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Journal articles on the topic 'Forward approximation'

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Author: Grafiati

Published: 14 March 2023

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1

Hao, Qi, and Alexey Stovas. "Analytic calculation of phase and group velocities of P-waves in orthorhombic media." GEOPHYSICS 81, no. 3 (May 2016): C79—C97. http://dx.doi.org/10.1190/geo2015-0156.1.

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We have developed an approximate method to calculate the P-wave phase and group velocities for orthorhombic media. Two forms of analytic approximations for P-wave velocities in orthorhombic media were built by analogy with the five-parameter moveout approximation and the four-parameter velocity approximation for transversely isotropic media, respectively. They are called the generalized moveout approximation (GMA)-type approximation and the Fomel approximation, respectively. We have developed approximations for elastic and acoustic orthorhombic media. We have characterized the elastic orthorhombic media in Voigt notation, and we can describe the acoustic orthorhombic media by introducing the modified Alkhalifah’s notation. Our numerical evaluations indicate that the GMA-type and Fomel approximations are accurate for elastic and acoustic orthorhombic media with strong anisotropy, and the GMA-type approximation is comparable with the approximation recently proposed by Sripanich and Fomel. Potential applications of the proposed approximations include forward modeling and migration based on the dispersion relation and the forward traveltime calculation for seismic tomography.

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2

Nguyen, Thi Ngoc Minh, Sylvain Le Corff, and Eric Moulines. "On the two-filter approximations of marginal smoothing distributions in general state-space models." Advances in Applied Probability 50, no. 01 (March 2018): 154–77. http://dx.doi.org/10.1017/apr.2018.8.

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AbstractA prevalent problem in general state-space models is the approximation of the smoothing distribution of a state conditional on the observations from the past, the present, and the future. The aim of this paper is to provide a rigorous analysis of such approximations of smoothed distributions provided by the two-filter algorithms. We extend the results available for the approximation of smoothing distributions to these two-filter approaches which combine a forward filter approximating the filtering distributions with a backward information filter approximating a quantity proportional to the posterior distribution of the state, given future observations.

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3

He, Chuanlin, Yi Zheng, Xu Xiang, and Yuanliang Ma. "Kirchhoff Approximations for the Forward-Scattering Target Strength of Underwater Objects." Journal of Theoretical and Computational Acoustics 28, no. 01 (October 14, 2019): 1950008. http://dx.doi.org/10.1142/s2591728519500087.

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Kirchhoff approximations for the forward-scattering target strength of underwater objects are developed by combining Babinet’s principle and the Kirchhoff integral, where theoretical formulations and a numerical implementation are given in detail. The Kirchhoff approximation is found to be a high-frequency physical acoustic approximation. The forward-scattering target strength versus frequency and the spatial angles for spherical objects, prolate spheroids and the Benchmark Target Strength Simulation Submarine (BeTSSi-Sub) model are obtained by the Kirchhoff approximation and compared with results from theory, the deformed cylinder method (DCM) and the boundary element method (BEM). The Kirchhoff approximation shows considerable agreement with the theoretical and numerical approaches in a region of [Formula: see text] from the rigorous forward-scattering direction. The forward-scattered field contour and the corresponding directivity for the BeTSSi-Sub model are also calculated as a demonstration. Mode coupling caused by the simulated target is clearly revealed. The results indicate that the Kirchhoff approximation can predict the forward-scattering target strength of complex underwater objects.

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4

Cheng, Yi. "Forward approximation and backward approximation in fuzzy rough sets." Neurocomputing 148 (January 2015): 340–53. http://dx.doi.org/10.1016/j.neucom.2014.06.062.

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5

Kratsios, Anastasis. "The Universal Approximation Property." Annals of Mathematics and Artificial Intelligence 89, no. 5-6 (January 22, 2021): 435–69. http://dx.doi.org/10.1007/s10472-020-09723-1.

