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ANALYSIS OF MULTISCALE METHODS
, 2004
"... The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasi-continuum method. ..."
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Cited by 277 (32 self)
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The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasi-continuum method.
Analysis of the heterogeneous multiscale method for ordinary differential equations
- Commun. Math. Sci
"... Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution. ..."
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Cited by 103 (11 self)
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Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.
Two-scale convergence
- Int. J. Pure Appl. Math
"... (Dedicated to the memory of Jacques-Louis Lions 1928-2001) This paper is devoted to the properties and applications of two-scale convergence introduced by Nguetseng in 1989. In a self-contained way, we present the details of the basic ideas in this theory. Moreover, we give an overview of the main h ..."
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Cited by 61 (3 self)
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(Dedicated to the memory of Jacques-Louis Lions 1928-2001) This paper is devoted to the properties and applications of two-scale convergence introduced by Nguetseng in 1989. In a self-contained way, we present the details of the basic ideas in this theory. Moreover, we give an overview of the main homogenization problems which have been studied by this technique. We also bridge gaps in previous presentations, make generalizations and give alternative proofs.
Solving the KPZ equation
- Ann. of Math
, 2013
"... We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a “universal ” measurable map from the probability space into an explicitly described auxiliary metric space, ..."
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Cited by 57 (9 self)
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We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a “universal ” measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction com-pletely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class. As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the Fokker-
Sparse grids and related approximation schemes for higher dimensional problems
"... The efficient numerical treatment of high-dimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach ..."
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Cited by 46 (12 self)
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The efficient numerical treatment of high-dimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energy-norm based sparse grids and demonstrate that, for functions with bounded mixed derivatives on the unit hypercube, the associated approximation rate in terms of the involved degrees of freedom shows no dependence on the dimension at all, neither in the approximation order nor in the order constant.
Bloch wave homogenization and spectral asymptotic analysis
- Journal de Mathematiques Pures et Appliquees
, 1998
"... ABSTRACT.- We consider a second-order elliptic equation in a bounded periodic heterogeneous medium and study the asymptotic behavior of its spectrum, as the structure period goes to zero. We use a new method of Eloch wave homogenization which, unlike the classical homogenization method, characterize ..."
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Cited by 40 (7 self)
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ABSTRACT.- We consider a second-order elliptic equation in a bounded periodic heterogeneous medium and study the asymptotic behavior of its spectrum, as the structure period goes to zero. We use a new method of Eloch wave homogenization which, unlike the classical homogenization method, characterizes a renormalized limit of the spectrum, namely sequences of eigenvalues of the order of the square of. the medium period. We prove that such a renormalized limit spectrum is made of two parts: the so-called Bloch spectrum, which is explicitly defined as the spectrum of a family of limit problems, and the so-called boundary layer spectrum, which is made of limit eigenvalues corresponding to sequences of eigenvectors concentrating on the boundary of the domain. This analysis relies also on a notion of Bloch measures which can be seen as ad hoc Wigner measures in the context of semi-classical analysis. Finally, for rectangular domains made of entire periodicity cells, a variant of the Bloch wave homogenization method gives an explicit characterization of the boundary layer spectrum too. 0 Elsevier, Paris Key words: Homogenization, Bloch waves, spectral analysis, boundary layers. RBsuMB.- On considere une equation elliptique du deuxitme ordre dans un milieu ptriodique heterogbne borne, et on dtudie le comportement asymptotique de son spectre lorsque la p&ode tend vers zero. On utilise une nouvelle methode d’homog&&ation par ondes de Bloch qui, contrairement aux mtthodes classiques d’homogenbisation, caracterise la limite renormalisee du spectre, et plus precisement les suites de valeurs propres de I’ordre du cam? de
Homogenization of the Vlasov equation and of the Vlasov- Poisson system with a strong external magnetic field
, 1998
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Invisibility and Inverse Problems
, 2008
"... We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that govern a var ..."
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Cited by 32 (18 self)
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We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that govern a variety of wave phenomena, including electrostatics, electromagnetism, acoustics and quantum mechanics, have transformation laws under changes of variables which allow one to design material parameters that steer waves around a hidden region, returning them to their original path on the far side. Not only are observers unaware of the contents of the hidden region, they are not even aware that something is being hidden; the object, which casts no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other devices having striking effects on wave propagation, unseen in nature, are also possible. These designs are initially based on the transformation laws of the relevant PDEs, but due to the singular transformations needed for the desired effects, care needs to be taken in formulating and analyzing physically meaningful solutions. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved.
Two-scale convergence on periodic surfaces and applications
- In Mathematical Modelling of Flow through Porous Media, Bourgeat AP, Carasso C, Luckhaus S, Mikelić A (eds). World Scientific
, 1995
"... This paper is concerned with the homogenization of model problems in periodic porous media when important phenomena occur on the boundaries of the pores. To this end, we generalize the notion of two-scale convergence for sequences of functions which are defined on periodic surfaces. We apply our res ..."
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Cited by 31 (2 self)
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This paper is concerned with the homogenization of model problems in periodic porous media when important phenomena occur on the boundaries of the pores. To this end, we generalize the notion of two-scale convergence for sequences of functions which are defined on periodic surfaces. We apply our results to two model problems: the first one is a diffusion equation in a porous medium with a Fourier boundary condition, the second one is a coupled system of diffusion equations inside and on the boundaries of the pores of a porous medium.
Homogenization of a Darcy-Stokes system modeling vuggy porous media
- Comput. Geosci
"... Abstract. We derive a macroscopic model for single phase, incompressible, viscous fluid flow in a porous medium with small cavities called vugs. We model the vuggy medium on the microscopic scale using Stokes equations within the vugular inclusions, Darcy’s law within the porous rock, and a Beavers- ..."
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Cited by 30 (2 self)
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Abstract. We derive a macroscopic model for single phase, incompressible, viscous fluid flow in a porous medium with small cavities called vugs. We model the vuggy medium on the microscopic scale using Stokes equations within the vugular inclusions, Darcy’s law within the porous rock, and a Beavers-Joseph-Saffman boundary condition on the interface between the two regions. We assume periodicity of the medium, and obtain uniform energy estimates independent of the period. Through a two-scale homogenization limit as the period tends to zero, we obtain a macroscopic Darcy’s law governing the medium on larger scales. We also develop some needed generalizations of the two-scale convergence theory needed for our bi-modal medium, including a two-scale convergence result on the Darcy-Stokes interface. The macroscopic Darcy permeability is computable from the solution of a cell problem. An analytic solution to this problem in a simple geometry suggests that: (1) flow along vug channels is primarily Poiseuille with a small perturbation related to the Beavers-Joseph slip, and (2) flow that alternates from vug to matrix behaves as if the vugs have infinite permeability. Key words. Homogenization, two-scale convergence, Darcy-Stokes system, vuggy porous media, Beavers-Joseph boundary condition 1. Introduction. A
