of the ALMESM is less than 1%, which indicates that the ALMESM has considerable potential for highly nonlinear problems and structural reliability analysis.

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3

Chen, Yuanqiang, H. Zheng, Wei Li, and Shan Lin. "MLS based local approximation in numerical manifold method." Engineering Computations 35, no. 7 (October 1, 2018): 2429–58. http://dx.doi.org/10.1108/ec-12-2017-0485.

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Purpose The purpose of this paper is to propose a new three-node triangular element in the framework of the numerical manifold method (NMM), which is designated by Trig3-MLScns. Design/methodology/approach The formulation uses the improved parametric shape functions of classical triangular elements (Trig3-0) to construct the partition of unity (PU) and the moving least square (MLS) interpolation method to construct the local approximation function. Findings Compared with the classical three-node element (Trig3-0), the Trig3-MLScns element has a higher order of approximations, much better accuracy and continuous nodal stress. Moreover, the linear dependence problem associated with many PU-based methods with high-order approximations is eliminated in the present element. A number of numerical examples indicate the high accuracy and robustness of the Trig3-MLScns element. Originality/value The proposed element inherits the individual merits of the NMM and the MLS.

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4

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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5

HESSE, KERSTIN, and Q. T. LE GIA. "LOCAL RADIAL BASIS FUNCTION APPROXIMATION ON THE SPHERE." Bulletin of the Australian Mathematical Society 77, no. 2 (April 2008): 197–224. http://dx.doi.org/10.1017/s0004972708000087.

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AbstractIn this paper we derive local error estimates for radial basis function interpolation on the unit sphere $\mathbb {S}^2\subset \mathbb {R}^3$. More precisely, we consider radial basis function interpolation based on data on a (global or local) point set $X\subset \mathbb {S}^2$ for functions in the Sobolev space $H^s(\mathbb {S}^2)$ with norm $\|\cdot \|_s$, where s>1. The zonal positive definite continuous kernel ϕ, which defines the radial basis function, is chosen such that its native space can be identified with $H^s(\mathbb {S}^2)$. Under these assumptions we derive a local estimate for the uniform error on a spherical cap S(z;r): the radial basis function interpolant ΛXf of $f\in H^s(\mathbb {S}^2)$ satisfies $\sup _{\mathbf {x}\in S(\mathbf {z};r)} |f(\mathbf {x})-\Lambda _X f(\mathbf {x})| \leq c h^{(s-1)/2} \|f\|_{s}$, where h=hX,S(z;r) is the local mesh norm of the point set X with respect to the spherical cap S(z;r). Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.

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6

Singh, Satwinder Jit, and Anindya Chatterjee. "Beyond fractional derivatives: local approximation of other convolution integrals." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2114 (October 29, 2009): 563–81. http://dx.doi.org/10.1098/rspa.2009.0378.

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Dynamic systems involving convolution integrals with decaying kernels, of which fractionally damped systems form a special case, are non-local in time and hence infinite dimensional. Straightforward numerical solution of such systems up to time t needs computations owing to the repeated evaluation of integrals over intervals that grow like t . Finite-dimensional and local approximations are thus desirable. We present here an approximation method which first rewrites the evolution equation as a coupled infinite-dimensional system with no convolution, and then uses Galerkin approximation with finite elements to obtain linear, finite-dimensional, constant coefficient approximations for the convolution. This paper is a broad generalization, based on a new insight, of our prior work with fractional order derivatives ( Singh & Chatterjee 2006 Nonlinear Dyn. 45 , 183–206). In particular, the decaying kernels we can address are now generalized to the Laplace transforms of known functions; of these, the power law kernel of fractional order differentiation is a special case. The approximation can be refined easily. The local nature of the approximation allows numerical solution up to time t with computations. Examples with several different kernels show excellent performance. A key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system, as distinct from numerically approximating just the solution for the given initial values; this allows non-standard uses of the approximation, e.g. in stability analyses.

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7

GATHERAL, JIM, and TAI-HO WANG. "THE HEAT-KERNEL MOST-LIKELY-PATH APPROXIMATION." International Journal of Theoretical and Applied Finance 15, no. 01 (February 2012): 1250001. http://dx.doi.org/10.1142/s021902491250001x.