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AbstractThe universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our understanding of our current models’ potential. This paper works towards overcoming these challenges by presenting a characterization, a representation, a construction method, and an existence result, each of which applies to any universal approximator on most function spaces of practical interest. Our characterization result is used to describe which activation functions allow the feed-forward architecture to maintain its universal approximation capabilities when multiple constraints are imposed on its final layers and its remaining layers are only sparsely connected. These include a rescaled and shifted Leaky ReLU activation function but not the ReLU activation function. Our construction and representation result is used to exhibit a simple modification of the feed-forward architecture, which can approximate any continuous function with non-pathological growth, uniformly on the entire Euclidean input space. This improves the known capabilities of the feed-forward architecture.

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6

Paronetto, Fabio. "Elliptic approximation of forward-backward parabolic equations." Communications on Pure & Applied Analysis 19, no. 2 (2020): 1017–36. http://dx.doi.org/10.3934/cpaa.2020047.

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7

Momoniat, E. "Matrix Exponentiation and the Frank-Kamenetskii Equation." Mathematical Problems in Engineering 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/713798.

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Long time solutions to the Frank-Kamenetskii partial differential equation modelling a thermal explosion in a vessel are obtained using matrix exponentiation. Spatial derivatives are approximated by high-order finite difference approximations. A forward difference approximation to the time derivative leads to a Lawson-Euler scheme. Computations performed with a BDF approximation to the time derivative and a fourth-order Runge-Kutta approximation to the time derivative are compared to results obtained with the Lawson-Euler scheme. Variation in the central temperature of the vessel corresponding to changes in the shape parameter and Frank-Kamenetskii parameter are computed and discussed.

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8

Jetta, Mahipal. "A highly stable explicit scheme for a fourth-order nonlinear diffusion filter." International Journal of Modeling, Simulation, and Scientific Computing 11, no. 04 (July 2, 2020): 2050030. http://dx.doi.org/10.1142/s1793962320500300.

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The standard finite difference scheme (forward difference approximation for time derivative and central difference approximations for spatial derivatives) for fourth-order nonlinear diffusion filter allows very small time-step size to obtain stable results. The alternating directional implicit (ADI) splitting scheme such as Douglas method is highly stable but compromises accuracy for a relatively larger time-step size. In this paper, we develop [Formula: see text] stencils for the approximation of second-order spatial derivatives based on the finite pointset method. We then make use of these stencils for approximating the fourth-order partial differential equation. We show that the proposed scheme allows relatively bigger time-step size than the standard finite difference scheme, without compromising on the quality of the filtered image. Further, we demonstrate through numerical simulations that the proposed scheme is more efficient, in obtaining quality filtered image, than an ADI splitting scheme.

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9

Lakshmanan, Valliappa, Robert Rabin, Jason Otkin, John S. Kain, and Scott Dembek. "Visualizing Model Data Using a Fast Approximation of a Radiative Transfer Model." Journal of Atmospheric and Oceanic Technology 29, no. 5 (May 1, 2012): 745–54. http://dx.doi.org/10.1175/jtech-d-11-00007.1.

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Abstract Visualizing model forecasts using simulated satellite imagery has proven very useful because the depiction of forecasts using cloud imagery can provide inferences about meteorological scenarios and physical processes that are not characterized well by depictions of those forecasts using radar reflectivity. A forward radiative transfer model is capable of providing such a visible-channel depiction of numerical weather prediction model output, but present-day forward models are too slow to run routinely on operational model forecasts. It is demonstrated that it is possible to approximate the radiative transfer model using a universal approximator whose parameters can be determined by fitting the output of the forward model to inputs derived from the raw output from the prediction model. The resulting approximation is very close to the result derived from the complex radiative transfer model and has the advantage that it can be computed in a small fraction of the time required by the forward model. This approximation is carried out on model forecasts to demonstrate its utility as a visualization and forecasting tool.

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10

Allen, K. Radway, and W. S. Hearn. "Some Procedures for use in Cohort Analysis and Other Population Simulations." Canadian Journal of Fisheries and Aquatic Sciences 46, no. 3 (March 1, 1989): 483–88. http://dx.doi.org/10.1139/f89-064.

