Father Wavelet Research Articles

Solving Differential Equations by New Optimized MRA & Invariant Solutions

The wavelets are important functions in the harmonic analysis‎. ‎Up to our knowledge‎, ‎apply wavelets to solve differential equations was limited to ODEs or PDEs with approximate and numerical solutions‎. ‎In this paper‎, ‎we design father wavelets with two independent variables according to differential invariants and propose the novel method based on the wavelets‎, ‎make new father wavelets‎, ‎apply multiresolution analysis (MRA) with these wavelets for solving DEs‎. ‎Our method can be used for ODEs and PDEs at every order‎. ‎This method will result in solution in the form of linear combination of father wavelets and corresponding mother wavelets.

Efficient extraction of equivalent circuit parameters for interconnect structures based on the Battle‐Lemarie scalet in the method of moments

AbstractBattle‐Lemarie father wavelet is used as a basis function in the method of moments (MoM) for solving interconnect and packaging problems and extracting circuit parameters needed in signal integrity. As the Battle‐Lemarie father wavelet or scalet is continuous, symmetric, and much more smooth compared with other wavelets like Franklin, Daubechies, and Coifman wavelets, it is more suitable as a basis function in the MoM for the electromagnetic modeling and simulation of interconnect structures because the unknown functions are involved in derivative operators in the integral equations. Also, the Battle‐Lemarie scalet is easier to construct and implement compared with multiwavelets, which are also continuous and smooth; hence, it requires less computational costs. The numerical examples for the circuit parameter extraction of single trapezoidal interconnect and asymmetric dual interconnects are presented to demonstrate the robustness of the scheme. Copyright © 2010 John Wiley & Sons, Ltd.

Efficient electromagnetic analysis for curved open thin-wire structures using intervallic wavelets in the method of moments

Efficient electromagnetic (EM) analysis for curved open thin-wire structures is developed by solving integral equations with intervallic wavelets in the method of moments (MoM). The electrical field integral equation for curved thin-wire structures is formulated based on the generalized Pocklington integral equation and the unknown current is expanded using the intervallic Coifman father wavelets as basis functions. In the current representation, an appropriate geometric map is established between thin-wire segments and bounded intervallic wavelets to match different domains. Since the Coifman wavelets (Coiflets) possess a unique feature, i.e. the vanishing moments which can yield a Dirac-δ-like sampling property, they are used to construct the intervallic wavelets. Also, the corresponding edge basis functions are built to represent the unknown current in the edge intervals for open structures. By using the intervallic wavelets in the MoM and performing a fast wavelet transform, sparse impedance matrices can be obtained and the drawbacks of dense matrices are overcome. Numerical examples for analyzing the scattering or radiation properties in different thin-wire structures are presented to demonstrate the robustness of the intervallic wavelets.

SUPPORT VECTOR MACHINES BASED ON THE ORTHOGONAL PROJECTION KERNEL OF FATHER WAVELET

Recently the study on the theory of wavelets shows that the wavelets have not only the multi-resolution property both in frequency and time domain, but also the good approximation ability. SVMs based on the statistical learning theory are a kind of general and effective learning machines, and have described for us the nice application blueprint in machine learning domain. There exists a bottleneck problem, or the pre-selection of kernel parameter for SVMs. In this paper, the orthogonal projection kernels of father wavelet (OPFW kernels) are introduced into SVMs. In doing so SVMs based on the OPFW kernels can have good performance in both approximation and generalisation. Simultaneously the parameter pre-selection of wavelet kernels can be implemented by discrete wavelet transform. Experiments on regression estimation illustrate the approximation and generalisation ability of our method.

Solitary Wave and Wavelet**The project supported by National Natural Science Foundation of China under Grant Nos. 40045016 and No. 40175016

In this paper, it is shown that the homoclinic orbits exist in iterated functional systems, so do the solitary wave structures. Moreover, Harr father wavelet, Mexican Cap wavelet, and other closed form wavelets have this solitary wave structure, too. So wavelet is a certain kind of solitary wave.

