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Exponential decay of Laplacian eigenfunctions in domains with branches of variable crosssectional profiles, Eur
 Phys. J. B
"... Abstract. We consider the eigenvalue problem for the Laplace operator in a planar domain which can be decomposed into a bounded domain of arbitrary shape and elongated “branches ” of variable crosssectional profiles. When the eigenvalue is smaller than a prescribed threshold, the corresponding eige ..."
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Abstract. We consider the eigenvalue problem for the Laplace operator in a planar domain which can be decomposed into a bounded domain of arbitrary shape and elongated “branches ” of variable crosssectional profiles. When the eigenvalue is smaller than a prescribed threshold, the corresponding eigenfunction decays exponentially along each branch. We prove this behavior for Robin boundary condition and illustrate some related results by numerically computed eigenfunctions. 1.
ON THE INTERPLAY OF BASIS SMOOTHNESS AND SPECIFIC RANGE CONDITIONS OCCURRING IN SPARSITY REGULARIZATION
, 2013
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Efficient Monte Carlo Methods for Simulating DiffusionReaction Processes in Complex Systems
, 2014
"... We present the principles, mathematical bases, numerical shortcuts and applications of fast random walk (FRW) algorithms. This Monte Carlo technique allows one to simulate individual trajectories of diffusing particles in order to study various probabilistic characteristics (harmonic measure, first ..."
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We present the principles, mathematical bases, numerical shortcuts and applications of fast random walk (FRW) algorithms. This Monte Carlo technique allows one to simulate individual trajectories of diffusing particles in order to study various probabilistic characteristics (harmonic measure, first passage/exit time distribution, reaction rates, search times and strategies, etc.) and to solve the related partial differential equations. The adaptive character and flexibility of FRWs make them particularly efficient for simulating diffusive processes in porous, multiscale, heterogeneous, disordered or irregularlyshaped media. 1.
Spectral Kernels for Probabilistic Analysis and Clustering of Shapes
"... Abstract. We propose a framework for probabilistic shape clustering based on kernelspace embeddings derived from spectral signatures. Our root motivation is to investigate practical yet principled clustering schemes that rely on geometrical invariants of shapes rather than explicit registration. T ..."
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Abstract. We propose a framework for probabilistic shape clustering based on kernelspace embeddings derived from spectral signatures. Our root motivation is to investigate practical yet principled clustering schemes that rely on geometrical invariants of shapes rather than explicit registration. To that end we revisit the use of the Laplacian spectrum and introduce a parametric family of reproducing kernels for shapes, extending WESD
Scientific jury:
, 2012
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Operator Based MultiScale Analysis of Simulation Bundles
, 2015
"... We propose a new mathematical data analysis approach, which is based on the mathematical principle of symmetry, for the postprocessing of bundles of finite element data from computeraided engineering. Since all those numerical simulation data stem from the numerical solution of the same partial d ..."
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We propose a new mathematical data analysis approach, which is based on the mathematical principle of symmetry, for the postprocessing of bundles of finite element data from computeraided engineering. Since all those numerical simulation data stem from the numerical solution of the same partial differential equation, there exists a set of transformations, albeit unknown, which map simulation to simulation. The transformations can be obtained indirectly by constructing a transformation invariant positive definitive operator valid for all simulations. The eigenbasis of such an operator turns out to be a convenient basis for the handled simulation set due to two reasons. First, the spectral coefficients decay very fast, depending on the smoothness of the function being represented, and therefore a reduced multiscale representation of all simulations can be obtained, which depends on the employed operator. Second, at each level of the eigendecomposition the eigenvectors can be seen to recover different independent variation modes like rotation, translation or local deformation. Furthermore, this representation enables the definition of a new distance measure between
Dynamic isoperimetry and the geometry of Lagrangian coherent structures
, 2014
"... The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. Barriers to transport, which mitigate mixing, describe a skeleton about which possibly turbulent flow evolves. We propose a novel, direct geometric ..."
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The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. Barriers to transport, which mitigate mixing, describe a skeleton about which possibly turbulent flow evolves. We propose a novel, direct geometric method to identify subsets of phase space that remain strongly coherent over a finite time duration. This new method is based on a dynamic extension of classical (static) isoperimetric problems; the latter are concerned with identifying submanifolds with the smallest boundary size relative to their volume. The present work introduces dynamic isoperimetric problems; the study of sets with small boundary size relative to volume as they are evolved by a general dynamical system. We formulate and prove dynamic versions of the fundamental (static) isoperimetric (in)equalities; a dynamic FedererFleming theorem and a dynamic Cheeger inequality. We introduce a new dynamic Laplacian operator and describe a computational method to identify coherent sets based on eigenfunctions of the dynamic Laplacian. Our results include formal mathematical statements concerning geometric properties of finitetime coherent sets, whose boundaries can be regarded as Lagrangian coherent structures. The computational advantages of our new approach are a wellseparated spectrum for the dynamic Laplacian, and flexibility in appropriate numerical approximation methods. Finally, we demonstrate that the dynamic Laplacian operator can be realised as a zerodiffusion limit of a newly advanced probabilistic transfer operator method [11] for finding coherent sets, which is based on small diffusion. Thus, the present approach sits naturally alongside the probabilistic approach [11], and adds a formal geometric interpretation. 1
PowerLaw Noises over General Spatial Domains and on
, 2014
"... Powerlaw noises abound in nature and have been observed extensively in both time series and spatially varying environmental parameters. Although, recent years have seen the extension of traditional stochastic partial differential equations to include systems driven by fractional Brownian motion, s ..."
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Powerlaw noises abound in nature and have been observed extensively in both time series and spatially varying environmental parameters. Although, recent years have seen the extension of traditional stochastic partial differential equations to include systems driven by fractional Brownian motion, spatially distributed scaleinvariance has received comparatively little attention, especially for parameters defined over nonstandard spatial domains. This paper discusses the generalization of powerlaw noises to general spatial domains by outlining their theoretical underpinnings as well as addressing their numerical simulation on arbitrary meshes. Three computational algorithms are presented for efficiently generating their sample paths, accompanied by numerous numerical illustrations. 1