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Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Version 1.0, Copyright MIT
, 2006
"... reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primaldual) Galerkin projection onto a lowdimensional space associated with a smooth “parametric ..."
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Cited by 208 (38 self)
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reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primaldual) Galerkin projection onto a lowdimensional space associated with a smooth “parametric manifold”—dimension reduction; efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence; a posteriori error estimation procedures—rigorous and sharp bounds for the linearfunctional outputs of interest; and OfflineOnline computational decomposition strategies—minimum marginal cost for high performance in the realtime/embedded (e.g., parameterestimation, con
Reduced Basis Approximation and A Posteriori Error Estimation for the TimeDependent Viscous Burgers Equation
 CALCOLO
, 2008
"... In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers equation in one space dimension. The key new ingredient is accurate solution–dependent (Online) calculation of the exponential–in–time stability factor by the Successive C ..."
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Cited by 54 (9 self)
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In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers equation in one space dimension. The key new ingredient is accurate solution–dependent (Online) calculation of the exponential–in–time stability factor by the Successive Constraint Method. Numerical results indicate that the a posteriori error bounds are practicable for reasonably large times — many convective scales — and reasonably large Reynolds numbers — O(100) or larger.
Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as ..."
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Cited by 40 (15 self)
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We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinitedimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the twodimensional stochastic NavierStokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic NavierStokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1
Markov solutions for the 3D stochastic NavierStokes equations with state dependent noise, available on the arXiv preprint archive at the web address http://www.arxiv.org/abs/math.AP/0512361
"... Abstract: We construct a Markov family of solutions for the 3D NavierStokes equation perturbed by a non degenerate noise. We improve the result of [3] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a ..."
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Cited by 32 (11 self)
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Abstract: We construct a Markov family of solutions for the 3D NavierStokes equation perturbed by a non degenerate noise. We improve the result of [3] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a state dependant noise. Another feature of this work is that we greatly simplify the proofs of [3].
A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs
"... We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this ..."
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Cited by 32 (14 self)
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We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator Mt can be obtained. Informally, this bound can be read as “Fix any finitedimensional projection Π on a subspace of sufficiently regular functions. Then the eigenfunctions of Mt with small eigenvalues have only a very small component in the image of Π.” We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of “Wiener polynomials, ” where the coefficients are possibly nonadapted stochastic processes satisfying a Lipschitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris ’ lemma, which is unavailable in the present context. We conclude by showing that the twodimensional stochastic NavierStokes equations and a large class of reactiondiffusion equations fit the framework of our theory. Contents 1
Some bound state problems in quantum mechanics, in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday
 Proc. Sympos. Pure Math., 76.1, American Mathematical Society
, 2007
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Global attractors and determining modes for the 3D NavierStokesVoight equations
, 2007
"... We investigate the longterm dynamics of the threedimensional NavierStokesVoight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D NavierStokesVoight equations and for the dimension of a global attractor of a semigro ..."
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Cited by 14 (4 self)
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We investigate the longterm dynamics of the threedimensional NavierStokesVoight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D NavierStokesVoight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view we consider the NavierStokesVoight model as a nonviscous (inviscid) regularization of the threedimensional NavierStokes equations. Furthermore, we also show that the weak solutions of the Navier StokesVoight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient ν → 0.
Stationary statistical properties of RayleighBénard convection at large Prandtl
"... This is the third in a series of our study of RayleighBénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for RayleighBénard convection at large Prandtl number are related to those of the infinite Prandtl number model fo ..."
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Cited by 14 (11 self)
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This is the third in a series of our study of RayleighBénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for RayleighBénard convection at large Prandtl number are related to those of the infinite Prandtl number model for convection that is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for RayleighBénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for RayleighBénard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form Ra1/3(ln Ra)1/3 + c Ra7/2 ln Ra
Global wellposedness for the 2D Boussinesq system without heat diffusion and with either anisotropic viscosity or inviscid voigtα regularization. Preprint arXiv:1010.5024v1 [math.AP
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Energy cascades and flux locality in physical scales of the 3D
 DEPARTMENT OF MATHEMATICS
, 2011
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