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In this article, we derive a new most-likely-path (MLP) approximation for implied volatility in terms of local volatility, based on time-integration of the lowest order term in the heat-kernel expansion. This new approximation formula turns out to be a natural extension of the well-known formula of Berestycki, Busca and Florent. Various other MLP approximations have been suggested in the literature involving different choices of most-likely-path; our work fixes a natural definition of the most-likely-path. We confirm the improved performance of our new approximation relative to existing approximations in an explicit computation using a realistic S&P500 local volatility function.

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8

Generowicz, Jacek, Chris Harvey-Fros, and Tim R. Morris. "C function representation of the Local Potential Approximation." Physics Letters B 407, no. 1 (August 1997): 27–32. http://dx.doi.org/10.1016/s0370-2693(97)00729-6.

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9

Islam, Md Monirul, and Shyamapada Modak. "Second approximation of local functions in ideal topological spaces." Acta et Commentationes Universitatis Tartuensis de Mathematica 22, no. 2 (January 2, 2019): 245–56. http://dx.doi.org/10.12697/acutm.2018.22.20.

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This paper gives a new dimension to discuss the local function in ideal topological spaces. We calculate error operators for various type of local functions and introduce more perfect approximation of the local functions for discussing their properties. We have also reached a topological space with the help of semi-closure.

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Qasim, Mohd, M. Mursaleen, Asif Khan, and Zaheer Abbas. "Approximation by Generalized Lupaş Operators Based on q-Integers." Mathematics 8, no. 1 (January 2, 2020): 68. http://dx.doi.org/10.3390/math8010068.

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The purpose of this paper is to introduce q-analogues of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing, and unbounded function ρ . Depending on the selection of q, these operators provide more flexibility in approximation and the convergence is at least as fast as the generalized Lupaş operators, while retaining their approximation properties. For these operators, we give weighted approximations, Voronovskaja-type theorems, and quantitative estimates for the local approximation.

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11

Burova, I. G. "The Local Nonpolynomial Splines and Solution of Integro-Differential Equations." WSEAS TRANSACTIONS ON MATHEMATICS 21 (October 24, 2022): 718–30. http://dx.doi.org/10.37394/23206.2022.21.84.

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The application of the local polynomial splines to the solution of integro-differential equations was regarded in the author’s previous papers. In a recent paper, we introduced the application of the local nonpolynomial splines to the solution of integro-differential equations. These splines allow us to approximate functions with a presribed order of approximation. In this paper, we apply the splines to the solution of the integro-differential equations with a smooth kernel. Applying the trigonometric or exponential spline approximations of the fifth order of approximation, we obtain an approximate solution of the integro-differential equation at the set of nodes. The advantages of using such splines include the ability to determine not only the values of the desired function at the grid nodes, but also the first derivative at the grid nodes. The obtained values can be connected by lines using the splines. Thus, after interpolation, we can obtain the value of the solution at any point of the considered interval. Several numerical examples are given.

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12

Romanchak, V. M. "Local transformations with a singular wavelet." Informatics 17, no. 1 (March 29, 2020): 39–46. http://dx.doi.org/10.37661/1816-0301-2020-17-1-39-46.

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The paper considers a local wavelet transform with a singular basis wavelet. The problem of nonparametric approximation of a function is solved by the use of the sequence of local wavelet transforms. Traditionally believed that the wavelet should have an average equal to zero. Earlier, the author considered singular wavelets when the average value is not equal to zero. As an example, the delta-shaped functions, participated in the estimates of Parzen – Rosenblatt and Nadara – Watson, were used as a wavelet. Previously, a sequence of wavelet transforms for the entire numerical axis and finite interval was constructed for singular wavelets. The paper proposes a sequence of local wavelet transforms, a local wavelet transform is defined, the theorems that formulate the properties of a local wavelet transform are proved. To confirm the effectiveness of the algorithm an example of approximating the function by use of the sum of discrete local wavelet transforms is given.

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13

Poggio, Tomaso, and Federico Girosi. "A Sparse Representation for Function Approximation." Neural Computation 10, no. 6 (August 1, 1998): 1445–54. http://dx.doi.org/10.1162/089976698300017250.

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We derive a new general representation for a function as a linear combination of local correlation kernels at optimal sparse locations (and scales) and characterize its relation to principal component analysis, regularization, sparsity principles, and support vector machines.

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14

Regis, R. G., and C. A. Shoemaker. "Local Function Approximation in Evolutionary Algorithms for the Optimization of Costly Functions." IEEE Transactions on Evolutionary Computation 8, no. 5 (October 2004): 490–505. http://dx.doi.org/10.1109/tevc.2004.835247.

