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Wavelet shrinkage: asymptopia
 Journal of the Royal Statistical Society, Ser. B
, 1995
"... Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators bein ..."
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Cited by 297 (36 self)
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Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators being obtained for a variety of interesting problems. Unfortunately, the results have often not been translated into practice, for a variety of reasons { sometimes, similarity to known methods, sometimes, computational intractability, and sometimes, lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coe cients towards the origin by an amount p p 2 log(n) = n. The method is di erent from methods in common use today, is computationally practical, and is spatially adaptive; thus it avoids a number of previous objections to minimax estimators. At the same time, the method is nearly minimax for a wide variety of loss functions { e.g. pointwise error, global error measured in L p norms, pointwise and global error in estimation of derivatives { and for a wide range of smoothness classes, including standard Holder classes, Sobolev classes, and Bounded Variation. This is amuch broader nearoptimality than anything previously proposed in the minimax literature. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and informationbased complexity.
Elliptic and Parabolic Problems in Unbounded Domains
"... We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semiexplicit representation formulas via operatorvalued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about t ..."
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Cited by 2 (1 self)
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We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semiexplicit representation formulas via operatorvalued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. 1.
MEAN VALUE THEOREMS FOR STOCHASTIC INTEGRALS
"... SUMMARY. The distributions of stochastic integrals are approximated by the distributions of stochastic integrals of piecewise constant processes. The rate of approximation in some negative Sobolev spaces is estimated. Generalizations are given for problems arising in control theory. ..."
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Cited by 1 (0 self)
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SUMMARY. The distributions of stochastic integrals are approximated by the distributions of stochastic integrals of piecewise constant processes. The rate of approximation in some negative Sobolev spaces is estimated. Generalizations are given for problems arising in control theory.
A Proof of the Consistency of the Finite Difference Technique on Sparse Grids
, 2000
"... In this paper, we give a proof of the consistency of the finite difference technique on regular sparse grids [7, 18]. We introduce an extrapolationtype discretization of differential operators on sparse grids based on the idea of the combination technique and we show the consistency of this discret ..."
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Cited by 1 (1 self)
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In this paper, we give a proof of the consistency of the finite difference technique on regular sparse grids [7, 18]. We introduce an extrapolationtype discretization of differential operators on sparse grids based on the idea of the combination technique and we show the consistency of this discretization. The equivalence of the new method with that of [7, 18] is established. Key words. finite differences, wavelets, interpolets, sparse grids, combination technique AMS subject classification. 33F05, 41A05, 41A25, 65B05, 65D05, 65M06, 65D25 1 Introduction Consider an univariate scaling function OE which induces a multiresolution analysis V l ae V l+1 ae ::: of L 2 (IR), see [4], such that [ l2IN V l = L 2 (IR) ; V l = spanfOE(2 l x \Gamma s) j s 2 ZZg : The wavelets / (l;t) are the basis functions of the complementary spaces W l : W l \Phi V l\Gamma1 = V l ; W l = spanf/ (l;t) j t 2 ZZg : In this paper, we consider scaling functions and wavelets of the interpolet ...
HÖRMANDER’S THEOREM FOR PARABOLIC EQUATIONS WITH COEFFICIENTS MEASURABLE IN THE TIME VARIABLE
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Preconditioners for Sparse Grid Discretizations
, 2001
"... In this paper we deal with preconditioners for sparse grid finitedifference and PetrovGalerkindiscretizations of the Poisson equation. We analyse the Jacobipreconditioner for the simple setting of nonadaptive grids and periodic boundary conditions. The analysis shows that the resulting conditi ..."
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In this paper we deal with preconditioners for sparse grid finitedifference and PetrovGalerkindiscretizations of the Poisson equation. We analyse the Jacobipreconditioner for the simple setting of nonadaptive grids and periodic boundary conditions. The analysis shows that the resulting condition numbers mainly depend on the underlying tensor product Wavelets. For example, high order Lifting Interpolets lead to l 2 condition numbers which are essentially independent of the finest mesh size. Based on this observation we introduce a socalled Liftingpreconditioner for discretizations which use Interpolets as trialfunctions. Numerical examples show the efficiency of the preconditioners for cases which are not covered by our analysis, e.g., adaptive grids.
BESOV SEMINORM PENALTY METHOD FOR SIGNAL RESTORATION
"... Abstract. In this paper, we propose an adaptive iteration algorithm for signal/image restoration. For digital signals, this algorithm minimizes a piecewise nonlinear lq, 0 < q ≤ 1, functional defined on wavelet domain with a single equality constraint. This minimization algorithm is based on the ..."
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Abstract. In this paper, we propose an adaptive iteration algorithm for signal/image restoration. For digital signals, this algorithm minimizes a piecewise nonlinear lq, 0 < q ≤ 1, functional defined on wavelet domain with a single equality constraint. This minimization algorithm is based on the Besov seminorm penalty method. Rudin et al’s BV seminorm based method is practically a piecewise linear l1 minimization algorithm on the spatial difference domain, so that it has a great success for reconstruction of piecewise constant signals. Our proposed method provides a nonoscillatory and edge preserving algorithm for reconstruction of piecewise smooth signals. 1.
Geophysical modelling with Colombeau functions: Microlocal properties and Zygmund regularity
, 2008
"... In global seismology Earth’s properties of fractal nature occur. Zygmund classes appear as the most appropriate and systematic way to measure this local fractality. For the purpose of seismic wave propagation, we model the Earth’s properties as Colombeau generalized functions. In one spatial dimensi ..."
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In global seismology Earth’s properties of fractal nature occur. Zygmund classes appear as the most appropriate and systematic way to measure this local fractality. For the purpose of seismic wave propagation, we model the Earth’s properties as Colombeau generalized functions. In one spatial dimension, we have a precise characterization of Zygmund regularity in Colombeau algebras. This is made possible via a relation between mollifiers and wavelets. 1