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What Is the Fractional Laplacian? [article]

Anna Lischke, Guofei Pang, Mamikon Gulian, Fangying Song, Christian Glusa, Xiaoning Zheng, Zhiping Mao, Wei Cai, Mark M. Meerschaert, Mark Ainsworth, George Em Karniadakis
2019 arXiv   pre-print
In this work, we provide new numerical methods as well as a self-contained discussion of state-of-the-art methods for discretizing the fractional Laplacian, and we present new results on the differences  ...  of the fractional Laplacian in bounded domains is most appropriate for a given application.  ...  Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc  ... 
arXiv:1801.09767v3 fatcat:nmzykkn2vzaqnpf272cgq6lypq

Fractional Laplacians : A short survey

Maha Daoud, El Haj Laamri
2021 Discrete and Continuous Dynamical Systems. Series S  
This paper describes the state of the art and gives a survey of the wide literature published in the last years on the fractional Laplacian.  ...  Also we give a rather long list of references : it is certainly not exhaustive but hopefully rich enough to track most connected results.  ...  The rest of paper is composed of four sections. Section 2 is devoted to a review of several (and equivalent) definitions of the fractional Laplacian (−∆) s in R N and its main properties.  ... 
doi:10.3934/dcdss.2021027 fatcat:42e4znuopjbblp7utk6snbycty

Laplacian Mesh Processing [article]

Olga Sorkine
2005 Eurographics State of the Art Reports  
The core of the framework is the mesh Laplacian operator, which allows to define differential coordinates and new bases for efficient mesh geometry representation.  ...  This framework is based on linear operators defined on polygonal meshes, and furnishes a variety of processing applications, such as shape approximation and compact representation, mesh editing, filtering  ...  Finally, the target mesh with the new coating is reconstructed via Eq. 3. Figure 13 shows some coating transfer results and mesh transplanting results from [SLCO * 04].  ... 
doi:10.2312/egst.20051044 fatcat:nmjjc5vhlvbhlg2akod3lzwq54

Fractional Laplacian in Conformal Geometry [article]

Sun-Yung Alice Chang, Maria del Mar Gonzalez
2010 arXiv   pre-print
In this note, we study the connection between the fractional Laplacian operator that appeared in the recent work of Caffarelli-Silvestre and a class of conformally covariant operators in conformal geometry  ...  The lemma is proven by comparing (3.13) together with (3.12), with the Caffarelli-Silvestre construction for the fractional Laplacian as given in (3.2) .  ...  The resulting problem (4.2) is still of divergence-type, degenerate elliptic, and with a weight in the Muckenhoupt class A 2 (c.f.  ... 
arXiv:1003.0398v1 fatcat:jmhaqhsox5ei5h53thgirclqia

Fractional Laplacian in conformal geometry

Sun-Yung Alice Chang, María del Mar González
2011 Advances in Mathematics  
In this note, we study the connection between the fractional Laplacian operator that appeared in the recent work of Caffarelli and Silvestre and a class of conformally covariant operators in conformal  ...  The lemma is proven by comparing (3.13) together with (3.12), with the Caffarelli and Silvestre construction for the fractional Laplacian as given in (3.2). 2 The next step is to generalize Theorem 3.1  ...  The resulting problem (4.2) is still of divergence-type, degenerate elliptic, and with a weight in the Muckenhoupt class A 2 (cf.  ... 
doi:10.1016/j.aim.2010.07.016 fatcat:hebzlpfazfdzvishxlo54zkioy

Direct Laplacian Center Gauge

Manfried Faber, Jeff Greensite, Stefan Olejník
2001 Journal of High Energy Physics  
We introduce a variation of direct maximal center gauge fixing: the "direct Laplacian" center gauge.  ...  The new procedure overcomes certain shortcomings of maximal center gauge, associated with Gribov copies, that were pointed out by Bornyakov et al. in hep-lat/0009035.  ...  Acknowledgments Our research is supported in part by Fonds zur Förderung der Wissenschaftlichen  ... 
doi:10.1088/1126-6708/2001/11/053 fatcat:dutrlorvj5ajlll76sjxgwpbpi

Laplacian Fractional Revival on Graphs

Ada Chan, Bobae Johnson, Mengzhen Liu, Malena Schmidt, Zhanghan Yin, Hanmeng Zhan
2021 Electronic Journal of Combinatorics  
We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian.  ...  In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional  ...  The authors would like to thank Gabriel Coutinho, Chris Godsil and Christino Tamon for useful discussions, and the anonymous reviewers for helpful comments.  ... 
doi:10.37236/10146 fatcat:mbbezlwnlfbu5fmpcvdjret234

On the fractional Laplacian of variable order [article]