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Yeh, T. J., and K. Youcef-Toumi. "Adaptive Control of Nonlinear, Uncertain Systems Using Local Function Estimation." Journal of Dynamic Systems, Measurement, and Control 120, no. 4 (December 1, 1998): 429–38. http://dx.doi.org/10.1115/1.2801483.

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An adaptive control scheme is proposed for controlling a certain class of nonlinear, uncertain systems. When a local approximation of the system function using its Taylor’s expansion is possible, this scheme provides an adaptation law to estimate such an approximation. With a proper sampling rule, the neighborhood of approximation can be moved from time to time in order to capture the fast changing system dynamics. Practical implementation issues are also considered to avoid exciting the unmodeled dynamics, to reduce the noise sensitivity, and accommodate the various signal levels in the system response. The important features and performance of the proposed controller are illustrated through simulations and experimental results associated with a magnetic bearing system.

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16

Bakunin, O. G. "Non-local velocity distribution function and one-flight approximation." Physics Letters A 330, no. 1-2 (September 2004): 22–27. http://dx.doi.org/10.1016/j.physleta.2004.07.046.

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17

Wedge, D., D. Ingram, D. Mclean, C. Mingham, and Z. Bandar. "On Global–Local Artificial Neural Networks for Function Approximation." IEEE Transactions on Neural Networks 17, no. 4 (July 2006): 942–52. http://dx.doi.org/10.1109/tnn.2006.875972.

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18

Sladek, J., and V. Sladek. "A Trefftz function approximation in local boundary integral equations." Computational Mechanics 28, no. 3-4 (April 1, 2002): 212–19. http://dx.doi.org/10.1007/s00466-001-0282-y.

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19

Ganguly, Arijit, and Anish Ghosh. "Remarks on Diophantine approximation in function fields." MATHEMATICA SCANDINAVICA 124, no. 1 (January 13, 2019): 5–14. http://dx.doi.org/10.7146/math.scand.a-109985.

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20

Wehrberger, K., and F. Beck. "Relativistic random-phase-approximation response function for quasielastic electron scattering in local density approximation." Physical Review C 35, no. 1 (January 1, 1987): 298–304. http://dx.doi.org/10.1103/physrevc.35.298.

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21

BHARALI, GAUTAM. "POLYNOMIAL APPROXIMATION, LOCAL POLYNOMIAL CONVEXITY, AND DEGENERATE CR SINGULARITIES — II." International Journal of Mathematics 22, no. 12 (December 2011): 1721–33. http://dx.doi.org/10.1142/s0129167x11007446.

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We provide some conditions for the graph of a Hölder-continuous function on [Formula: see text], where [Formula: see text] is a closed disk in ℂ, to be polynomially convex. Almost all sufficient conditions known to date — provided the function (say F) is smooth — arise from versions of the Weierstrass Approximation Theorem on [Formula: see text]. These conditions often fail to yield any conclusion if rank ℝDF is not maximal on a sufficiently large subset of [Formula: see text]. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in ℂ2 at an isolated complex tangency.

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Burova, I. G., E. G. Ivanova, and V. A. Kostin. "Interval estimation of trigonometrical splines." MATEC Web of Conferences 292 (2019): 03001. http://dx.doi.org/10.1051/matecconf/201929203001.

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Quite often, it is necessary to quickly determine variation range of the function. If the function values are known at some points, then it is easy to construct the local spline approximation of this function and use the interval analysis rules. As a result, we get the area within which the approximation of this function changes. It is necessary to take into account the approximation error when studying the obtained area of change of function approximation. Thus, we get the range of changing the function with the approximation error. This paper discusses the features of using polynomial and trigonometrical splines of the third order approximation to determine the upper and lower boundaries of the area (domain) in which the values of the approximation are contained. Theorems of approximation by these local trigonometric and polynomial splines are formulated. The values of the constants in the estimates of the errors of approximation by the trigonometrical and polynomial splines are given. It is shown that these constants cannot be reduced. An algorithm for constructing the variation domain of the approximation of the function is described. The results of the numerical experiments are given.

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MAYORGA, RENÉ V., and MARIANO ARRIAGA. "NON-LINEAR GLOBAL OPTIMIZATION VIA PARAMETERIZATION AND INVERSE FUNCTION APPROXIMATION: AN ARTIFICIAL NEURAL NETWORKS APPROACH." International Journal of Neural Systems 17, no. 05 (October 2007): 353–68. http://dx.doi.org/10.1142/s0129065707001202.

