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Computing Nonlinear Eigenfunctions via Gradient Flow Extinction [article]

Leon Bungert, Martin Burger, Daniel Tenbrinck
2019 arXiv   pre-print
In this work we investigate the computation of nonlinear eigenfunctions via the extinction profiles of gradient flows.  ...  We discuss results of numerical experiments in which we use extinction profiles and the gradient flow for the task of spectral graph clustering as used, e.g., in machine learning applications.  ...  (GF) (top row) and the proposed extinction profile scheme (S) (bottom row) Fig. 3 . 3 Comparison between the computed 1D TV eigenfunctions of the gradient flow scheme (GF) and the proposed extinction  ... 
arXiv:1902.10414v1 fatcat:z6hmoignrzhadoo2av46m5xrny

Modes of Homogeneous Gradient Flows [article]

Ido Cohen, Omri Azencot, Pavel Lifshitz, Guy Gilboa
2020 arXiv   pre-print
Here our aim is to establish a consistent theory for gradient flows ψ_t = P(ψ), where P is a nonlinear homogeneous operator.  ...  We then proceed to show that in the general case the orthogonal modes {ϕ_i } are approximately nonlinear eigenfunctions P(ϕ_i) ≈λ_i ϕ_i.  ...  Firstly, we show that DMD is not applicable for γ-homogeneous flow (γ = 1), initialized with a nonlinear eigenfunction.  ... 
arXiv:2007.01534v3 fatcat:xqxykdlbrfeuvajazguto6u2ye

Spectral Decompositions using One-Homogeneous Functionals [article]

Martin Burger and Guy Gilboa and Michael Moeller and Lina Eckardt and Daniel Cremers
2016 arXiv   pre-print
Additionally, results on the orthogonality of the decomposition, a Parseval-type identity and the notion of generalized (nonlinear) eigenvectors closely link our nonlinear multiscale decompositions to  ...  This paper discusses the use of absolutely one-homogeneous regularization functionals in a variational, scale space, and inverse scale space setting to define a nonlinear spectral decomposition of input  ...  MB acknowledges support by ERC via Grant EU FP 7 -ERC Consolidator Grant 615216 LifeInverse.  ... 
arXiv:1601.02912v1 fatcat:d4ck2hdhzrckzfrrdzglwdp4zi

Adaptive Anisotropic Total Variation - A Nonlinear Spectral Analysis [article]

Shai Biton, Guy Gilboa
2018 arXiv   pre-print
We have that eigenfunction sets, admitting λ u ∈∂ J_A^2TV(u), are perfectly preserved under A^2TV-flow or minimization with L^2 square fidelity.  ...  They have shown that a TV regularizer can spatially preserve perfectly sets which are nonlinear eigenfunctions of the form λ u ∈∂ J_TV(u), where ∂ J_TV(u) is the TV subdifferential.  ...  The time of the flow chosen for the computation is % of the approximated extinction time. The value of = 0.05 · R a was used (for R a = 200, = 10).  ... 
arXiv:1811.11281v1 fatcat:4hf2cngsqvamthhcvy2luk5sdm

Nonlinear Spectral Geometry Processing via the TV Transform [article]

Marco Fumero, Michael Moeller, Emanuele Rodolà
2020 arXiv   pre-print
We introduce a novel computational framework for digital geometry processing, based upon the derivation of a nonlinear operator associated to the total variation functional.  ...  Our computational framework is flexible, can be applied to a variety of signals, and is easily adapted to different geometry representations, including triangle meshes and point clouds.  ...  MF and ER are supported by the ERC Starting Grant No. 802554 (SPECGEO) and the MIUR under grant "Dipartimenti di eccellenza 2018-2022" of the Department of Computer Science of Sapienza University.  ... 
arXiv:2009.03044v1 fatcat:woo7h2zzizhrrm43h5ieudwnii

Spectral Representations of One-Homogeneous Functionals [article]

Martin Burger, Lina Eckardt, Guy Gilboa, Michael Moeller
2015 arXiv   pre-print
We discuss three meaningful definitions of spectral representations by scale space and variational methods and prove that (nonlinear) eigenfunctions of the regularization functionals are indeed atoms in  ...  Acknowledgements MB acknowledges support by ERC via Grant EU FP 7 -ERC Consolidator Grant 615216 LifeInverse.  ...  Let us start with the gradient flow, for which it is natural to investigate the energy dissipation, i.e., we compute time derivatives of J.  ... 
arXiv:1503.05293v1 fatcat:ci56h7q7pvaafd3ywfawxzmzce

Examining the Limitations of Dynamic Mode Decomposition through Koopman Theory Analysis [article]

Ido Cohen, Guy Gilboa
2021 arXiv   pre-print
These conditions lay the foundations for system reconstruction, global controllability, and observability for nonlinear dynamics. DMD can be interpreted as a finite dimension approximation of KMD.  ...  Eli Appelboim from Electrical and Computer Engineering Department for stimulating discussions.  ...  we would like to obtain global controllability via a Koopman eigenfunction according to Remark 3.19.  ... 
arXiv:2107.07456v1 fatcat:fpmzp6b2kre3jk3upaorlgskme

Nonlinear Spectral Decompositions by Gradient Flows of One-Homogeneous Functionals [article]

Leon Bungert, Martin Burger, Antonin Chambolle, Matteo Novaga
2020 arXiv   pre-print
results on finite-dimensional polyhedral semi-norms, where gradient flows can yield spectral decompositions into eigenvectors.  ...  If these are eigenvectors, this implies an interesting orthogonality relation and the equivalence of the gradient flow to a variational regularization method and an inverse scale space flow.  ...  Due to the invertibility of R, the gradient flows with respect to J andJ are fully equivalent and the respective solutions are connected via the rotation R.  ... 
arXiv:1901.06979v2 fatcat:mowotx73xrcrdhyphkkgjbkj2q

Unsteady rotating laminar flow: analytical solution of Navier-Stokes equations [article]

Alessio Bocci and Giovanni Mingari Scarpello and Daniele Ritelli
2016 arXiv   pre-print
Basic physical assumptions are that the pressure axial gradient keeps itself on its hydrostatic value and that no radial velocity exists.  ...  We provide a integration of Navier-Stokes equations concerning the unsteady-state laminar flow of an incompressible, isothermal (newtonian) fluid in a cylindrical vessel spinning about its symmetry axis  ...  The problem will be solved by expanding in a series of eigenfunctions the function Ωr: the relevant coefficients C n have to be computed founding on the fact that the family of the eigenfunctions forms  ... 
arXiv:1609.05392v2 fatcat:gy44lf76j5a7hphaen25bbqch4

A noise-controlled free shear flow

2005 Journal of Fluid Mechanics  
However, decomposition of the flow into empirical eigenfunctions, as surrogates for Fourier modes in the non-periodic streamwise direction, shows that the turbulence structures advect downstream more uniformly  ...  The adjoint of the perturbed and linearized compressible viscous flow equations is formulated in such a way that its solution can be used to optimize control actuation in order to reduce flow-generated  ...  We also gratefully acknowledge the financial support from AFOSR and the computer resources provided by NPACI and NCSA.  ... 
doi:10.1017/s0022112005007093 fatcat:fkeovnoib5g2hitf6x7cw32cea

Second-order flows for computing the ground states of rotating Bose-Einstein condensates [article]

Haifan Chen, Guozhi Dong, Wei Liu, Ziqing Xie
2022 arXiv   pre-print
complexity in comparison with the respective gradient flow type approaches.  ...  Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order  ...  In [32, Theorem 2.8], it is proven that there exists no finite extinction time of second-order flows for homogeneous functional, whereas the extinction time is finite for gradient flows of homogeneous  ... 
arXiv:2205.00805v1 fatcat:l5pre44r3rfdrcq5hhdmb7rwzu

Classification and Analysis of Mean Curvature Flow Self-Shrinkers [article]

Caleb Hussey
2013 arXiv   pre-print
We investigate Mean Curvature Flow self-shrinking hypersurfaces with polynomial growth. It is known that such self shrinkers are unstable. We focus mostly on self-shrinkers of the form S^k×^n-k⊂^n+1.  ...  We use a connection between the stability operator and the quantum harmonic oscillator Hamiltonian to find all eigenvalues and eigenfunctions of the stability operator on these self-shrinkers.  ...  MCF is a nonlinear parabolic flow, so its solutions satisfy a maximum principle [N] .  ... 
arXiv:1303.0354v1 fatcat:uqv2sneoq5b2rn2xggv2qhoini

Lagrangian model simulations of molecular mixing, including finite rate chemical reactions, in a temporally developing shear layer

Chester H. H. Chang, Werner J. A. Dahm, Grétar Tryggvason
1991 Physics of Fluids A Fluid Dynamics  
The results indicate that the model is able to accurately follow even highly sensitive nonlinear measures of the mixing and reaction progress such as the local extinction phenomenon characteristic of large  ...  diffusion and chemical reaction scales of the flow.  ...  In particular, these comparisons test the ability of the model to accurately follow even highly sensitive nonlinear measures of the mixing and reaction progress such as the local extinction phenomenon  ... 
doi:10.1063/1.858058 fatcat:3kyvxdtphvanrdpgb7m7yjz2si

Effect of combustion on near-critical swirling flow

Zvi Rusak, A K Kapila, Jung J Choi
2002 Combustion theory and modelling  
In the absence of combustion the columnar solution loses stability via a transcritical bifurcation as the level of swirl rises beyond a critical value.  ...  As a result the critical value of swirl for a combusting flow is smaller than that for the cold flow.  ...  Using a standard ordinary differential equation solver from Maple for (51) (with the numerical condition Φ y (0) = 1 ), the eigenfunction Φ(y) is computed.  ... 
doi:10.1088/1364-7830/6/4/305 fatcat:fbdbhrpor5hofiskbkx63bzz4y

Tomographic absorption spectroscopy for the study of gas dynamics and reactive flows

Weiwei Cai, Clemens F. Kaminski
2017 Progress in Energy and Combustion Science  
Optical imaging techniques are ubiquitous for the resolution of non-uniformities in gas flows.  ...  Planar imaging techniques such as laser-induced fluorescence are well established and applied extensively in turbulent reactive flows, offering both high temporal and spatial resolutions.  ...  Since the computational cost for each function evaluation also increases with pixel number, the computational cost increases exponentially with the size of the nonlinear tomographic inversion problem.  ... 
doi:10.1016/j.pecs.2016.11.002 fatcat:nzehtszawncprekxevvc2na77q
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