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76
Data compression and harmonic analysis
 IEEE Trans. Inform. Theory
, 1998
"... In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory... ..."
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Cited by 142 (24 self)
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In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory...
Tree Methods for Moving Interfaces
, 1999
"... Fast adaptive numerical methods for solving moving interface problems are presented. The methods combine a level set approach with frequent redistancing and semiLagrangian time stepping schemes which are explicit yet unconditionally stable. A quadtree mesh is used to concentrate computational effor ..."
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Cited by 56 (7 self)
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Fast adaptive numerical methods for solving moving interface problems are presented. The methods combine a level set approach with frequent redistancing and semiLagrangian time stepping schemes which are explicit yet unconditionally stable. A quadtree mesh is used to concentrate computational effort on the interface, so the methods moves an interface with N degrees of freedom in O(N log N) work per time step. Efficiency is increased by taking large time steps even for parabolic curvature flows. The methods compute accurate viscosity solutions to a wide variety of difficult moving interface problems involving merging, anisotropy, faceting and curvature.
Neural Networks for Optimal Approximation of Smooth and Analytic Functions
 Neural Computation
, 1996
"... . We prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives, if the activation function evaluated by each principal element satisfies certain technical conditions. Under these con ..."
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Cited by 43 (5 self)
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. We prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives, if the activation function evaluated by each principal element satisfies certain technical conditions. Under these conditions, it is also possible to construct networks that provide a geometric order of approximation for analytic target functions. The permissible activation functions include the squashing function (1 + e x ) 1 as well as a variety of radial basis functions. Our proofs are constructive. The weights and thresholds of our networks are chosen independently of the target function; we give explicit formulas for the coe#cients as simple, continuous, linear functionals of the target function. 1. Introduction. In recent years, there has been a great deal of research in the theory of approximation of real valued functions using artificial neural networks with one or more hidden layers, with each pr...
Kinetic Models for Chemotaxis and their DriftDiffusion Limits
, 2003
"... Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemoattractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a driftdiusion model is pro ..."
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Cited by 42 (11 self)
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Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemoattractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a driftdiusion model is proven.
Fast Treebased Redistancing for Level Set Computations
, 1999
"... Level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface, such as the signed distance to the interface. This paper presents ecient algorithms for this "redistancing" problem. The algorith ..."
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Cited by 37 (6 self)
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Level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface, such as the signed distance to the interface. This paper presents ecient algorithms for this "redistancing" problem. The algorithms use quadtrees and triangulation to compute global approximate signed distance functions. A quadtree mesh is built to resolve the interface and the vertex distances are evaluated exactly with a robust search strategy to provide both continuous and discontinuous interpolants. Given a polygonal interface with N elements, our algorithms run in O(N) space and O(N log N) time. Twodimensional numerical results show they are highly efficient in practice.
Curvelets and Curvilinear Integrals
, 1999
"... Let C(t):I↦ → R² be a simple closed unitspeed C² curve in R 2 with normal ⃗n(t). The curve C generates a distribution Γ which acts on vector fields ⃗v(x1,x2):R² ↦ → R² by line integration according to Γ(⃗v) = ⃗v(C(t)) · ⃗n(t)dt. We consider the problem of efficiently approximating such functional ..."