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In this article, a novel technique for non-linear global optimization is presented. The main goal is to find the optimal global solution of non-linear problems avoiding sub-optimal local solutions or inflection points. The proposed technique is based on a two steps concept: properly keep decreasing the value of the objective function, and calculating the corresponding independent variables by approximating its inverse function. The decreasing process can continue even after reaching local minima and, in general, the algorithm stops when converging to solutions near the global minimum. The implementation of the proposed technique by conventional numerical methods may require a considerable computational effort on the approximation of the inverse function. Thus, here a novel Artificial Neural Network (ANN) approach is implemented to reduce the computational requirements of the proposed optimization technique. This approach is successfully tested on some highly non-linear functions possessing several local minima. The results obtained demonstrate that the proposed approach compares favorably over some current conventional numerical (Matlab functions) methods, and other non-conventional (Evolutionary Algorithms, Simulated Annealing) optimization methods.

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Wu, Sheng-Shiung, Sing-Jie Jong, Kai Hu, and Jiann-Ming Wu. "Learning Neural Representations and Local Embedding for Nonlinear Dimensionality Reduction Mapping." Mathematics 9, no. 9 (April 30, 2021): 1017. http://dx.doi.org/10.3390/math9091017.

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This work explores neural approximation for nonlinear dimensionality reduction mapping based on internal representations of graph-organized regular data supports. Given training observations are assumed as a sample from a high-dimensional space with an embedding low-dimensional manifold. An approximating function consisting of adaptable built-in parameters is optimized subject to given training observations by the proposed learning process, and verified for transformation of novel testing observations to images in the low-dimensional output space. Optimized internal representations sketch graph-organized supports of distributed data clusters and their representative images in the output space. On the basis, the approximating function is able to operate for testing without reserving original massive training observations. The neural approximating model contains multiple modules. Each activates a non-zero output for mapping in response to an input inside its correspondent local support. Graph-organized data supports have lateral interconnections for representing neighboring relations, inferring the minimal path between centroids of any two data supports, and proposing distance constraints for mapping all centroids to images in the output space. Following the distance-preserving principle, this work proposes Levenberg-Marquardt learning for optimizing images of centroids in the output space subject to given distance constraints, and further develops local embedding constraints for mapping during execution phase. Numerical simulations show the proposed neural approximation effective and reliable for nonlinear dimensionality reduction mapping.

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Ding, Shuo, Xiao Heng Chang, and Qing Hui Wu. "A Study on Approximation Performances of Improved Bp Neural Networks Based on LM Algorithms." Applied Mechanics and Materials 411-414 (September 2013): 1935–38. http://dx.doi.org/10.4028/www.scientific.net/amm.411-414.1935.

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When approximating nonlinear functions, standard BP algorithms and traditional improved BP algorithms have low convergence rate and tend to be stuck in local minimums. In this paper, standard BP algorithm is improved by numerical optimization algorithm. Firstly, the principle of Levenberg-Marquardt algorithm is introduced. Secondly, to test its approximation performance, LMBP neural network is programmed via MATLAB7.0 taking specific nonlinear function as an example. Thirdly, its approximation result is compared with those of standard BP algorithm and adaptive learning rate algorithm. Simulation results indicate that compared with standard BP algorithm and adaptive learning rate algorithm, LMBP algorithm overcomes deficiencies ranging from poor convergence ability, prolonged convergence time, increasing iteration steps to nonconvergence. Thus with its good approximation ability, LMBP algorithm is the most suitable for medium-sized networks.

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26

Van Neck, D., A. E. L. Dieperink, and E. Moya de Guerra. "Spectral function for finite nuclei in the local-density approximation." Physical Review C 51, no. 4 (April 1, 1995): 1800–1808. http://dx.doi.org/10.1103/physrevc.51.1800.

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27

Brookes, R. G., and A. W. McInnes. "The existence and local behaviour of the quadratic function approximation." Journal of Approximation Theory 62, no. 3 (September 1990): 383–95. http://dx.doi.org/10.1016/0021-9045(90)90061-t.

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28

Wu, G. Y. "Analysis of the local approximation in the Wigner function theory." Solid State Communications 90, no. 6 (May 1994): 397–400. http://dx.doi.org/10.1016/0038-1098(94)90807-9.

