Efficient computation of oscillatory integrals by exponential transformations

2021 ◽  
Author(s):  
Hassan Majidian
Keyword(s):  
Oscillatory Integrals ◽  
Efficient Computation

scholarly journals Efficient Computation of Highly Oscillatory Fourier Transforms with Nearly Singular Amplitudes over Rectangle Domains

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1930
Author(s):  
Zhen Yang ◽  
Junjie Ma
Keyword(s):  
Adaptive Mesh Refinement ◽  
Numerical Experiments ◽  
Singular Integrals ◽  
Fourier Transforms ◽  
Adaptive Mesh ◽  
Oscillatory Integrals ◽  
Efficient Computation ◽  
New Approach ◽  
Highly Oscillatory ◽  
Nearly Singular Integrals

In this paper, we consider fast and high-order algorithms for calculation of highly oscillatory and nearly singular integrals. Based on operators with regard to Chebyshev polynomials, we propose a class of spectral efficient Levin quadrature for oscillatory integrals over rectangle domains, and give detailed convergence analysis. Furthermore, with the help of adaptive mesh refinement, we are able to develop an efficient algorithm to compute highly oscillatory and nearly singular integrals. In contrast to existing methods, approximations derived from the new approach do not suffer from high oscillatory and singularity. Finally, several numerical experiments are included to illustrate the performance of given quadrature rules.

On efficient computation of multidimensional oscillatory integrals with local Fourier bases

2001 ◽  
Vol 47 (5) ◽  
pp. 3491-3502 ◽  
Author(s):  
A. Averbuch ◽  
E. Braverman ◽  
M. Israeli ◽  
R. Coifman
Keyword(s):  
Oscillatory Integrals ◽  
Efficient Computation

scholarly journals Efficient Computation of Oscillatory Integrals via Adaptive Multiscale Local Fourier Bases

2000 ◽  
Vol 9 (1) ◽  
pp. 19-53 ◽  
Author(s):  
A. Averbuch ◽  
E. Braverman ◽  
R. Coifman ◽  
M. Israeli ◽  
A. Sidi
Keyword(s):  
Oscillatory Integrals ◽  
Efficient Computation

scholarly journals Efficient Computation of Highly Oscillatory Integrals by Using QTT Tensor Approximation

2016 ◽  
Vol 16 (1) ◽  
pp. 145-159 ◽  
Author(s):  
Boris Khoromskij ◽  
Alexander Veit
Keyword(s):  
Oscillatory Integrals ◽  
Integration Scheme ◽  
Efficient Computation ◽  
Oscillatory Function ◽  
Numerical Integration Scheme ◽  
Wide Range ◽  
Spatial Dimensions ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals ◽  
Logarithmic Complexity

AbstractWe propose a new method for the efficient approximation of a class of highly oscillatory weighted integrals where the oscillatory function depends on the frequency parameter ${\omega \ge 0}$, typically varying in a large interval. Our approach is based, for a fixed but arbitrary oscillator, on the pre-computation and low-parametric approximation of certain ω-dependent prototype functions whose evaluation leads in a straightforward way to recover the target integral. The difficulty that arises is that these prototype functions consist of oscillatory integrals which makes them difficult to evaluate. Furthermore, they have to be approximated typically in large intervals. Here we use the quantized-tensor train (QTT) approximation method for functional M-vectors of logarithmic complexity in M in combination with a cross-approximation scheme for TT tensors. This allows the accurate approximation and efficient storage of these functions in the wide range of grid and frequency parameters. Numerical examples illustrate the efficiency of the QTT-based numerical integration scheme on various examples in one and several spatial dimensions.

No fear of George Kingsley Zipf

10.1558/37291 ◽  
2018 ◽  
Vol 2 (2) ◽  
pp. 242-263
Author(s):  
Stefano Rastelli ◽  
Kook-Hee Gil
Keyword(s):  
Minimalist Program ◽  
Classroom Research ◽  
Language Design ◽  
Efficient Computation ◽  
Generative Linguistics ◽  
The Third ◽  
Factor Effect ◽  
Language Classroom ◽  
Recent Developments ◽  
Insight Into

This paper offers a new insight into GenSLA classroom research in light of recent developments in the Minimalist Program (MP). Recent research in GenSLA has shown how generative linguistics and acquisition studies can inform the language classroom, mostly focusing on what linguistic aspects of target properties should be integrated as a part of the classroom input. Based on insights from Chomsky’s ‘three factors for language design’ – which bring together the Faculty of Language, input and general principles of economy and efficient computation (the third factor effect) for language development – we put forward a theoretical rationale for how classroom research can offer a unique environment to test the learnability in L2 through the statistical enhancement of the input to which learners are exposed.

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