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Concept

Spectral method

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a Fourier series which is a sum of sinusoids) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible. Spectral methods and finite element methods are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains (compact support). Consequently, spectral methods connect variables globally while finite elements do so locally. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possib

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Chargement

Controlling oscillations in spectral schemes using Artificial Neural Networks

Lukas Schwander

Global Fourier spectral methods are excellent tools to solve conserva- tion laws. They enable fast convergence rates and highly accurate solutions. However, being high-order methods, they suffer from the Gibbs phenomenon, which leads to spurious numerical oscillations in the vicinity of discontinuities. This can have a detrimental effect on the solution quality and lead to unphysical results. While local approxima- tion techniques allow for local limiting or reconstruction, there are no such possibilities for global methods. This thesis proposes a neural net- work based method that adds artificial viscosity around discontinuities of the solution to the conservation law. This enables the transforma- tion of discontinuities into steep but continuous jumps. Test cases in one and two spatial dimensions as well as systems of conservation laws (Euler equations) are solved. Furthermore, the method is generalized to other global basis approaches on non-uniform grids and reduced ba- sis methods. The proposed method delivers satisfactory results in all test cases. On the one hand, it is able to detect and handle discontinu- ities. On the other hand, it stays highly accurate for smooth data.

2020

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While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods as a local estimate of solution regularity generally is required. In this work we demonstrate how to train and use a local artificial neural network to estimate the local solution regularity and demonstrate the efficiency of nonlinear artificial viscosity methods based on this in the context of Fourier spectral methods. We compare with entropy viscosity techniques and illustrate the promise of the neural network based estimators when solving one- and two-dimensional conservation laws, including the Euler equations.

2020

Chargement

Controlling oscillations in spectral methods by local artificial viscosity governed by neural networks

Jan Sickmann Hesthaven, Deep Ray, Lukas Schwander

While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods since a local estimate of the solution regularity is generally required. In this work we demonstrate how to train and use a local artificial neural network to estimate the local solution regularity and demonstrate the efficiency of nonlinear artificial viscosity methods based on this, in the context of Fourier spectral methods. We compare with entropy viscosity techniques and illustrate the promise of the neural network based estimators when solving one- and two-dimensional conservation laws, including the Euler equations. (C) 2021 Elsevier Inc. All rights reserved.

ACADEMIC PRESS INC ELSEVIER SCIENCE2021

Chargement

Transformation de Fourier rapide

La transformation de Fourier rapide (sigle anglais : FFT ou fast Fourier transform) est un algorithme de calcul de la transformation de Fourier discrète (TFD). Sa complexité varie en O(n log n) ave

Transformation de Fourier

thumb|Portrait de Joseph Fourier. En mathématiques, plus précisément en analyse, la transformation de Fourier est une extension, pour les fonctions non périodiques, du développement en série de Fou

Sciences numériques

Les sciences numériques (traduction de l'anglais computational sciences), autrement dénommées calcul scientifique ou informatique scientifique, ont pour objet la construction de modèles mathématiques

ME-474: Numerical flow simulation

This course provides practical experience in the numerical simulation of fluid flows. Numerical methods are presented in the framework of the finite volume method. A simple solver is developed with Matlab, and a commercial software is used for more complex problems.

ME-372: Finite element method

L'étudiant acquiert une initiation théorique à la méthode des éléments finis qui constitue la technique la plus courante pour la résolution de problèmes elliptiques en mécanique. Il apprend à appliquer cette méthode à des cas simples et à l'exploiter pour résoudre les problèmes de la pratique.

PHYS-512: Statistical physics of computation

This course covers the statistical physics approach to computer science problems ranging from graph theory and constraint satisfaction to inference and machine learning. In particular the replica and cavity methods, message passings algorithms, and analysis of the related phase transitions.