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Chengde, Zheng, and Wang Renhong. "Existence and local behavior of nondiagonal bivariate quadratic function approximation." Applied Mathematics-A Journal of Chinese Universities 18, no. 4 (December 2003): 442–52. http://dx.doi.org/10.1007/s11766-003-0071-9.

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Sturm, K. "Dynamic Structure Factor: An Introduction." Zeitschrift für Naturforschung A 48, no. 1-2 (February 1, 1993): 233–42. http://dx.doi.org/10.1515/zna-1993-1-244.

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Abstract The doubly differential cross-section for weak inelastic scattering of waves or particles by manybody systems is derived in Born approximation and expressed in terms of the dynamic structure factor according to van Hove. The application of this very general scheme to scattering of neutrons, x-rays and high-energy electrons is discussed briefly. The dynamic structure factor, which is the space and time Fourier transform of the density-density correlation function, is a property of the many-body system independent of the external probe and carries information on the excitation spectrum of the system. The relation of the electronic structure factor to the density-density response function defined in linear-response theory is shown using the fluctuation-dissipation theorem. This is important for calculations, since the response function can be calculated approximately from the independent-particle response function in self-consistent field approximations, such as the random-phase approximation or the local-density approximation of the density functional theory. Since the density-density response function also determines the dielectric function, the dynamic structure can be expressed by the dielectric function.

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Wang, Wei, Xin Chen, and Jianxin He. "Adaptive Critic Design with Local Gaussian Process Models." Journal of Advanced Computational Intelligence and Intelligent Informatics 20, no. 7 (December 20, 2016): 1135–40. http://dx.doi.org/10.20965/jaciii.2016.p1135.

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In this paper, local Gaussian process (GP) approximation is introduced to build the critic network of adaptive dynamic programming (ADP). The sample data are partitioned into local regions, and for each region, an individual GP model is utilized. The nearest local model is used to predict a given state-action point. With the two-phase value iteration method for a Gaussian-kernel (GK)-based critic network which realizes the update of the hyper-parameters and value functions simultaneously, fast value function approximation can be achieved. Combining this critic network with an actor network, we present a local GK-based ADP approach. Simulations were carried out to demonstrate the feasibility of the proposed approach.

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Wehrberger, K., and F. Beck. "Erratum: Relativistic random-phase-approximation response function for quasielastic electron scattering in local density approximation." Physical Review C 35, no. 6 (June 1, 1987): 2337. http://dx.doi.org/10.1103/physrevc.35.2337.

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Burova, I. G., and Yu K. Demyanovich. "Nonlenear Integro-differential Equations and Splines of the Fifth Order of Approximation." WSEAS TRANSACTIONS ON MATHEMATICS 21 (September 23, 2022): 691–700. http://dx.doi.org/10.37394/23206.2022.21.81.

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In this paper, we consider the solution of nonlinear Volterra–Fredholm integro-differential equation, which contains the first derivative of the function. Our method transforms the nonlinear Volterra-Fredholm integro-differential equations into a system of nonlinear algebraic equations. The method based on the application of the local polynomial splines of the fifth order of approximation is proposed. Theorems about the errors of the approximation of a function and its first derivative by these splines are given. With the help of the proposed splines, the function and the derivative are replaced by the corresponding approximation. Note that at the beginning, in the middle and at the end of the interval of the definition of the integro-differential equation, the corresponding types of splines are used: the left, the right or the middle splines of the fifth order of approximation. When using the spline approximations, we also obtain the corresponding formulas for numerical differentiation. which we also apply for the solution of integro-differential equations. The formulas for approximation of the function and its derivative are presented. The results of the numerical solution of several integro-differential equations are presented. The proposed method is shown that it can be applied to solve integro-differential equations containing the second derivative of the solution.

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Liu, G. R. "An Overview on Meshfree Methods: For Computational Solid Mechanics." International Journal of Computational Methods 13, no. 05 (August 31, 2016): 1630001. http://dx.doi.org/10.1142/s0219876216300014.

