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Register a new userSignal Analysis based on Complex Wavelet Signs
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Added by
Martin Storath
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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We propose a signal analysis tool based on the sign (or the phase) of complex wavelet coefficients, which we call a signature. The signature is defined as the fine-scale limit of the signs of a signals complex wavelet coefficients. We show that the signature equals zero at sufficiently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. At such feature points, the orientation of the signature in the complex plane can be interpreted as an indicator of local symmetry and antisymmetry. We establish that the signature rotates in the complex plane under fractional Hilbert transforms. We show that certain random signals, such as white Gaussian noise and Brownian motions, have a vanishing signature. We derive an appropriate discretization and show the applicability to signal analysis.
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In current work, non-familiar shifted Lucas polynomials are introduced. We have constructed a computational wavelet technique for solution of initial/boundary value second order differential equations. For this numerical scheme, we have developed weight function and Rodrigues formula for Lucas polynomials. Further, Lucas polynomials and their properties are used to propose shifted Lucas polynomials and then utilization of shifted Lucas polynomials provides us shifted Lucas wavelet. We furnished the operational matrix of differentiation and the product operational matrix of the shifted Lucas wavelets. Moreover, convergence and error analysis ensure accuracy of the proposed method. Illustrative examples show that the present method is numerically fruitful, effective and convenient for solving differential equations
We introduce a new efficient algorithm for Helmholtz problems in perforated domains with the design of the scheme allowing for possibly large wavenumbers. Our method is based upon the Wavelet-based Edge Multiscale Finite Element Method (WEMsFEM) as proposed recently in [14]. For a regular coarse mesh with mesh size H, we establish O(H) convergence of this algorithm under the resolution assumption, and with the level parameter being sufficiently large. The performance of the algorithm is demonstrated by extensive 2-dimensional numerical tests including those motivated by photonic crystals.
We present a general -- i.e., independent of the underlying equation -- registration procedure for parameterized model order reduction. Given the spatial domain $Omega subset mathbb{R}^2$ and the manifold $mathcal{M}= { u_{mu} : mu in mathcal{P} }$ associated with the parameter domain $mathcal{P} subset mathbb{R}^P$ and the parametric field $mu mapsto u_{mu} in L^2(Omega)$, our approach takes as input a set of snapshots ${ u^k }_{k=1}^{n_{rm train}} subset mathcal{M}$ and returns a parameter-dependent bijective mapping ${Phi}: Omega times mathcal{P} to mathbb{R}^2$: the mapping is designed to make the mapped manifold ${ u_{mu} circ {Phi}_{mu}: , mu in mathcal{P} }$ more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition ${ Omega_{q} }_{q=1} ^{N_{rm dd}}$ of the domain $Omega$ such that $Omega_1,ldots,Omega_{N_{rm dd}}$ are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain, a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.
We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis. In a first step we give a self-contained characterization of tensor product Sobolev-Besov spaces on the $d$-torus with arbitrary smoothness in terms of the decay of such wavelet coefficients. In the second part we perform and analyze scattered-data approximation using a hyperbolic cross type truncation of the basis expansion for the associated least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. In case of i.i.d. samples we can even bound the approximation error with high probability by loosing only $log$-terms that do not depend on $d$ compared to the best approximation. In addition, if the function has low effective dimension (i.e. only interactions of few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to increase the accuracy. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.
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