Convergence of a first order scheme for a non-local eikonal equation

2006 ◽  
Vol 56 (9) ◽  
pp. 1136-1146 ◽  
Author(s):  
O. Alvarez ◽  
E. Carlini ◽  
R. Monneau ◽  
E. Rouy
Keyword(s):  
Eikonal Equation ◽  
First Order ◽  
Order Scheme ◽  
Non Local

Kirchhoff migration using eikonal-based computation of traveltime source derivatives

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. S211-S219 ◽  
Author(s):  
Siwei Li ◽  
Sergey Fomel
Keyword(s):  
Differential Equation ◽  
Eikonal Equation ◽  
Order Partial Differential Equation ◽  
Fast Marching ◽  
First Order ◽  
Velocity Models ◽  
Kirchhoff Migration ◽  
Input Source ◽  
First Arrival ◽  
Synthetic Models

The computational efficiency of Kirchhoff-type migration can be enhanced by using accurate traveltime interpolation algorithms. We addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location. We adopted a first-order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently. Unlike methods that rely on finite-difference estimations, the accuracy of the eikonal-based derivative did not depend on input source sampling. For smooth velocity models, the first-order traveltime source derivatives enabled a cubic Hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into Kirchhoff antialiasing schemes. We provided an implementation of the proposed method to first-arrival traveltimes by modifying the fast-marching eikonal solver. Several simple synthetic models and a semirecursive Kirchhoff migration of the Marmousi model demonstrated the applicability of the proposed method.

scholarly journals A temporal fourth-order scheme for the first-order acoustic wave equations

2013 ◽  
Vol 194 (3) ◽  
pp. 1473-1485 ◽  
Author(s):  
Guihua Long ◽  
Yubo Zhao ◽  
Jun Zou
Keyword(s):  
Acoustic Wave ◽  
Fourth Order ◽  
Wave Equations ◽  
First Order ◽  
Order Scheme

A Non-Local Convective Flux Limiter for Upwind-Biased Finite Volume Simulations

2010 ◽  
Author(s):  
Nicole M. W. Poe ◽  
D. Keith Walters
Keyword(s):  
Finite Volume ◽  
Finite Volume Methods ◽  
Second Order ◽  
Numerical Dissipation ◽  
Ansys Fluent ◽  
First Order ◽  
Low Dissipation ◽  
Solution Convergence ◽  
Improve Accuracy ◽  
Non Local

Finite volume methods on structured and unstructured meshes often utilize second-order, upwind-biased linear reconstruction schemes to approximate the convective terms, in an attempt to improve accuracy over first-order methods. Limiters are employed to reduce the inherent variable over- and under-shoot of these schemes; however, they also can significantly increase the numerical dissipation of a solution. This paper presents a novel non-local, non-monotonic (NLNM) limiter developed by enforcing cell minima and maxima on dependent variable values projected to cell faces. The minimum and maximum values for a cell are determined primarily through the recursive reference to the minimum and maximum values of its upwind neighbors. The new limiter is implemented using the User Defined Function capability available in the commercial CFD solver Ansys FLUENT. Various simple test cases are presented which exhibit the NLNM limiter’s ability to eliminate non-physical oscillations while maintaining relatively low dissipation of the solution. Results from the new limiter are compared with those from other limited and unlimited second-order upwind (SOU) and first-order upwind (FOU) schemes. For the cases examined in the study, the NLNM limiter was found to improve accuracy without significantly increasing solution convergence rate.

scholarly journals Classification of bi-Hamiltonian pairs extended by isometries

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210185
Author(s):  
Maxim V. Pavlov ◽  
Pierandrea Vergallo ◽  
Raffaele Vitolo
Keyword(s):  
Type System ◽  
Local Part ◽  
First Order ◽  
Hydrodynamic Type System ◽  
Dependent Variables ◽  
Particular Solution ◽  
Non Local ◽  
Hamiltonian Operators ◽  
Hamiltonian Pair

The aim of this article is to classify pairs of the first-order Hamiltonian operators of Dubrovin–Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations.

scholarly journals A measure theoretic approach to traffic flow optimisation on networks

2018 ◽  
Vol 30 (6) ◽  
pp. 1187-1209 ◽  
Author(s):  
SIMONE CACACE ◽  
FABIO CAMILLI ◽  
RAUL DE MAIO ◽  
ANDREA TOSIN
Keyword(s):  
Traffic Flow ◽  
Gradient Descent ◽  
Optimal Control Problems ◽  
Theoretic Approach ◽  
Nonlinear Transport ◽  
Traffic Lights ◽  
First Order ◽  
Flow Problems ◽  
Non Local ◽  
Smart Traffic

We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective is to minimise/maximise macroscopic quantities, such as traffic volume or average speed, controlling few agents, e.g. smart traffic lights and automated cars. The measure theoretic approach allows to study in a same setting local and non-local drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. We also propose a gradient descent adjoint-based optimisation method, obtained by deriving first-order optimality conditions for the control problem, and we provide some numerical experiments in the case of smart traffic lights for a 2–1 junction.

High order scheme for the non-local transport in ICF plasmas

2006 ◽  
Vol 133 ◽  
pp. 205-207
Author(s):  
J.-L. Feugeas ◽  
Ph. Nicolaï ◽  
G. Schurtz ◽  
P. Charrier ◽  
E. Ahusborde
Keyword(s):  
High Order ◽  
High Order Scheme ◽  
Order Scheme ◽  
Non Local ◽  
Local Transport

scholarly journals Explicitly Imposing Constraints in Deep Networks via Conditional Gradients Gives Improved Generalization and Faster Convergence

2019 ◽  
Vol 33 ◽  
pp. 4772-4779
Author(s):  
Sathya N. Ravi ◽  
Tuan Dinh ◽  
Vishnu Suresh Lokhande ◽  
Vikas Singh
Keyword(s):  
Numerical Optimization ◽  
Gradient Descent ◽  
Deep Neural Networks ◽  
Image Inpainting ◽  
Stochastic Gradient Descent ◽  
Rigorous Analysis ◽  
First Order ◽  
General Constraints ◽  
Limited Applicability ◽  
Order Scheme

A number of results have recently demonstrated the benefits of incorporating various constraints when training deep architectures in vision and machine learning. The advantages range from guarantees for statistical generalization to better accuracy to compression. But support for general constraints within widely used libraries remains scarce and their broader deployment within many applications that can benefit from them remains under-explored. Part of the reason is that Stochastic gradient descent (SGD), the workhorse for training deep neural networks, does not natively deal with constraints with global scope very well. In this paper, we revisit a classical first order scheme from numerical optimization, Conditional Gradients (CG), that has, thus far had limited applicability in training deep models. We show via rigorous analysis how various constraints can be naturally handled by modifications of this algorithm. We provide convergence guarantees and show a suite of immediate benefits that are possible — from training ResNets with fewer layers but better accuracy simply by substituting in our version of CG to faster training of GANs with 50% fewer epochs in image inpainting applications to provably better generalization guarantees using efficiently implementable forms of recently proposed regularizers.

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