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This review paper presents a methodological study on possible and existing meshfree methods for solving the partial differential equations (PDEs) governing solid mechanics problems, based mainly on the research work in the past two decades at the authors group. We start with a discussion on the general steps in a meshfree method based on nodes, with the displacements as the primary variables. We then examine the major techniques used in each of these steps: (1) techniques for displacement function approximations using nodes, (2) approximation of the gradient of the displacements or strains based on nodes and a background T-cells that can be automatically generated and refined, and (3) formulation techniques for producing algebraic equations. The function approximation techniques include node-based interpolation methods, cell-based interpolation methods, function smoothing techniques, and moving least squares approximation techniques. The gradient approximation includes direct differentiation, gradient smoothing, and special strain construction. Formulation techniques include strong-form, weakform, local weakform, weak-strong-form, and weakened weakform (W2). In theory, a meshfree method can be developed using a combination of function approximation, gradient approximation, and formulation techniques, which can lead to matrix of a large number of possible methods. This review attempts to provide an overall methodological review, rather than a usual review of comparing different methods. We hope to show readers the differences between the forests, and just between the trees.

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Павлов, А. В., and А. О. Гаугель. "Моделирование эквивалентной передаточной функции и импульсного отклика схемы голографии Фурье для высокочастотных голограмм." Оптика и спектроскопия 131, no. 7 (2023): 926. http://dx.doi.org/10.21883/os.2023.07.56127.4540-23.

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An approach for modeling the transfer function and approximating the impulse response in the +1 diffraction order of the 4f holography scheme of the Fourier holography under usage of high-frequency holograms characterized by the presence of an inverse section of the dependence of the local diffraction efficiency on the spatial frequency in the frequency range below the frequency of equality of local amplitudes of the reference spectrum and the reference beam by the model of “Difference of Gaussians” is proposed and justified. The expediency of using, when implementing processing models that involve working only with the global maximum of the circuit response, a transfer function that is equivalent in terms of the minimum mean square error of the impulse response, as providing a more accurate approximation of the impulse response compared to the approximation of the direct transfer function, is shown. The validity of the approach is confirmed by comparing the simulation results with experimental data.

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36

RAMAKANTH, A., W. NOLTING, G. G. REDDY, D. MEYER, and S. SCHWIEGER. "LOCAL MOMENT ORDERING IN PERIODIC ANDERSON MODEL." International Journal of Modern Physics B 15, no. 19n20 (August 10, 2001): 2583–94. http://dx.doi.org/10.1142/s0217979201006434.

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Strongly correlated electron systems are studied with the help of periodic Anderson model (PAM). The PAM in which highly correlated nondegenerate localized states form a subsystem is considered and the focus of study is on magnetic ordering of electrons in these localized states. In order to study the PAM, which is not amenable to exact solution, two approximate schemes are proposed. The first one is called the spectral density approximation (SDA). Guided by the atomic limit, a two-pole ansatz is made for the localized electron spectral density. The spectral weights and the quasiparticle energies are determined by a moment method. From the spectral density, the spin and energy dependent self-energy is evaluated. A principal limitation of this method is that per ansatz, the quasiparticles are of infinite lifetime. To introduce a finite lifetime, a second approximation scheme is proposed where coherent potential approximation (CPA) is applied to PAM. In order to do CPA, an alloy analogy (AA) is required. In the conventional AA, the concentrations α and the atomic levels E of the fictitious alloy are taken from the atomic limit. Since the interest is in the magnetic properties, this AA is not appropriate. Therefore, a modified AA (MAA) is proposed. In MAA, α and E are obtained using the high energy expansion of the Green's function and the self-energy. In both the approximations, the density of states and the magnetization are selfconsistently evaluated and a phase diagram is obtained. Comparison of the results of the two schemes brings out the effect of quasiparticle damping on the magnetic properties.

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Nguyen-Truong, Hieu T., Tan-Tien Pham, Nam H. Vu, Dang H. Ngo, and Hung M. Le. "Energy-loss Function for Lead." Communications in Physics 27, no. 1 (May 17, 2017): 65. http://dx.doi.org/10.15625/0868-3166/27/1/9201.

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We study the energy-loss function for lead in the framework of the time-dependent density functional theory, using the full-potential linearized augmented plane-wave plus local orbitals method. The ab initio calculations are performed in the adiabatic local density approximation. The comparison between the obtained energy-loss function for zero momentum transfer with those from reflection electron energy loss spectroscopy measurements and from first-principles calculations shows good agreement.

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38

Li, Sen, Linquan Yao, Shichao Yi, and Wei Wang. "A Meshless Radial Basis Function Based on Partition of Unity Method for Piezoelectric Structures." Mathematical Problems in Engineering 2016 (2016): 1–17. http://dx.doi.org/10.1155/2016/7632176.

