arXiv subjects

Albert Cohen

Publications and source records attributed to Albert Cohen.

59 records · Page 4Linked to original sources

arXiv2011

Greedy bisection generates optimally adapted triangulations

We study the properties of a simple greedy algorithm for the generation of data-adapted anisotropic triangulations. Given a function f, the algorithm produces nested triangulations and corresponding piecewise polynomial approximations of f. The refinement procedure picks the triangle which maximizes the local Lp approximation error, and bisect it in a direction which is chosen so to minimize this error at the next step. We study the approximation error in the Lp norm when the algorithm is applied to C2 functions with piecewise linear approximations. We prove that as the algorithm progresses, the triangles tend to adopt an optimal aspect ratio which is dictated by the local hessian of f. For convex functions, we also prove that the adaptive triangulations satisfy a convergence bound which is known to be asymptotically optimal among all possible triangulations.

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arXiv2011

Adaptive multiresolution analysis based on anisotropic triangulations

A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions).

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arXiv2011

Anisotropic smoothness classes : from finite element approximation to image models

We propose and study quantitative measures of smoothness which are adapted to anisotropic features such as edges in images or shocks in PDE's. These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest examples of such quantities are based on the determinant of the hessian of the function to be approximated. Since they are not semi-norms, these quantities cannot be used to define linear function spaces. We show that they can be well defined by mollification when the function to be approximated has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used record variation semi-norm which does not account for the geometric smoothness of the edges.

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arXiv2008

Approximation and learning by greedy algorithms

We consider the problem of approximating a given element $f$ from a Hilbert space $\mathcal{H}$ by means of greedy algorithms and the application of such procedures to the regression problem in statistical learning theory. We improve on the existing theory of convergence rates for both the orthogonal greedy algorithm and the relaxed greedy algorithm, as well as for the forward stepwise projection algorithm. For all these algorithms, we prove convergence results for a variety of function classes and not simply those that are related to the convex hull of the dictionary. We then show how these bounds for convergence rates lead to a new theory for the performance of greedy algorithms in learning. In particular, we build upon the results in [IEEE Trans. Inform. Theory 42 (1996) 2118--2132] to construct learning algorithms based on greedy approximations which are universally consistent and provide provable convergence rates for large classes of functions. The use of greedy algorithms in the context of learning is very appealing since it greatly reduces the computational burden when compared with standard model selection using general dictionaries.

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arXiv2002

Adaptive methods for PDE's: wavelets or mesh refinement?

Adaptive mesh refinement techniques are nowadays an established and powerful tool for the numerical discretization of PDE's. In recent years, wavelet bases have been proposed as an alternative to these techniques. The main motivation for the use of such bases in this context is their good performances in data compression and the approximation theoretic foundations which allow to analyze and optimize these performances. We shall discuss these theoretical foundations, as well as one of the approaches which has been followed in developing efficient adaptive wavelet solvers. We shall also discuss the similarities and differences between wavelet methods and adaptive mesh refinement.

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