Terms Of Wavelet Bases Research Articles

Nuclear Embeddings in Weighted Function Spaces

We study nuclear embeddings for weighted spaces of Besov and Triebel–Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results concerning the compactness of corresponding embeddings. The concept of nuclearity was introduced by A. Grothendieck in 1955. Recently there is a refreshed interest to study such questions. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a discretisation in terms of wavelet bases, operator ideal techniques, as well as a very useful result of Tong about the nuclearity of diagonal operators acting in ell _p spaces. In that way we can further contribute to the characterisation of nuclear embeddings of function spaces on domains.

Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases

We study compact embeddings for weighted spaces of Besov and Triebel‐Lizorkin type where the weight belongs to some Muckenhoupt class. This extends our previous results [25] to more general weights of logarithmically disturbed polynomial growth, both near some singular point and at infinity. We obtain sharp asymptotic estimates for the entropy numbers of this embedding. Essential tools are a discretisation in terms of wavelet bases, as well as a refined study of associated embeddings in sequence spaces and interpolation arguments in endpoint situations.

Entropy and Approximation Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, I

We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt Ap class. For weights of purely polynomial growth, both near some singular point and at infinity, we obtain sharp asymptotic estimates for the entropy numbers and approximation numbers of this embedding. The main tool is a discretization in terms of wavelet bases.

Multi-resolution approximation to the stochastic inverse problem

The linear inverse problem of estimating the input random field in a first-kind stochastic integral equation relating two random fields is considered. For a wide class of integral operators, which includes the positive rational functions of a self-adjoint elliptic differential operator on L2(ℝd), the ill-posed nature of the problem disappears when such operators are defined between appropriate fractional Sobolev spaces. In this paper, we exploit this fact to reconstruct the input random field from the orthogonal expansion (i.e. with uncorrelated coefficients) derived for the output random field in terms of wavelet bases, transformed by a linear operator factorizing the output covariance operator. More specifically, conditions under which the direct orthogonal expansion of the output random field coincides with the integral transformation of the orthogonal expansion derived for the input random field, in terms of an orthonormal wavelet basis, are studied.

Multi-resolution approximation to the stochastic inverse problem

The linear inverse problem of estimating the input random field in a first-kind stochastic integral equation relating two random fields is considered. For a wide class of integral operators, which includes the positive rational functions of a self-adjoint elliptic differential operator on L 2(ℝ d ), the ill-posed nature of the problem disappears when such operators are defined between appropriate fractional Sobolev spaces. In this paper, we exploit this fact to reconstruct the input random field from the orthogonal expansion (i.e. with uncorrelated coefficients) derived for the output random field in terms of wavelet bases, transformed by a linear operator factorizing the output covariance operator. More specifically, conditions under which the direct orthogonal expansion of the output random field coincides with the integral transformation of the orthogonal expansion derived for the input random field, in terms of an orthonormal wavelet basis, are studied.

Backscattering Properties of Multi-Scale Rough Surfaces

In this paper a novel multi-scale roughness description is presented and its impact on surface backscattering is investigated. The analysed rough surfaces are synthesised by exploiting a Karhunen-Lo6ve decomposition of 1/ f random processes in terms of wavelet bases. For these surfaces, relevant roughness parameters are not traditional quantities like the height profile std (s) and the profile correlation length (l) but they are a new set of parameters related to the surface multi-scale properties. Besides, the paper investigates the possibility of matching the multi-scale roughness description with e.m. asymptotic models such as the Integral Equation Method (IEM). Under appropriate hypothesis, it is shown that the IEM model can be adapted to the multi-scale roughness description and a sensitive analysis of copolarised backscattering coefficients to relevant roughness parameters is carried out in the case of one dimensional surfaces.

Improving computational efficiency of model predictive control algorithm using wavelet transformation

We propose an MPC algorithm with desirable stability and constraint-handling properties, yet which retains computational simplicity. The closed-loop stability and constraint-handling are achieved by formulating an infinite-horizon objective and then solving an equivalent finite-horizon problem. On-line computational requirements for multi-time-scale systems, are reduced substantially by techniques called ‘blocking’ and ‘condensation’. We show that the blocking and condensation techniques are conveniently applied after expressing the MPC objective and constraint equations in terms of wavelet bases.