Eric Darve, Marta D'Elia, Roberto Garrappa, Andrea Giusti, Natalia L. Rubio
2021 arXiv   pre-print
We present a novel definition of variable-order fractional Laplacian on Rn based on a natural generalization of the standard Riesz potential.  ...  Our definition holds for values of the fractional parameter spanning the entire open set (0, n/2).  ...  In Section 2 we review some basic results on the standard fractional Laplacian, Riesz potential, and the fractional Poisson's equation.  ... 
arXiv:2109.01060v1 fatcat:yclb2z5zsrh7foonfbptxdenqi

Laplacian modes probing gauge fields

Falk Bruckmann, Ernst-Michael Ilgenfritz
2005 Physical Review D  
We also investigate periodic and antiperiodic modes of the adjoint Laplacian for comparison. In the second part we introduce a new Fourier-like low-pass filter method.  ...  It provides link variables by truncating a sum involving the Laplacian eigenmodes.  ...  EMI gratefully appreciates the hospitality at the Instituut Lorentz of Leiden University and FB that of the Theoretical Particle Physics group of Humboldt University Berlin.  ... 
doi:10.1103/physrevd.72.114502 fatcat:ip2msbxdanhaxkyydo5nafnueu

Discrete Laplacian operator and its applications in signal processing

Waseem Waheed, Guang Deng, Bo Liu.
2020 IEEE Access  
The construction of the proposed fractional Laplacian utilizes the DCT transform avoiding the complexity associated with the discretization step which is typical in the constructions based on signal processing  ...  Furthermore, a consensus on the most suitable definition for a given task is yet to be reached.  ...  Compared with the Laplacian operator, the fractional Laplacian operator permits the user to ''tune'' the parameter α to achieve the desired result.  ... 
doi:10.1109/access.2020.2993577 fatcat:tqiqubafujh77n4vmjog2xyqby

Unbiased `walk-on-spheres' Monte Carlo methods for the fractional Laplacian [article]

Andreas E. Kyprianou and Ana Osojnik and Tony Shardlow
2017 arXiv   pre-print
In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic alpha-stable process with alpha in (0, 2).  ...  We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet boundary-value problem in which the Laplacian is replaced by the fractional Laplacian and boundary conditions are  ...  Acknowledgements We would like to thank Mateusz Kwaśniki for pointing out a number of references to us and Alexander Freudenberg for a close reading of an earlier version of this manuscript.  ... 
arXiv:1609.03127v3 fatcat:tqsjw4ndunenran64g2xpe4idi

Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights

Piotr Garbaczewski, Vladimir Stephanovich
2019 Physical review. E  
On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data respecting fractional Laplacian should actually be.  ...  The fractional Laplacian (- Δ)^α /2, α∈ (0,2) has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of α -stable stochastic processes in R^n.  ...  Dybiec for explanations concerning the stopping scenario of Refs. [29, 30] , R. Metzler for reference suggestion and A. Pakes for providing  ... 
doi:10.1103/physreve.99.042126 fatcat:foaz2yoywna7xpthesqnql7e5y

Iterated conformal dynamics and Laplacian growth

Felipe Barra, Benny Davidovitch, Itamar Procaccia
2002 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
In particular we show that the fractal dimension of Laplacian growth patterns is much higher than the fractal dimension of DLA, with the possibility of dimension 2 for the former not excluded.  ...  With this we stress that the difference between DLA and Laplacian growth is NOT in the manner of ultraviolet regularization, but rather in their deeply different growth rules.  ...  Since this is the fraction of the unit circle which is covered in each layer, the limit of Laplacian Growth is obtained with C = 1. DLA is asymptotically consistent with C = 0.  ... 
doi:10.1103/physreve.65.046144 pmid:12005963 fatcat:mnccu6tqkjbe7fk45emrraaj5a

A walk outside spheres for the fractional Laplacian: fields and first eigenvalue [article]

Tony Shardlow
2018 arXiv   pre-print
A criteria is derived for the variable accuracy and a comparison is given with analytical results of Dyda (2012).  ...  Finally, we show how to couple the method with the variable-accuracy Arnoldi iteration to compute the smallest eigenvalue of the fractional Laplacian.  ...  A review of the fractional Laplacian and its importance in the applied sciences is given by [LPG + 18], which includes a description of WOS and other numerical approaches.  ... 
arXiv:1803.03921v2 fatcat:eo2pire3unhhxowm2w6nyzsqeq

Recent progress on the fractional Laplacian in conformal geometry [article]

Maria del Mar Gonzalez
2016 arXiv   pre-print
The aim of this paper is to report on recent development on the conformal fractional Laplacian, both from the analytic and geometric points of view, but especially towards the PDE community.  ...  Thus we first review the classical construction for the conformal fractional Laplacian on the sphere coming from representation theory, which yields its Fourier symbol, and then prove some new results  ...  ) where (−∆) s is the standard fractional Laplacian on R n with respect to the Euclidean metric.  ... 
arXiv:1609.08988v1 fatcat:i3en3ep6vzdcdfjmxj23wxmrvi
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