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Optimal observability of the multidimensional wave and Schrödinger equations in quantum ergodic domains
, 2013
"... We consider the wave and Schrödinger equations on a bounded open connected subset Ω of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset ω of Ω during a time interval [0, T ..."
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Cited by 6 (4 self)
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We consider the wave and Schrödinger equations on a bounded open connected subset Ω of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset ω of Ω during a time interval [0, T] with T> 0. It is well known that, if the pair (ω, T) satisfies the Geometric Control Condition (ω being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be estimated in terms of the energy localized in ω × (0, T). We address the problem of the optimal location of the observation subset ω among all possible subsets of a given measure or volume fraction. A priori this problem can be modeled in terms of maximizing the observability constant, but from the practical point of view it appears more relevant to model it in terms of maximizing an average either over random initial data or over large time. This leads us to define a new notion of observability constant, either randomized, or asymptotic in time. In both cases we come up with a spectral functional that can be viewed as a measure of eigenfunction concentration. Roughly speaking, the subset
A DOUBLE SMOOTHING TECHNIQUE FOR CONSTRAINED CONVEX OPTIMIZATION PROBLEMS AND APPLICATIONS TO OPTIMAL CONTROL
, 2011
"... In this paper, we propose an efficient approach for solving a class of convex optimization problems in Hilbert spaces. Our feasible region is a (possibly infinitedimensional) simple convex set, i.e. we assume that projections on this set are computationally easy to compute. The problem we conside ..."
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Cited by 4 (0 self)
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In this paper, we propose an efficient approach for solving a class of convex optimization problems in Hilbert spaces. Our feasible region is a (possibly infinitedimensional) simple convex set, i.e. we assume that projections on this set are computationally easy to compute. The problem we consider is the minimization of a convex function over this region under the additional constraint Au ∈ T, where A is a linear operator and T is a (finitedimensional) convex set whose dimension is small as compared to the dimension of the feasible region. In our approach, we dualize the linear constraints, solve the resulting dual problem with a purely dual gradient method and show how to reconstruct an approximate primal solution. In order to accelerate our scheme, we introduce a novel double smoothing technique that involves regularization of the dual problem to allow the use of a fast gradient method. As a result, we obtain a method with complexity O ( 1 1 ln) gradient iterations, where ɛ is the desired accuracy for the primaldual ɛ ɛ solution. Our approach covers, in particular, optimal control problems with trajectory governed by a system of linear differential equations, where the additional constraints can for example force the trajectory to visit some convex sets at certain moments of time.
Double smoothing technique for infinitedimensional optimization problems with applications to optimal control
, 2010
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FINITE ELEMENT EIGENVALUE ENCLOSURES FOR THE MAXWELL OPERATOR
"... Abstract. We propose employing the extension of the LehmannMaehlyGoerisch method developed by Zimmermann and Mertins, as a highly effective tool for the pollutionfree finite element computation of the eigenfrequencies of the resonant cavity problem on a bounded region. This method gives compleme ..."
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Cited by 1 (1 self)
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Abstract. We propose employing the extension of the LehmannMaehlyGoerisch method developed by Zimmermann and Mertins, as a highly effective tool for the pollutionfree finite element computation of the eigenfrequencies of the resonant cavity problem on a bounded region. This method gives complementary bounds for the eigenfrequencies which are adjacent to a given parameter t ∈ R. We present a concrete numerical scheme which provides certified enclosures in a suitable asymptotic regime. We illustrate the applicability of this scheme by means of some numerical experiments on benchmark data using Lagrange elements and unstructured meshes. Contents
EIGENVALUE ENCLOSURES AND APPLICATIONS TO THE MAXWELL OPERATOR
, 2013
"... Abstract. This paper is concerned with methods for computing certified bounds for the isolated eigenvalues of selfadjoint operators. We examine in close detail the connections between an extension of the TempleLehmannGoerisch method developed a few years ago by Zimmermann and Mertins, and a gener ..."
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Abstract. This paper is concerned with methods for computing certified bounds for the isolated eigenvalues of selfadjoint operators. We examine in close detail the connections between an extension of the TempleLehmannGoerisch method developed a few years ago by Zimmermann and Mertins, and a general framework considered by Davies and Plum. We propose employing the former as a highly effective tool for the pollutionfree numerical estimation of the eigenfrequencies and field phasors of the resonant cavity problem on a bounded region filled with a generally anisotropic medium, by means of finite elements. Contents
EIGENVALUE ENCLOSURES
"... Abstract. This paper is concerned with methods for numerical computation of eigenvalue enclosures. We examine in close detail the equivalence between an extension of the LehmannMaehlyGoerisch method developed a few years ago by Zimmermann and Mertins, and a geometrically motivated method develope ..."
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Abstract. This paper is concerned with methods for numerical computation of eigenvalue enclosures. We examine in close detail the equivalence between an extension of the LehmannMaehlyGoerisch method developed a few years ago by Zimmermann and Mertins, and a geometrically motivated method developed more recently by Davies and Plum. We extend various previously known results in the theory and establish explicit convergence estimates in both settings. The theoretical results are supported by two benchmark numerical experiments on the isotropic Maxwell eigenvalue problem. Contents
The Differentiability of the Upper Envelop
, 2012
"... We present the proof of the DanskinValadier theorem, i.e. when the directional derivative of the supremum of a collection of functions admits a natural representation. 1 Preliminary Consider a collection of extended realvalued functions fi: X 7 → R̄, where i ∈ I is some index set, X is some real v ..."
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We present the proof of the DanskinValadier theorem, i.e. when the directional derivative of the supremum of a collection of functions admits a natural representation. 1 Preliminary Consider a collection of extended realvalued functions fi: X 7 → R̄, where i ∈ I is some index set, X is some real vector space, and R ̄: = R ∪ {±∞}. Define the supremum (i.e. upper envelop) of the collection as f(x): = sup i∈I fi(x). (1) We are interested in studying the directional derivative of f, hopefully relating it to the directional derivatives of fi. Recall that the directional derivative of g, along direction d, is defined as g′(x; d): = lim t↓0 g(x+ td) − g(x)