Optimum design of thin plates via frequency optimization using BEM

Babouskos, Nick G.; Katsikadelis, John T.

doi:10.1007/s00419-014-0962-7

“…Numerical differentiation is highly sensitive to noise [34], especially for the higher order derivatives. After an extensive examination the literature utilizing IRBFs [35, 45, 46] for the derivatives approximation, or for the solution of specific problems [47, 48] as well as other differentiation methods [49, 50] and a variety of formulations for IRBFs (14a, 14b, 14c, 14d, 14e, 14f), the proposed procedure accomplished striking accuracy, with error magnitude ε of O (10 −100 ) or less (Figures - and supplementary database), for the derivatives' computation. This finally permitted the implementation of the Taylor method for significant extrapolation extents (Figures 2 and 3 and Data ).…”

Section: Discussionmentioning

confidence: 99%

Numerical Solution for the Extrapolation Problem of Analytic Functions

Bakas

¹

2019

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In this work, a numerical solution for the extrapolation problem of a discrete set of n values of an unknown analytic function is developed. The proposed method is based on a novel numerical scheme for the rapid calculation of higher order derivatives, exhibiting high accuracy, with error magnitude of O(10−100) or less. A variety of integrated radial basis functions are utilized for the solution, as well as variable precision arithmetic for the calculations. Multiple alterations in the function’s direction, with no curvature or periodicity information specified, are efficiently foreseen. Interestingly, the proposed procedure can be extended in multiple dimensions. The attained extrapolation spans are greater than two times the given domain length. The significance of the approximation errors is comprehensively analyzed and reported, for 5832 test cases.

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“…In empirical practice, one may select among the available in literature RBFs, try some new, or optimize their shape parameter c. In Appendix I, we also provide a simple computer code for the symbolic differentiation of any selected RBF, using SymPy (Meurer et al 2017) package. Particular interest is exhibited in the Integrated RBFs (IRBFs) (Bakas 2019;Babouskos and Katsikadelis 2015;Yiotis and Katsikadelis 2015), which are formulated from the indefinite integration of the kernel, such that its derivative is the RBF u. Accordingly, we may integrate the kernel more than once, to approximate higher-order derivatives.…”

Section: Annbn For the Approximation Of Derivativesmentioning

confidence: 99%

Gradient free stochastic training of ANNs, with local approximation in partitions

Bakas

¹

,

Langousis

²

,

Nicolaou

³

et al. 2023

Stoch Environ Res Risk Assess

4

0

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We present a numerical scheme for computation of Artificial Neural Networks (ANN) weights, which stems from the Universal Approximation Theorem, avoiding costly iterations. The proposed algorithm adheres to the underlying theory, is highly fast, and results in remarkably low errors when applied to regression and classification problems of complex data sets with $${\textbf{x}} \in {\mathbb {R}}^{n}$$ x ∈ R n (e.g. Griewank, Gomez-Levy, Shekel, and Polynomial functions) with random noise addition (i.e. Uniform, Normal, Generalized Pareto, Log-Normal, and a mixture of Log-Normal, Exponential, and Frechet), as well as the database for handwritten digits recognition MNIST (Modified National Institute of Standards and Technology) with $$7\times 10^4$$ 7 × 10 4 images. The same mathematical formulation was found capable of approximating highly nonlinear functions in multiple dimensions, with low errors (e.g. $$10^{-10}$$ 10 - 10 ) for the test set of the unknown functions, their higher-order partial derivatives, as well as numerically solving Partial Differential Equations, such as those appearing in Physics, Engineering, Environmental Sciences, etc. The method is based on the calculation of the weights of each neuron in small neighbourhoods of the data. Accordingly, optimization of hyperparameters is not necessary, as the number of neurons stems directly from the dimensionality of the data, further improving the algorithmic speed. Under this setting, overfitting is inherently avoided, and the results are interpretable and reproducible. The complexity of the proposed algorithm is of class P with $${\mathcal {O}}(mNni_{cl} + Nmn^2+Nn^3 + mN^2+N^3)$$ O ( m N n i cl + N m n 2 + N n 3 + m N 2 + N 3 ) computing time, with respect to the observations m, features n, and Neurons N, contrary to the NP-Complete class of standard algorithms for ANN training. The performance of the method is high, irrespective of the size of the data set, and the test set errors are similar or smaller than the training errors, indicating the generalization efficiency of the algorithm. A supplementary computer code in Julia and Python Languages is provided, which can be used to reproduce the validation examples, and/or apply the algorithm to other data sets.

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“…Utilizing a high arithmetic precision, we demonstrate that such need, which arose to address the computational inaccuracies, does not exist. Taking into account the high extrapolation spans attained in [1] and obtained with integrated radial basis functions [22,23] and some hundreds or even thousands of digits for the calculations, we apply a high arithmetic precision utilizing the "BigFloat" structure of Julia language [24], using the GMP [9] library to truncate the Taylor series, known as Taylor polynomials or partial sums.…”

Section: Introductionmentioning

confidence: 99%

Taylor Polynomials in a High Arithmetic Precision as Universal Approximators

Bakas

2024

Computation

0

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Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code.

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Optimum design of thin plates via frequency optimization using BEM

Cited by 7 publications

References 21 publications

Numerical Solution for the Extrapolation Problem of Analytic Functions

Numerical Solution for the Extrapolation Problem of Analytic Functions

Gradient free stochastic training of ANNs, with local approximation in partitions

Taylor Polynomials in a High Arithmetic Precision as Universal Approximators

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