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The catch equation [Formula: see text] must often be solved either forwards or backwards. Because Ft is an implicit function of catch, population size, and M, the solution must be either iterative or approximate. In this paper an improved approximation to Nt+1 is developed of the form[Formula: see text]for the forward solution and a corresponding equation for backward solution. Appropriate values of A are presented for a range of combinations of M and F; and A = 0.585 gives approximations to Nt+1 with an error of less than 1% provided that (Ft + M) is less than about 1.5. Much closer approximations are obtained if A is calculated for each case as a polynomial in M and either Ct/Nt for the forward solution or Ct−1/Nt for the backward solution. The appropriate coefficients for these polynomials are derived both by truncating Taylor's expansion and by statistical methods, and are presented here. One such polynomial for A gives errors of less than 0.02% where (F + M) is less than 1.1. Similarly good approximations can also be obtained by interpolating into a table of Nt+1/Nt against M and C/N.

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11

Fischbach, F. A. "Variable Gaussian approximation for near forward Mie scattering." Applied Optics 25, no. 24 (December 15, 1986): 4525. http://dx.doi.org/10.1364/ao.25.004525.

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12

Fadin, V. "Non-forward BFKL at next-to-leading approximation." Nuclear Physics B - Proceedings Supplements 99, no. 1-2 (April 2001): 204–12. http://dx.doi.org/10.1016/s0920-5632(01)01335-4.

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13

I-Ming Pao and Ming-Ting Sun. "Approximation of calculations for forward discrete cosine transform." IEEE Transactions on Circuits and Systems for Video Technology 8, no. 3 (June 1998): 264–68. http://dx.doi.org/10.1109/76.678620.

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14

Stefanica, Dan, and Radoš Radoičić. "A sharp approximation for ATM-forward option prices and implied volatilites." International Journal of Financial Engineering 03, no. 01 (March 2016): 1650002. http://dx.doi.org/10.1142/s242478631650002x.

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In this paper, we provide an approximation formula for at-the-money forward options based on a Pólya approximation of the cumulative density function of the standard normal distribution, and prove that the relative error of this approximation is uniformly bounded for options with arbitrarily large (or small) maturities and implied volatilities. This approximation is viable in practice: for options with implied volatility less than 95% and maturity less than three years, which includes the large majority of traded options, the values given by the approximation formula fall within the tightest typical implied vol bid–ask spreads. The relative errors of the corresponding approximate option values are also uniformly bounded for all maturities and implied volatilities. The error bounds established here are the first results in the literature holding for all integrated volatilities, and are vastly superior to those of two other approximation formulas analyzed in this paper, including the Brenner–Subrahmanyam formula.

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15

Matić, Ivan, Radoš Radoičić, and Dan Stefanica. "Pólya-based approximation for the ATM-forward implied volatility." International Journal of Financial Engineering 04, no. 02n03 (June 2017): 1750032. http://dx.doi.org/10.1142/s2424786317500323.

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We introduce a closed form approximation for the implied volatility of ATM-forward options. The relative error of this approximation is uniformly bounded for all option maturities and implied volatilities. The approximation is extremely precise, having relative error less than [Formula: see text] for all options with integrated volatility less than [Formula: see text], such as options with maturity less than three years and implied volatility less than 100%. Moreover, the approximate implied volatilities fall within the implied volatility bid–ask spread for all the liquid options, such as options with volatility less than 200% and maturity less than nine years.

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16

Seri, Raffaello, and Christine Choirat. "COMPARISON OF APPROXIMATIONS FOR COMPOUND POISSON PROCESSES." ASTIN Bulletin 45, no. 3 (June 25, 2015): 601–37. http://dx.doi.org/10.1017/asb.2015.7.