The shiftability of some wavelet bases

An explicit functional called the shiftability value of wavelet basis, which measures a deviation from translation-invariance, is studied. For functions φ with ‖φ‖ = 1, it lies in (0,1] with the value 1 being best possible. The relation between the shiftability value r V of the father wavelet and the corresponding value r W of the mother wavelet is given. The shiftability value of Meyer's wavelet and the B-spline wavelets is computed. For Meyer's wavelet, we prove that r W = 3 r V − 2 and demonstrate how to control the shiftability value by properly designing the wavelet. For the B-spline wavelet of order n, we give the asymptotical relation r ( n) W ∼ 3 r ( n) V − 2, and show r ( n) V → 1 as n → ∞. The family of B-spline wavelets contains several well-known wavelets such as Franklin's wavelet, Battle's wavelet, and Lemarié's wavelet, which makes our results potentially useful.

Wavelets and distributed approximating functionals

A general procedure is proposed for constructing father and mother wavelets that have excellent time-frequency localization and can be used to generate entire wavelet families for use as wavelet transforms. One interesting feature of our father wavelets (scaling functions) is that they belong to a class of generalized delta sequences, which we refer to as distributed approximating functionals (DAFs). We indicate this by the notation wavelet-DAFs. Correspondingly, the mother wavelets generated from these wavelet-DAFs are appropriately called DAF-wavelets. Wavelet-DAFs can be regarded as providing a pointwise (localized) spectral method, which furnishes a bridge between the traditional global methods and local methods for solving partial differential equations. They are shown to provide extremely accurate numerical solutions for a number of nonlinear partial differential equations, including the Korteweg-de Vries (KdV) equation, for which a previous method has encountered difficulties (J. Comput. Phys. 132 (1997) 233).

Die Hutlets — eine biorthogonale Wavelet-Familie: Effiziente Realisierung durch multipliziererfreie, perfekt rekonstruierende Quadratur Mirror Filter

Article Die Hutlets — eine biorthogonale Wavelet-Familie: Effiziente Realisierung durch multipliziererfreie, perfekt rekonstruierende Quadratur Mirror Filter was published on January 1, 1997 in the journal Frequenz (volume 51, issue 1-2).

Regularization of a non-characteristic Cauchy problem for a parabolic equation

In this paper the non-characteristic Cauchy problem ut- alpha (x)uxx-b(x)ux-c(x)u=0, x in (0,l), t in R; u(0,t)= phi (t), t in R; ux(0,t)=0, t in R; is considered. The problem is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. In this paper the following mollification method is suggested for this problem: if the Cauchy data are given inexactly then we mollify them by elements of well-posedness classes of the problem, namely by elements of an appropriate co-regular multiresolution approximation {Vj}j in Z of L2(R) which is generated by the father wavelet of Meyer (1992). Within VJ the problem is well posed, and we can find a mollification parameter J depending on the noise level epsilon in the Cauchy data such that the error estimation between the exact solution and the mollified solution is of Holder type. The method can be numerically implemented using fundamental results by Beylkin, Coifman and Rokhlin (1991) on representing (pseudo)differential operators in wavelet bases. A stable marching difference scheme based on this method is suggested. Several numerical examples are given.

A necessary and sufficient condition for the existence of a father wavelet

It is proved that if ${2^{-m/2} ψ(2^{-m} • - k)}_{m,k ∈ ℤ}$ is an orthonormal basis in $L^2(ℝ;ℂ)$, then the mother wavelet ψ is obtained from a multiresolution generated by a father wavelet if and only if $∑_{p=1}^{∞| ∑_{k ∈ ℤ} |ψ̂(2^{p}(• + k))|^2 > 0$ a

Unconditional Bases of Wavelets for Sobolev Spaces

It is shown that an orthonormal wavelet basis for $L^2 (\mathbb{R})$ associated with a multiresolution is an unconditional basis for the Sobolev space , $\mathcal{H}^t (\mathbb{R})$, $t \in [ - s,s]$, provided the father wavelet belongs to $\mathcal{H}^s (\mathbb{R})$ and the square of its absolute value has finite moments. A criterion for when the wavelet belongs to $\mathcal{H}^s (\mathbb{R})$ is given.