A wavelet collocation method for nonlinear Hammerstein equations is formulated. A sparsity in the Jacobian matrix is obtained which gives rise to a fast algorithm for nonlinear solvers such as the Newton’s method and the quasi-Newton method. A fast multilevel augmentation method is developed on a transformed nonlinear equation which gives an additional saving in computational time.

The theory of localized frames is refined to include quasi-Banach spaces and spaces with multiple generators. Applications are given to non-linear approximation with frames and to the convergence of the iterative frame algorithm in finer norms, and to the characterization of Besov spaces with wavelet frames.

In a celebrated construction, Chen and Skriganov gave explicit examples of point sets achieving the best possible L2-norm of the discrepancy function. We consider the discrepancy function of the ChenSkriganov point sets in Besov spaces with dominating mixed smoothness and show that they also achieve the best possible rate in this setting. The proof uses a b-adic generalization of the Haar syste...

If G is a locally compact groupoid with a Haar system λ, then a positive definite function p on G has a form p(x) = 〈L(x)ξ(d(x)), ξ(r(x))〉, where L is a representation of G on a Hilbert bundle H = (G, {Hu}, μ), μ is a quasi invariant measure on G 0 and ξ ∈ L(G,H). [10]. In this paper firt we prove that if μ is a quasi invariant ergodic measure on G, then two corresponding representations of G a...

Many multivariate Gaussian models can conveniently be split into independent, block-wise problems. Common settings where this situation arises are balanced ANOVA models, balanced longitudinal models, and certain block-wise shrinkage estimators in nonparametric regression estimation involving orthogonal bases such as Fourier or wavelet bases. It is well known that the standard, least squares est...

We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers (dilations) of Rn. The wavelets we construct are C∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance induced by the cover. The dual wavelets are also C∞, with the same number of vanishing moments,...

In this article we analyze some residual-based stabilization techniques for the transient Stokes problem when considering anisotropic time-space discretizations. We define an anisotropic time-space discretization as a family of time-space partitions that does not satisfy the condition h ≤ Cδt with C uniform with respect to h and δt. Standard residual-based stabilization techniques are motivated...

We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers (dilations) of R. The wavelets we construct are C∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance induced by the cover. The dual wavelets are also C∞, with the same number of vanishing moments, ...

This paper proposes a novel method to detect fire and/or flames in real-time by processing the video data generated by an ordinary camera monitoring a scene. In addition to ordinary motion and color clues, flame and fire flicker is detected by analyzing the video in the wavelet domain. Quasi-periodic behavior in flame boundaries is detected by performing temporal wavelet transform. Color variat...

In this paper we design a new family of biorthogonal wavelet transforms that are based on polynomial and discrete splines. The wavelet transforms are constructed via lifting steps, where the prediction and update filters are derived from various types of interpolatory and quasi-interpolatory splines. The transforms use finite and infinite impulse response (IIR) filters and are implemented in a ...