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A meshless radial basis function based on partition of unity method (RBF-PUM) is proposed to analyse static problems of piezoelectric structures. The methods of radial basis functions (RBFs) possess some merits: the shape functions have high order continuity;h-adaptivity is simpler than mesh-based methods; the shape functions are easily implemented in high dimensional space. The partition of unity method (PUM) easily constructs local approximation. The character of local approximate space can be varied and regarded asp-adaptivity. Considering the good properties of the two methods, the RBFs are used for local approximation and the local supported weight functions are used in the partition of unity method. The system equations of the RBF-PUM are derived using the variational principle. The field variables are approximated using the RBF-PUM shape functions which inherit all the advantages of the RBF shape functions such as the delta function property. The boundary conditions can be implemented easily. Numerical examples of piezoelectric structures are investigated to illustrate the efficiency of the proposed method and the obtained results are compared with analytical solutions and available numerical solutions. The behaviors of some parameters that probably influenced numerical results are also studied in detail.

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Zeng, Mudong, Yujie Liao, Runze Li, and Agus Sudjianto. "Local Linear Approximation Algorithm for Neural Network." Mathematics 10, no. 3 (February 3, 2022): 494. http://dx.doi.org/10.3390/math10030494.

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This paper aims to develop a new training strategy to improve efficiency in estimation of weights and biases in a feedforward neural network (FNN). We propose a local linear approximation (LLA) algorithm, which approximates ReLU with a linear function at the neuron level and estimate the weights and biases of one-hidden-layer neural network iteratively. We further propose the layer-wise optimized adaptive neural network (LOAN), in which we use the LLA to estimate the weights and biases in the LOAN layer by layer adaptively. We compare the performance of the LLA with the commonly-used procedures in machine learning based on seven benchmark data sets. The numerical comparison implies that the proposed algorithm may outperform the existing procedures in terms of both training time and prediction accuracy.

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Luo, Mao, and Shao Yun Song. "Realization of RBF Neural Network with Local and Semi-Local Transfer Function Approximation and Classification." Applied Mechanics and Materials 687-691 (November 2014): 1628–32. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.1628.

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Incremental Neural Network (IncNet) structure is controlled by the growth and pruning, and the complexity of the match and training data. Dual radial transfer function is more flexible than other commonly transfer function used in artificial neural network. Recent improvements in the multi-dimensional space (having the N-1 parameters) to increase the rotation of the transfer function of the constant value. Based on the results of the benchmark approach and psychological classification analysis clearly shows than any other classification network model has a stronger generalization.

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41

Wirbeleit, Frank. "Non-Gaussian Diffusion of Phosphorus and Arsenic in Silicon with Local Density Diffusivity Model." Defect and Diffusion Forum 303-304 (July 2010): 21–29. http://dx.doi.org/10.4028/www.scientific.net/ddf.303-304.21.

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In the light of published phosphorus and arsenic diffusion profiles [1,2] a non-Gaussian mathematical diffusion model is developed in this work based on separate forward and reflected impurity diffusion flows and called local density diffusion (LDD) model. The LDD model includes the rational function diffusion (RFD) model published in [3] and represents an improvement for near surface and tail concentration profile slope approximation by introducing just one single empirical fit parameter “r”. This single fit parameter is related to the given combination of impurity species (phosphorus: r=0.18; arsenic: r=0.43) in the applied host lattice system (silicon), but does not vary while approximating different experiments with different impurity surface concentrations and penetration depths [1,2]. Based on the LDD approximation in this work a surface enhanced diffusivity for phosphorus and a tail decelerated diffusion for arsenic is suggested in comparison to RFD model approximation only. The local density diffusivity is found to be non-linear along the penetration path and reaches its maximum at a distance LLDD from the surface.

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Morishita, Kentaro, and Eitaro Aiyoshi. "Global Function Approximation by Fuzzy-Connection of Plural Local Neural Networks." IEEJ Transactions on Electronics, Information and Systems 123, no. 10 (2003): 1839–46. http://dx.doi.org/10.1541/ieejeiss.123.1839.

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43

Kuntz, Ludwig, and Stefan Scholtes. "Qualitative aspects of the local approximation of a piecewise differentiable function." Nonlinear Analysis: Theory, Methods & Applications 25, no. 2 (July 1995): 197–215. http://dx.doi.org/10.1016/0362-546x(94)00202-s.

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44

Narimani, M., and Z. Nourbakhsh. "Topological band order, structural, electronic and optical properties of XPdBi (X = Lu, Sc) compounds." Modern Physics Letters B 30, no. 14 (May 29, 2016): 1650159. http://dx.doi.org/10.1142/s0217984916501591.

