Topic
Biorthogonal system
About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.
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TL;DR: In this paper, a numerical solution of weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method is presented.
Abstract: Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B-spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples.
22 citations
TL;DR: In this article, the effects of dissipation on periodic lattice systems are effectively described by the potentials of the Berry phases of the system's non-Hermitian Hamiltonian, and a general numerical gauge smoothing procedure is developed to calculate complex Berry phases from the biorthogonal basis of the Hamiltonian.
Abstract: We numerically investigate topological phases of periodic lattice systems in tight-binding description under the influence of dissipation. The effects of dissipation are effectively described by $\mathcal{PT}$-symmetric potentials. In this framework we develop a general numerical gauge smoothing procedure to calculate complex Berry phases from the biorthogonal basis of the system's non-Hermitian Hamiltonian. Further, we apply this method to a one-dimensional $\mathcal{PT}$-symmetric lattice system and verify our numerical results by an analytical calculation.
22 citations
TL;DR: In this paper, the concept of quasi-basis was introduced for non-self-adjoint Hamiltonians with purely point real spectra, and a series of conditions under which such a definition is still possible.
Abstract: In some recent papers, studies on biorthogonal Riesz bases have found renewed motivation because of their connection with pseudo-Hermitian quantum mechanics, which deals with physical systems described by Hamiltonians that are not self-adjoint but may still have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed in some previous papers. However, in many physical models, one has to deal not with orthonormal bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of -quasi basis, and we show a series of conditions under which a definition of non-self-adjoint Hamiltonian with purely point real spectra is still possible.
22 citations
17 May 2004
TL;DR: This work investigates the implementation of two lifting coefficient sets, rational and irrational, for the biorthogonal 9/7 wavelet and finds that the best hardware and PSNR performance is obtained using the rational coefficient set quantized with gain compensation and lumped scaling.
Abstract: The lifting structure has been shown to be computationally efficient for implementing filter banks. The hardware implementation of a filter bank requires that the lifting coefficients be quantized. The quantization method determines compression performance, hardware size, hardware speed and energy. We investigate the implementation of two lifting coefficient sets, rational and irrational, for the biorthogonal 9/7 wavelet. Six different approaches are used to find optimal quantized lifting coefficients from these sets. We find that the best hardware and PSNR performance is obtained using the rational coefficient set quantized with gain compensation and lumped scaling.
22 citations
TL;DR: In this paper, a new family of global A -biorthogonal methods by using short two-term recurrences and formal orthogonal polynomials, which contain the global bi-conjugate residual (Gl-BCR) algorithm and its improved version, was presented.
22 citations