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Cited by 29 (2 self)
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Let C(t):I↦ → R² be a simple closed unitspeed C² curve in R 2 with normal ⃗n(t). The curve C generates a distribution Γ which acts on vector fields ⃗v(x1,x2):R² ↦ → R² by line integration according to Γ(⃗v) = ⃗v(C(t)) · ⃗n(t)dt. We consider the problem of efficiently approximating such functionals. Suppose we have a vector basis or frame Φ = ( ⃗ φµ); then an mterm approximation to Γ can be formed by selecting m terms (µi:1≤i≤m) and taking ˜Γm(⃗v) = m∑ i=1 Γ ( ⃗ φµ i)[⃗v, ⃗ φµ i Here the µi can be chosen adaptively based on the curve C. We are interested in finding a vector basis or frame for which the above scheme yields the highestquality mterm approximations. Here performance is measured by considering worstcase error on vector fields which are smooth in an L 2 Sobolev sense: Err(Γ, ˜ Γm) =sup{Γ(⃗v) − ˜ Γm(⃗v)  : ‖Div(⃗v)‖2 ≤ 1}. We establish an isometry between this problem and the problem of approximating objects with edges in L 2 norm. Starting from the recentlyintroduced tight frames of scalar curvelets, we construct a vector frame of curvelets for this problem. Invoking results on the nearoptimality of scalar curvelets in representing objects with edges, we argue that vector curvelets provide nearoptimal quality mterm approximations. We show that they dramatically outperform both wavelet and Fourierbased representations in terms of mterm approximation error. The mterm approximations to Γ are built from terms with support approaching more and more closely the curve C with increasing m; the terms have support obeying the scaling law width ≈ length 2. Comparable results can be developed, with additional work, for scalar curvelet approximation in the case of scalar integrands I(f) = f(C(t))dt.
TimeFrequency Analysis of Localization Operators
 J. FUNCT. ANAL
, 2002
"... We study a class of pseudodifferential operators known as timefrequency localization operators, AntiWick operators, GaborToeplitz operators or wave packets. Given a symbol a and two windows ' 1 ; ' 2 , we investigate the multilinear mapping from (a; ' 1 ; ' 2 ) 2 S ) to the localization op ..."
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Cited by 26 (16 self)
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We study a class of pseudodifferential operators known as timefrequency localization operators, AntiWick operators, GaborToeplitz operators or wave packets. Given a symbol a and two windows ' 1 ; ' 2 , we investigate the multilinear mapping from (a; ' 1 ; ' 2 ) 2 S ) to the localization operator A a and we give sufficient and necessary conditions for a to be bounded or to belong to a Schatten class. Our results are formulated in terms of timefrequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.
On Leray’s selfsimilar solutions of the NavierStokes equations
 Acta Math
, 1996
"... In the 1934 paper [L] Leray raised the question of the existence of selfsimilar solutions of the Navier{Stokes equations ut u +(u r)u+rp=0 div u =0 ..."
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Cited by 22 (0 self)
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In the 1934 paper [L] Leray raised the question of the existence of selfsimilar solutions of the Navier{Stokes equations ut u +(u r)u+rp=0 div u =0
A Fast Modular SemiLagrangian Method for Moving Interfaces
, 2000
"... A fast modular numerical method for solving general moving interface problems is presented. It simplifies code development by providing a blackbox solver which moves a given interface one step with given normal velocity. The method combines an efficiently redistanced level set approach, a problemi ..."
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Cited by 18 (4 self)
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A fast modular numerical method for solving general moving interface problems is presented. It simplifies code development by providing a blackbox solver which moves a given interface one step with given normal velocity. The method combines an efficiently redistanced level set approach, a problemindependent velocity extension, and a secondorder semiLagrangian time stepping scheme which reduces numerical error by exact evaluation of the signed distance function. Adaptive quadtree meshes are used to concentrate computational effort on the interface, so the method moves an Nelement interface in O(N log N) work per time step. Efficiency is increased by taking large time steps even for parabolic curvature flows. Numerical results show that the method computes accurate viscosity solutions to a wide variety of difficult geometric moving interface problems involving merging, anisotropy, faceting, nonlocality and curvature.
Regularity of Horizons and The Area Theorem
"... We prove that the area of sections of future event horizons in spacetimes satisfying the null energy condition is nondecreasing towards the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperboli ..."
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Cited by 16 (12 self)
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We prove that the area of sections of future event horizons in spacetimes satisfying the null energy condition is nondecreasing towards the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic spacetime and there exists a conformal completion with a "regular" I + ; 3) the horizon is a black hole event horizon in a spacetime which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends ...