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: This paper extends the simplified sampling biorthogonal direction implicit method for 2-D Maxwell's equations to a 3-D SSB-ADI method, which partially uses the sampling property and compact support of the SBTD algorithm to display spatial discretization.
Abstract: We previously introduced the simplified sampling biorthogonal direction implicit (SSB-ADI) method for 2-D Maxwell's equations. This paper extends it to a 3-D SSB-ADI method. Similar to the original method, the 3-D SSB-ADI method also partially uses the sampling property and compact support of the SBTD algorithm to display spatial discretization. Dispersion and stability analysis are presented. A numerical example shows that the SSB-ADI method is superior to the SB-ADI method in terms of reducing the CPU time and memory requirement of the numerical simulation.
6 citations
TL;DR: The analysis and comparison of computational complexity of related algorithms have indicated that the proposed algorithm provided a faster computational approach for computing analysis/dual windows as compared with that of the existing algorithms, which can save amount of computation and memory.
Abstract: By using the biorthogonal analysis approach, an effective algorithm based on factorization for solving the analysis/dual windows in multiwindow discrete Gabor transform (M-DGT) is presented for arbitrary given synthesis windows. The constraint condition matrix of the M-DGT between analysis/dual windows and synthesis windows is proved to be equivalent to a fixed number of independent orthogonal relationship matrixes of discrete Gabor transform (DGT), which can be quickly and efficiently solved by using sub-equation sets. The analysis and comparison of computational complexity of related algorithms have indicated that the proposed algorithm provided a faster computational approach for computing analysis/dual windows as compared with that of the existing algorithms, which can save amount of computation and memory.
6 citations
30 Nov 2020
TL;DR: The biorthogonal rational functions of the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials.
Abstract: The biorthogonal rational functions of the ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials These properties are described by three difference operators $X,Y,Z$ which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space The pairwise commutators of the operators $X,Y,Z$ generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials
6 citations
Journal Article•
TL;DR: In this article, a new numerical method for solving fractional optimal control problems (FOCPs) is presented based upon biorthogonal cubic Hermite spline multiwavelets approximations.
Abstract: In this paper, a new numerical method for solving fractional optimal control problems (FOCPs) is presented. The fractional derivative in the dynamic system is described in the Caputo sense. The method is based upon biorthogonal cubic Hermite spline multiwavelets approximations. The properties of biorthogonal multiwavelets are first given. The operational matrix of fractional Riemann-Lioville integration and multiplication are then utilized to reduce the given optimization problem to the system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain algebraic equations. Illustrative examples are provided to confirm the applicability of the new method.
6 citations
6 citations