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AbstractIn this paper, we compare the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this model, it is usually supposed that a portfolio is at risk for a time period of length t. The occurrences of the claims are governed by a Poisson process of intensity μ so that the number of claims in [0,t] is a Poisson random variable with parameter λ = μ t. Each single claim is an independent replication of the random variable X, representing the claim severity. The aggregate claim or total claim amount process in [0,t] is represented by the random sum of N independent replications of X, whose cumulative distribution function (cdf) is the object of study. Due to its computational complexity, several approximation methods for this cdf have been proposed. In this paper, we consider 15 approximations put forward in the literature that only use information on the lower order moments of the involved distributions. For each approximation, we consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure their accuracy as λ = μ t diverges to infinity. Using these expansions, several statements concerning the quality of approximations for the distribution of the aggregate claim process can find theoretical support. Other statements can be disproved on the same grounds. Finally, we investigate numerically the accuracy of the proposed formulas.

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17

Bilbiie, Florin O. "Optimal Forward Guidance." American Economic Journal: Macroeconomics 11, no. 4 (October 1, 2019): 310–45. http://dx.doi.org/10.1257/mac.20170335.

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Optimal forward guidance is the simple policy of keeping interest rates low for some optimally determined number of periods after the liquidity trap ends and moving to normal-times optimal policy thereafter. I solve for the optimal duration in closed form in a new Keynesian model and show that it is close to fully optimal Ramsey policy. The simple rule “announce a duration of half of the trap’s duration times the disruption” is a good approximation, including in a medium-scale dynamic stochastic general equilibrium (DSGE) model. By anchoring expectations of Delphic agents (who mistake commitment for bad news), the simple rule is also often welfare-preferable to Odyssean commitment. (JEL D84, E12, E43, E52, E56)

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18

KÜMMEL, HERMANN G. "A BIOGRAPHY OF THE COUPLED CLUSTER METHOD." International Journal of Modern Physics B 17, no. 28 (November 10, 2003): 5311–25. http://dx.doi.org/10.1142/s0217979203020442.

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The origins of the coupled cluster method are described. Special attention is paid to the arguments put forward for the exponential structure of the wave functions. Various approximation schemes invented during the last 40 years are presented. The problems arising from these approximations necessarily truncating or destroying the exponential form are discussed and ways to deal with them are described.

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19

Fornasier, Massimo, Jan Vybíral, and Ingrid Daubechies. "Robust and resource efficient identification of shallow neural networks by fewest samples." Information and Inference: A Journal of the IMA 10, no. 2 (January 19, 2021): 625–95. http://dx.doi.org/10.1093/imaiai/iaaa036.

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Abstract We address the structure identification and the uniform approximation of sums of ridge functions $f(x)=\sum _{i=1}^m g_i(\langle a_i,x\rangle )$ on ${\mathbb{R}}^d$, representing a general form of a shallow feed-forward neural network, from a small number of query samples. Higher order differentiation, as used in our constructive approximations, of sums of ridge functions or of their compositions, as in deeper neural network, yields a natural connection between neural network weight identification and tensor product decomposition identification. In the case of the shallowest feed-forward neural network, second-order differentiation and tensors of order two (i.e., matrices) suffice as we prove in this paper. We use two sampling schemes to perform approximate differentiation—active sampling, where the sampling points are universal, actively and randomly designed, and passive sampling, where sampling points were preselected at random from a distribution with known density. Based on multiple gathered approximated first- and second-order differentials, our general approximation strategy is developed as a sequence of algorithms to perform individual sub-tasks. We first perform an active subspace search by approximating the span of the weight vectors $a_1,\dots ,a_m$. Then we use a straightforward substitution, which reduces the dimensionality of the problem from $d$ to $m$. The core of the construction is then the stable and efficient approximation of weights expressed in terms of rank-$1$ matrices $a_i \otimes a_i$, realized by formulating their individual identification as a suitable nonlinear program. We prove the successful identification by this program of weight vectors being close to orthonormal and we also show how we can constructively reduce to this case by a whitening procedure, without loss of any generality. We finally discuss the implementation and the performance of the proposed algorithmic pipeline with extensive numerical experiments, which illustrate and confirm the theoretical results.

