Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour

2019 ◽  
Vol 144 ◽  
pp. 234-252 ◽  
Author(s):  
Raimund Bürger ◽  
Daniel Inzunza ◽  
Pep Mulet ◽  
Luis M. Villada
Keyword(s):  
Gradient Flow ◽  
Collective Behaviour ◽  
Explicit Methods ◽  
Flow Equations

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

scholarly journals A Flow Perspective on Nonlinear Least-Squares Problems

2020 ◽  
Vol 48 (4) ◽  
pp. 987-1003
Author(s):  
Hans Georg Bock ◽  
Jürgen Gutekunst ◽  
Andreas Potschka ◽  
María Elena Suaréz Garcés
Keyword(s):  
Least Squares ◽  
Newton Method ◽  
Large Scale ◽  
Gradient Flow ◽  
Nonlinear Least Squares ◽  
Bundle Adjustment ◽  
Least Squares Problems ◽  
Flow Equations ◽  
Backward Euler ◽  
Forward Euler

AbstractJust as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß–Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the corresponding Gauß–Newton flow equations. We highlight the advantages of the Gauß–Newton flow and the Gauß–Newton method from a statistical and a numerical perspective in comparison with the Newton method, steepest descent, and the Levenberg–Marquardt method, which are respectively equivalent to Newton flow forward Euler, gradient flow forward Euler, and gradient flow backward Euler. We finally show an unconditional descent property for a generalized Gauß–Newton flow, which is linked to Krylov–Gauß–Newton methods for large-scale nonlinear least squares problems. We provide numerical results for large-scale problems: An academic generalized Rosenbrock function and a real-world bundle adjustment problem from 3D reconstruction based on 2D images.

scholarly journals Multiscale Image Representation and Texture Extraction Using Hierarchical Variational Decomposition

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Liming Tang ◽  
Chuanjiang He
Keyword(s):  
Finite Difference ◽  
Energy Minimization ◽  
Scale Parameter ◽  
Image Representation ◽  
Gradient Flow ◽  
Hierarchical Decomposition ◽  
Multiscale Representation ◽  
Flow Equations ◽  
Proposed Model ◽  
Texture Extraction

In order to achieve a mutiscale representation and texture extraction for textured image, a hierarchical(BV,Gp,L2)decomposition model is proposed in this paper. We firstly introduce the proposed model which is obtained by replacing the fixed scale parameter of the original(BV,Gp,L2)decomposition with a varying sequence. And then, the existence and convergence of the hierarchical decomposition are proved. Furthermore, we show the nontrivial property of this hierarchical decomposition. Finally, we introduce a simple numerical method for the hierarchical decomposition, which utilizes gradient decent for energy minimization and finite difference for the associated gradient flow equations. Numerical results show that the proposed hierarchical(BV,Gp,L2)decomposition is very appropriate for multiscale representation and texture extraction of textured image.

scholarly journals Nonminimal gradient flows in QCD-like theories

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Marco Boers
Keyword(s):  
Degrees Of Freedom ◽  
Anomalous Dimension ◽  
Gradient Flow ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Flow Equation ◽  
Gradient Flows ◽  
Leading Order ◽  
Flow Equations ◽  
Yang Mills

Abstract The Yang-Mills gradient flow for QCD-like theories is generalized by including a fermionic matter term in the gauge field flow equation. We combine this with two different flow equations for the fermionic degrees of freedom. The solutions for the different gradient flow setups are used in the perturbative computations of the vacuum expectation value of the Yang-Mills Lagrangian density and the field renormalization factor of the evolved fermions up to next-to-leading order in the coupling. We find a one-parameter family of flow systems for which there exists a renormalization scheme in which the evolved fermion anomalous dimension vanishes to all orders in perturbation theory. The fermion number dependence of different flows is studied and applications to lattice studies are anticipated.

A finite-volume scheme for gradient-flow equations with non-homogeneous diffusion

2021 ◽  
Vol 89 ◽  
pp. 150-162
Author(s):  
Julien Mendes ◽  
Antonio Russo ◽  
Sergio P. Perez ◽  
Serafim Kalliadasis
Keyword(s):  
Finite Volume ◽  
Gradient Flow ◽  
Finite Volume Scheme ◽  
Flow Equations ◽  
Volume Scheme

scholarly journals Forward Roll Coating of a Williamson’s Material onto a Moving Web: A Theoretical Study

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
M. Zahid ◽  
I. Siddique ◽  
S. Saleem ◽  
A. Al-Zubaidi ◽  
M. A. Rana ◽  
...  
Keyword(s):  
Coating Thickness ◽  
Gradient Flow ◽  
Separation Point ◽  
Maximum Pressure ◽  
Perturbation Approach ◽  
Regular Perturbation ◽  
Roll Coating ◽  
Flow Equations ◽  
Operating Variables ◽  
Roll Separating Force

This paper presents a mathematical model for the thin film roll coating process of an incompressible Williamson material, passing through a closed passage between a rotating roll and a web. In light of lubrication approximation theory, the flow equations are nondimensionalized. The regular perturbation approach is used to provide solutions for the velocity profile, pressure gradient, flow rate per unit width, and shear stress at the roll surface. Important engineering quantities such as coating thickness, maximum pressure, separation point, roll/sheet separating force, and roll-transmitted power to the fluid are also obtained. The effects of several factors are graphically projected. The study shows that the material factors that are involved determine the operating variables. Coating thickness and separation point are controlled by Weissenberg’s number, therefore acting as a controlling parameter for the rate of flow, thickness in coating, power contribution, pressure, roll separating force, and separation point. In comparison to the existing results in the literature, the current results are broader and zero-order results are more accurate.

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