Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements can be dramatically smaller than the size. Thus, certain natural classes of images with pixels need only = ( 1 4 log 5 2 ()) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual pixel samples. More specifically, suppose has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)—so the coefficients belong to an ball for 0 1. The most important coefficients in that expansion allow reconstruction with 2 error ( 1 2 1

Abstract not found

Abstract. This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of N wavelets (so called N-term approximation), which would be obtained by keeping the N largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N −s)in the energy norm, whenever such a rate is possible by N-term approximation. The range of s>0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to N. The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization. 1.

We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) n-term approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space Br q(Lp(Ω)) and Ω ⊂ Rd is a bounded Lipschitz domain; the error is always measured in the Hs-norm. The respective widths are the linear widths (or approximation numbers), the nonlinear widths, the Gelfand widths, and the manifold widths. As a technical tool, we also study the Bernstein numbers. Our main results are the following. If p ≥ 2 then the order of convergence is the same for all four classes of approximations. In particular, the best linear approximations are of the same order as the best nonlinear ones. The best linear approximation can be quite difficult to realize as a numerical algorithm since the optimal Galerkin space usually depends on the operator and of the shape of the domain Ω. For p < 2 there is a difference, nonlinear approximations are better than linear ones. However, in this case, it turns out that linear information about the right hand side f is again optimal. Our main theoretical tool is the best n-term approximation with respect to an optimal Riesz basis and related nonlinear widths. These general results are used to study the Poisson equation in a polygonal domain. It turns out that best n-term wavelet approximation is (almost) optimal. The main results of

Abstract not found

This paper is concerned with the Besov regularity of the solutions to interface problems in a segment S of the unit disk in R 2 : We investigate the smoothness of the solutions as measured in the specific scale B s ø (L ø (S)); 1=ø = s=2+1=p; of Besov spaces which determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. The proofs are based on representations of the solution spaces which were derived by Kellogg [15] and on characterizations of Besov spaces by wavelet expansions. Key Words: Interface problems, adaptive methods, nonlinear approximation, Besov spaces, wavelets. AMS Subject classification: Primary 35B65, secondary 41A46, 46E35, 65N30. 1 Introduction In recent years, the use of adaptive schemes has become a widespread strategy in numerical analysis. In particular, adaptive algorithms have been successfully implemented for the numerical treatment of boundary value problems of the form Au = f on\Omega ae R d ; (1.1) Bu...

We consider a Gaussian process X with smoothness comparable to the Brownian motion. We analyze reconstructions of X which are based on observations at finitely many points. For each realization of X the error is defined in a weighted supremum norm; the overall error of a reconstruction is defined as the pth moment of this norm. We determine the rate of the minimal errors and provide different reconstruction methods which perform asymptotically optimal. In particular, we show that linear interpolation at the quantiles of a certain density is asymptotically optimal.

Abstract not found

Abstract not found

It is well known, that pseudodifferential equations of negative order considered in the Sobolev space with a small smoothness index are ill--posed. On the other hand, it is known that efficient discretization schemes with properly chosen discretization parameter allow to obtain a regularization effect for such equations. The main accomplishment of the present paper is the principle for the adaptive choice of the discretization parameter directly from noisy discrete data. We argue that the combination of this principle with wavelet--based matrix compression techniques leads to algorithms which are order--optimal in the sense of complexity. 1.