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20

Arthur, R. Martin. "Spherical-harmonic approximation to the forward problem of electrocardiology." Journal of Electrocardiology 32, no. 2 (April 1999): 103–14. http://dx.doi.org/10.1016/s0022-0736(99)90089-4.

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21

Lappi, T., and H. Mäntysaari. "Forward dihadron correlations in the Gaussian approximation of JIMWLK." Nuclear Physics A 910-911 (August 2013): 498–501. http://dx.doi.org/10.1016/j.nuclphysa.2012.12.057.

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22

Xu, Feng, Xiaoshu Cai, and Kuanfang Ren. "Geometrical-optics approximation of forward scattering by coated particles." Applied Optics 43, no. 9 (March 19, 2004): 1870. http://dx.doi.org/10.1364/ao.43.001870.

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23

Thomas, Jan, Vinupritha P, and Kathirvelu D. "Neuromuscular Control With Forward Dynamic Approximation In Human Arm." Biomedical and Pharmacology Journal 10, no. 02 (June 25, 2017): 895–906. http://dx.doi.org/10.13005/bpj/1183.

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24

Tsao, Chueh-Yung, Chuang-Chang Chang, and Chung-Gee Lin. "Analytic approximation formulae for pricing forward-starting Asian options." Journal of Futures Markets 23, no. 5 (March 21, 2003): 487–516. http://dx.doi.org/10.1002/fut.10070.

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25

Yun, Beong. "A Neural Network Approximation Based on a Parametric Sigmoidal Function." Mathematics 7, no. 3 (March 14, 2019): 262. http://dx.doi.org/10.3390/math7030262.

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It is well known that feed-forward neural networks can be used for approximation to functions based on an appropriate activation function. In this paper, employing a new sigmoidal function with a parameter for an activation function, we consider a constructive feed-forward neural network approximation on a closed interval. The developed approximation method takes a simple form of a superposition of the parametric sigmoidal function. It is shown that the proposed method is very effective in approximation of discontinuous functions as well as continuous ones. For some examples, the availability of the presented method is demonstrated by comparing its numerical results with those of an existing neural network approximation method. Furthermore, the efficiency of the method in extended application to the multivariate function is also illustrated.

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26

Helin, Tapio, and Remo Kretschmann. "Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems." Numerische Mathematik 150, no. 2 (January 15, 2022): 521–49. http://dx.doi.org/10.1007/s00211-021-01266-9.

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AbstractIn this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. 10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.

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27

Suzuki, Tomoyuki, Keisuke Takasao, and Noriaki Yamazaki. "Remarks on Numerical Experiments of the Allen-Cahn Equations with Constraint via Yosida Approximation." Advances in Numerical Analysis 2016 (June 7, 2016): 1–16. http://dx.doi.org/10.1155/2016/1492812.

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We consider a one-dimensional Allen-Cahn equation with constraint from the viewpoint of numerical analysis. The constraint is provided by the subdifferential of the indicator function on the closed interval, which is the multivalued function. Therefore, it is very difficult to perform a numerical experiment of our equation. In this paper we approximate the constraint by the Yosida approximation. Then, we study the approximating system of the original model numerically. In particular, we give the criteria for the standard forward Euler method to give the stable numerical experiments of the approximating equation. Moreover, we provide the numerical experiments of the approximating equation.

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28

Zhdanov, Michael S., Vladimir I. Dmitriev, Sheng Fang, and Gábor Hursán. "Quasi‐analytical approximations and series in electromagnetic modeling." GEOPHYSICS 65, no. 6 (November 2000): 1746–57. http://dx.doi.org/10.1190/1.1444859.

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The quasi‐linear approximation for electromagnetic forward modeling is based on the assumption that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background (normal) field through an electrical reflectivity tensor λ⁁. In the original formulation of the quasi‐linear approximation, λ⁁ was determined by solving a minimization problem based on an integral equation for the scattering currents. This approach is much less time‐consuming than the full integral equation method; however, it still requires solution of the corresponding system of linear equations. In this paper, we present a new approach to the approximate solution of the integral equation using λ⁁ through construction of quasi‐analytical expressions for the anomalous electromagnetic field for 3-D and 2-D models. Quasi‐analytical solutions reduce dramatically the computational effort related to forward electromagnetic modeling of inhomogeneous geoelectrical structures. In the last sections of this paper, we extend the quasi‐analytical method using iterations and develop higher order approximations resulting in quasi‐analytical series which provide improved accuracy. Computation of these series is based on repetitive application of the given integral contraction operator, which insures rapid convergence to the correct result. Numerical studies demonstrate that quasi‐analytical series can be treated as a new powerful method of fast but rigorous forward modeling solution.

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29

Andreev, Vyacheslav V., Leonid A. Slavutskii, and Elena V. Slavutskaya. "FEED FORWARD NEURAL NET SIGNAL PROCESSING: APPROXIMATION AND DECISION MAKING." Vestnik Chuvashskogo universiteta, no. 1 (March 30, 2022): 14–22. http://dx.doi.org/10.47026/1810-1909-2022-1-14-22.

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Feed forward artificial neural networks (multilayer perceptron) allow solving a wide range of regression (approximation) and classification (data splitting into subsets) problems. The corresponding algorithms are applied in electrical and power engineering. The peculiarity of such an artificial neural network is that the training sample can be submitted to the input in an arbitrary sequence. Therefore, the signals themselves during artificial neural network training should be formed taking into account their time form. The paper proposes the use of artificial neural network in a sliding time window. Using simple examples of transients and relay protection, the paper analyses the possibilities of the signals’ time form approximation and the problems of recognition using the artificial neural network of the signal parameters near zero and threshold values. It is shown that errors in the operation of the artificial neural network can be compensated. The choice of the duration of the sliding window must necessarily take into account the need for additional processing of data from the output of the neural network in the form of their statistical analysis, the obtained dependencies approximation or smoothing.

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VAN APPEL, JACQUES, and THOMAS A. MCWALTER. "EFFICIENT LONG-DATED SWAPTION VOLATILITY APPROXIMATION IN THE FORWARD-LIBOR MODEL." International Journal of Theoretical and Applied Finance 21, no. 04 (June 2018): 1850020. http://dx.doi.org/10.1142/s0219024918500206.

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We provide efficient swaption volatility approximations for longer maturities and tenors under the lognormal forward-LIBOR model (LFM). In particular, we approximate the swaption volatility with a mean update of the spanning forward rates. Since the joint distribution of the forward rates is not known under a typical pricing measure, we resort to numerical discretization techniques. More specifically, we approximate the mean forward rates with a multi-dimensional weak order 2.0 Itō–Taylor scheme. The higher-order terms allow us to more accurately capture the state dependence in the drift terms and compute conditional expectations with second-order accuracy. We test our approximations for longer maturities and tenors using a quasi-Monte Carlo (QMC) study and find them to be substantially more effective when compared to the existing approximations, particularly for calibration purposes.

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31

Uhlmann, Hermann, and Olaf Michelsson. "A fast forward solution with a boundary element method for eddy current nondestructive testing." Facta universitatis - series: Electronics and Energetics 15, no. 2 (2002): 205–16. http://dx.doi.org/10.2298/fuee0202205u.

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Eddy current non-destructive testing is used to determine position and size of cracks or other defects in conducting materials. The presence of a crack normal to the excited eddy currents distorts the magnetic field; so for the identification of defects a very accurate and fast 3D-computation of the magnetic field is necessary. A computation scheme for 3D quasistatic electromagnetic fields by means of the Boundary Element Method is presented. Although the use of constant field approximations on boundary elements is the easiest way, it often provides an insufficient accuracy. This can be overcome by higher order approximation schemes. The numerical results are compared against some analytically solvable arrangements.

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32

Qureshi, Sania, Norodin A. Rangaig, and Dumitru Baleanu. "New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator." Mathematics 7, no. 4 (April 24, 2019): 374. http://dx.doi.org/10.3390/math7040374.

