Biorthogonal system

About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.

Biorthogonal rational Krylov subspace methods

Abstract: A general framework for oblique projections of non-Hermitian matrices onto rational Krylov subspaces is developed. To obtain this framework we revisit the classical rational Krylov subspace algorithm and prove that the projected matrix can be written efficiently as a structured pencil, where the structure can take several forms such as Hessenberg or inverse Hessenberg. One specific instance of the structures appearing in this framework for oblique projections is a tridiagonal pencil. This is a direct generalization of the classical biorthogonal Krylov subspace method, where the projection becomes a single non-Hermitian tridiagonal matrix and of the Hessenberg pencil representation for rational Krylov subspaces. Based on the compact storage of this tridiagonal pencil in the biorthogonal setting, we can develop short recurrences. Numerical experiments confirm the validity of the approach.

Stabilization of a solution for two-dimensional loaded parabolic equation

Abstract: In this paper we consider the stabilization problem of the solution of a boundary value problem for the heat equation with a loaded two-dimensional Laplace operator. The loaded terms represent the values of the required function and traces of the first-order partial derivatives of the required function at fixed points. An algorithm for constructing boundary control functions is proposed.

Extracting long basic sequences from systems of dispersed vectors

Abstract: We study Banach spaces satisfying some geometric or structural properties involving tightness of transfinite sequences of nested linear subspaces. These properties are much weaker than WCG and closely related to Corson's property (C). Given a transfinite sequence of normalized vectors, which is dispersed or null in some sense, we extract a subsequence which is a biorthogonal sequence, or even a weakly null monotone basic sequence, depending on the setting. The Separable Complementation Property is established for spaces with an M-basis under rather weak geometric properties. We also consider an analogy of the Baire Category Theorem for the lattice of closed linear subspaces.

Investigation of boundary algorithms for multiresolution analysis

Abstract: An investigation on the multiresolution time-domain (MRTD) method utilizing different wavelet levels in one mesh is presented. Contrary to adaptive thresholding techniques, only a rigid addition of higher order wavelets in certain critical cells is considered. Their effect is discussed analytically and verified by simulations of plain and dielectrically filled cavities with Daubechies' and Battle-Lemarie orthogonal, as well as Cohen-Daubechies-Feauveau (CDF) biorthogonal wavelets, showing their insufficiency unless used as a full set of expansion. It is pointed out that improvements cannot be expected from these fixed mesh refinements. Furthermore, an advanced treatment concerning thin metallization layers in CDF algorithms is presented, leading to a reduction in cell number by a factor of three per space dimension compared to conventional finite difference time domain (FDTD), but limited to very special structures with infinitely thin irises. All MRTD results are compared to those of conventional FDTD approaches.

Biorthogonal expansion of non-symmetric Jack functions

Abstract: We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner-Pollaczek type polynomials. This is done by computing the Cherednik-Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function.