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In this paper, the structural, electronic and optical properties of LuPdBi and ScPdBi compounds are investigated using the density functional theory by WIEN2K package within the generalized gradient approximation, local density approximation, Engel–Vosco generalized gradient approximations and modified Becke–Johnson potential approaches. The topological phases and band orders of these compounds are studied. The effect of pressure on band inversion strength, electron density of states and the linear coefficient of the electronic specific heat of these compounds is investigated. Furthermore, the effect of pressure on real and imaginary parts of dielectric function, absorption and reflectivity coefficients of these compounds is studied.

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Bilinskas, Mykolas J., Gintautas Dzemyda, and Martynas Sabaliauskas. "Speeding-up the Fitting of the Model Defining the Ribs-bounded Contour." Applied Computer Systems 21, no. 1 (May 24, 2017): 66–70. http://dx.doi.org/10.1515/acss-2017-0009.

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Abstract The method for analysing transversal plane images from computer tomography scans is considered in the paper. This method allows not only approximating ribs-bounded contour but also evaluating patient rotation around the vertical axis during a scan. In this method, a mathematical model describing the ribs-bounded contour was created and the problem of approximation has been solved by finding the optimal parameters of the mathematical model using least-squares-type objective function. The local search has been per-formed using local descent by quasi-Newton methods. The benefits of analytical derivatives of the function are disclosed in the paper.

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XU, GUOPING, and HARRY ZHENG. "LOWER BOUND APPROXIMATION TO BASKET OPTION VALUES FOR LOCAL VOLATILITY JUMP-DIFFUSION MODELS." International Journal of Theoretical and Applied Finance 17, no. 01 (February 2014): 1450007. http://dx.doi.org/10.1142/s0219024914500071.

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In this paper, we derive an easily computed approximation to European basket call prices for a local volatility jump-diffusion model. We apply the asymptotic expansion method to find the approximate value of the lower bound of European basket call prices. If the local volatility function is time independent then there is a closed-form expression for the approximation. Numerical tests show that the suggested approximation is fast and accurate in comparison with the Monte Carlo (MC) and other approximation methods in the literature.

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47

Al-Sa'di, Sa'ud, and Salti Samarah. "Bernstein and Bernstein-Like Inequalities for Modulation Spaces." International Journal of Emerging Multidisciplinaries: Mathematics 1, no. 1 (January 14, 2022): 91–101. http://dx.doi.org/10.54938/ijemdm.2022.01.1.3.

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It is shown that the modulation spaces Mwp can be characterized by the approximation behavior of their elements using Local Fourier bases. In analogy to the Local Fourier bases, we show that the modulation spaces can also be characterized by the approximation behavior of their elements using Gabor frames. We derive direct and inverse approximation theorems that describe the best approximation by linear combinations of N terms of a given function using its modulates and translates.

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48

Karami, A., Saeid Abbasbandy, and E. Shivanian. "Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation." Mathematical and Computational Applications 24, no. 4 (December 10, 2019): 101. http://dx.doi.org/10.3390/mca24040101.

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In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

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Abedi, Erfan, Salman Beigi, and Leila Taghavi. "Quantum Lazy Training." Quantum 7 (April 27, 2023): 989. http://dx.doi.org/10.22331/q-2023-04-27-989.

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In the training of over-parameterized model functions via gradient descent, sometimes the parameters do not change significantly and remain close to their initial values. This phenomenon is called lazy training and motivates consideration of the linear approximation of the model function around the initial parameters. In the lazy regime, this linear approximation imitates the behavior of the parameterized function whose associated kernel, called the tangent kernel, specifies the training performance of the model. Lazy training is known to occur in the case of (classical) neural networks with large widths. In this paper, we show that the training of geometrically local parameterized quantum circuits enters the lazy regime for large numbers of qubits. More precisely, we prove bounds on the rate of changes of the parameters of such a geometrically local parameterized quantum circuit in the training process, and on the precision of the linear approximation of the associated quantum model function; both of these bounds tend to zero as the number of qubits grows. We support our analytic results with numerical simulations.

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Schlather, Martin, and Tilmann Gneiting. "Local approximation of variograms by covariance functions." Statistics & Probability Letters 76, no. 12 (July 2006): 1303–4. http://dx.doi.org/10.1016/j.spl.2006.02.002.

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