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In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f ( t ) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration.

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33

Geiss, Christel, Céline Labart, and Antti Luoto. "Mean square rate of convergence for random walk approximation of forward-backward SDEs." Advances in Applied Probability 52, no. 3 (September 2020): 735–71. http://dx.doi.org/10.1017/apr.2020.17.

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AbstractLet (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

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34

Beard, Les P., Gerald W. Hohmann, and Alan C. Tripp. "Fast resistivity/IP inversion using a low‐contrast approximation." GEOPHYSICS 61, no. 1 (January 1996): 169–79. http://dx.doi.org/10.1190/1.1443937.

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By computing only the diagonal terms of the volume integral equation forward solution of the 3-D DC resistivity problem, we have achieved a fast forward solution accurate at low to moderate resistivity contrasts. The speed and accuracy of the solution make it practical for use in 2-D or 3-D inversion algorithms. The low‐contrast approximation is particularly well‐suited to the smooth nature of minimum structure inversion, since complete forward solutions may be computationally expensive. By using this approximate 3-D solution as the forward model in an inversion algorithm, and by constraining the resistivities and polarizabilities along any row of cells in the strike direction to be held constant, we effect a fast 2-D resistivity inversion that contains end corrections. Because the low‐contrast solution is inaccurate for cells near the electrodes, we employ a full solution to compute the response of the near‐surface when the near‐surface environment is substantially different from the host rock. This response is stored and used in the iterative resistivity inversion in conjunction with the approximate solution. Once an adequate estimated resistivity model has been found, derivatives from this model are used with Seigel’s formula to compute the inverse solution to the linear polarizability problem in a single iteration.

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Iqbal, Sajad, and Yujie Wei. "Recovery of the time-dependent implied volatility of time fractional Black–Scholes equation using linearization technique." Journal of Inverse and Ill-posed Problems 29, no. 4 (January 22, 2021): 599–610. http://dx.doi.org/10.1515/jiip-2020-0105.

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Abstract This paper tries to examine the recovery of the time-dependent implied volatility coefficient from market prices of options for the time fractional Black–Scholes equation (TFBSM) with double barriers option. We apply the linearization technique and transform the direct problem into an inverse source problem. Resultantly, we get a Volterra integral equation for the unknown linear functional, which is then solved by the regularization method. We use L 1 {L_{1}} -forward difference implicit approximation for the forward problem. Numerical results using L 1 {L_{1}} -forward difference implicit approximation ( L 1 {L_{1}} -FDIA) for the inverse problem are also discussed briefly.

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Li, Xiangzhen, Xiang'e Han, Renxian Li, and Huifen Jiang. "A Faster Approximation of Forward Scattering by Gradient-index Particles." PIERS Online 3, no. 4 (2007): 382–86. http://dx.doi.org/10.2529/piers061023013549.

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Frederickx, Roald, and Philip Dutré. "A forward scattering dipole model from a functional integral approximation." ACM Transactions on Graphics 36, no. 4 (July 20, 2017): 1–13. http://dx.doi.org/10.1145/3072959.3073681.

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Li, Xiangzhen, Xiang'e Han, Renxian Li, and Huifen Jiang. "Geometrical-optics approximation of forward scattering by gradient-index spheres." Applied Optics 46, no. 22 (July 9, 2007): 5241. http://dx.doi.org/10.1364/ao.46.005241.

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Bouchard, Bruno, and Romuald Elie. "Discrete-time approximation of decoupled Forward–Backward SDE with jumps." Stochastic Processes and their Applications 118, no. 1 (January 2008): 53–75. http://dx.doi.org/10.1016/j.spa.2007.03.010.

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ARTHUR, R. "Spherical-harmonic approximation to the forward problem of electrocardiology*1." Journal of Cardiothoracic and Vascular Anesthesia 32, no. 2 (April 1999): 103–14. http://dx.doi.org/10.1016/s1053-0770(99)90047-2.

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Xie, Tingfan, and Feilong Cao. "The errors in simultaneous approximation by feed-forward neural networks." Neurocomputing 73, no. 4-6 (January 2010): 903–7. http://dx.doi.org/10.1016/j.neucom.2009.09.014.

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Da Silva, Carmen, and René Escalante. "Segmented Tau approximation for a forward–backward functional differential equation." Computers & Mathematics with Applications 62, no. 12 (December 2011): 4582–91. http://dx.doi.org/10.1016/j.camwa.2011.10.040.

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Ye, Xingde, and Jinsheng Jiang. "A NEW PSEUDOSPECTRAL APPROXIMATION FOR THE FORWARD-BACKWARD HEAT EQUATION." Acta Mathematica Scientia 16, no. 2 (April 1996): 121–28. http://dx.doi.org/10.1016/s0252-9602(17)30787-7.

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Fouque, J. P., G. Papanicolaou, and Y. Samuelides. "Forward and Markov approximation: the strong-intensity-fluctuations regime revisited." Waves in Random Media 8, no. 3 (July 1998): 303–14. http://dx.doi.org/10.1088/0959-7174/8/3/003.

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RIEMANN, TORD, and ZBIGNIEW WAS. "SIMPLE APPROXIMATION FOR QED O(α) CORRECTED FORWARD BACKWARD ASYMMETRY." Modern Physics Letters A 04, no. 25 (November 30, 1989): 2487–91. http://dx.doi.org/10.1142/s0217732389002781.

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It is shown that the simple semi-analytical formula which involves a convolution of the Born asymmetry with the photon spectrum agrees to a precision better than LEP’s experimental error with exact O(α) result for the forward-backward asymmetry. The exact O(α) result differs from the convolution by 0.5×10−3 at the Z peak and by less than 3 ×10−3 close to Z peak. The result was obtained without kinematical cut-offs.

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Poon, H. C., D. Snider, and S. Y. Tong. "Small-atom approximation in forward- and back-scattering photoelectron spectroscopies." Physical Review B 33, no. 4 (February 15, 1986): 2198–206. http://dx.doi.org/10.1103/physrevb.33.2198.

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Zhang, Chong, Da-Nian Huang, Kai Zhang, Yi-Tao Pu, and Ping Yu. "Magnetic interface forward and inversion method based on Padé approximation." Applied Geophysics 13, no. 4 (December 2016): 712–20. http://dx.doi.org/10.1007/s11770-016-0591-8.

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Rosenau da Costa, M., H. Westfahl, and A. O. Caldeira. "Forward scattering approximation and bosonization in integer quantum Hall systems." Annals of Physics 323, no. 3 (March 2008): 673–704. http://dx.doi.org/10.1016/j.aop.2007.04.018.

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Alonso, José M., Javier Ibáñez, Emilio Defez, and Fernando Alvarruiz. "Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials." Mathematics 11, no. 3 (January 18, 2023): 520. http://dx.doi.org/10.3390/math11030520.

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This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is based on the approximation of the matrix exponential by means of Bernoulli matrix polynomials. We carry out an analysis of the absolute and relative forward errors incurred in the approximations, deriving corresponding suitable values for the matrix polynomial degree and the scaling factor to be used. Finally, we use a comprehensive matrix testbed to perform a thorough comparison of the alternative approximations, also taking into account other current state-of-the-art approaches. The most accurate and efficient options are identified as results.

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CHEN, ZHIXIANG, and FEILONG CAO. "APPROXIMATION BY SPHERICAL NEURAL NETWORKS WITH ZONAL FUNCTIONS." ANZIAM Journal 58, no. 3-4 (April 2017): 238–46. http://dx.doi.org/10.1017/s1446181117000104.

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We address the construction and approximation for feed-forward neural networks (FNNs) with zonal functions on the unit sphere. The filtered de la Vallée-Poussin operator and the spherical quadrature formula are used to construct the spherical FNNs. In particular, the upper and lower bounds of approximation errors by the FNNs are estimated, where the best polynomial approximation of a spherical function is used as a measure of approximation error.

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