Biorthogonal Functions Research Articles

High Order Biorthogonal Functions in \({H}\)(curl)

High Order Biorthogonal Functions in \({H}\)(curl)

Fourier transform of biorthogonal polynomials in one variable

In this paper, some new families of biorthogonal functions in one variable produced by using Fourier transforms of biorthogonal polynomials, which are suggested by Laguerre, Jacobi and Szegö–Hermite polynomials, and their biorthogonality relations obtained via Parseval identity are derived.

A bivariational, stable, and convergent hierarchy for time-dependent coupled cluster with adaptive basis sets.

We propose a new formulation of time-dependent coupled cluster with adaptive basis functions and division of the one-particle space into active and secondary subspaces. The formalism is fully bivariational in the sense of a real-valued time-dependent bivariational principle and converges to the complete-active-space solution, a property that is obtained by the use of biorthogonal basis functions. A key and distinguishing feature of the theory is that the active bra and ket functions span the same space by construction. This ensures numerical stability and is achieved by employing a split unitary/non-unitary basis set transformation: the unitary part changes the active space itself, while the non-unitary part transforms the active basis. The formulation covers vibrational as well as electron dynamics. Detailed equations of motion are derived and implemented in the context of vibrational dynamics, and the numerical behavior is studied and compared to related methods.

A unified mixed finite element method for fourth-order time-dependent problems using biorthogonal systems

This article introduces a unified mixed finite element framework based on a saddle-point formulation that applies to time-dependent fourth order linear and nonlinear problems with clamped, simply supported, and Cahn-Hilliard type boundary conditions. The classical mixed formulations lead to large matrix systems that demand huge storage and computational time making the schemes expensive, especially for the time-dependent problems. The proposed scheme circumvents this by employing biorthogonal basis functions that lead to sparse and positive-definite systems. The article discusses a mixed finite element method for the biharmonic problem and the time-dependent linear and nonlinear versions of the extended Fisher-Kolmogorov equations equipped with the aforementioned boundary conditions. The wellposedness of the scheme is discussed and a priori error estimates are presented for the semi-discrete and fully discrete finite element schemes. The numerical experiments validate the theoretical estimates derived in the paper.

On the solvability of direct and inverse problems for a generalized diffusion equation

This paper delves into both direct and two inverse source problems associated with a diffusion equation featuring integral convolution over time, while considering non-classical boundary conditions. The inverse source problems are shown to exhibit ill-posed characteristics in accordance with Hadamard’s definition. A bi-orthogonal function system is employed to express series solutions for the inverse source problems. By imposing specific conditions on the provided data, we establish the existence of unique series solutions. Several special cases of the diffusion equation are presented, depending on the nature of the memory kernel. Furthermore, to illustrate the findings regarding inverse source problems, we provide specific examples.

Biorthogonal Functions for Complex Exponentials and an Application to the Controllability of the Kawahara Equation Via a Moment Approach

Biorthogonal Functions for Complex Exponentials and an Application to the Controllability of the Kawahara Equation Via a Moment Approach

A stable and mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations

The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability.The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Numerical analysis of the new basis functions reveals that their higher dual regularity does not only lead to conservation, but also yields an L2-stable basis for the combination technique. Accordingly, further numerical experiments applying the combination technique to a semi-Lagrangian Vlasov–Poisson solver in six dimensions show a stabilizing effect of the biorthogonal and full weighting bases on the simulations.

Solvability of a Nonlocal Inverse Problem for a Fourth-Order Equation

В данной работе для уравнения в частных производных четвертого порядка в прямоугольной области рассмотрена нелокальная обратная задача по поиску неизвестной правой части, которая зависит от одной переменной. Собственные функции и присоединенные функции соответствующей спектральной задачи, и их биортогональные функции полны и образуют базис Рисса в пространстве L2​(0,1). Установлены критерии единственности и существования решения рассматриваемой нелокальной обратной задачи для уравнения четвертого порядка. Единственность решения нелокальной обратной задачи вытекает из полноты системы биортогональных функций. Решение задачи построено в виде суммы ряда по собственным и присоединенным функциям, соответствующей спектральной задачи. Установлены достаточные условия на граничные функции, которые гарантируют теоремы существования и устойчивости решения рассматриваемой задачи. В замкнутой области показана абсолютная и равномерная сходимость найденного решения обратной задачи в виде ряда, а также рядов, полученных почленным дифференцированием по t и x соответственно два и четыре раза, в зависимости от гладкости функции заданными начальными условиями. При этом возникают малые знаменатели, затрудняющие сходимость этих рядов. Доказано, что в зависимости от размера области, множество ненулевых решений выражения в знаменателе не пусто. А также, показано, что если этот знаменатель равен нулю, то данная задача будет иметь нетривиальное решение при однородных условиях. Доказана также, устойчивость решения обратной задачи по нормам пространств L2​,W2n​ и C(Ω±), относительно изменения входных данных. In this paper, we consider a nonlocal inverse problem of finding an unknown right-hand side with one variable for a fourth-order partial differential equation in a rectangular domain. The eigenfunctions and functions of the corresponding spectral problem and its biorthogonal functions are complete and form a Riesz basis in the space L2​(0,1). Criteria for the uniqueness and existence of a solution to the considered nonlocal inverse problem for a fourth-order equation are established. The uniqueness of the solution of the nonlocal inverse problem follows from the completeness of the system of biorthogonal functions. The solution of the problem is constructed as the sum of a series in terms of eigenfunctions and associated functions of the corresponding spectral problem. Sufficient conditions are established for the boundary functions that guarantee existence and stability theorems for the solution of the problem under consideration. In a closed domain, absolute and uniform convergence of the found solution of the inverse problem is shown in the form of a series, as well as series obtained by term-by-term differentiation with respect to t and x two and four times, respectively, depending on the smoothness of the function given the initial conditions. In this case, small denominators arise, which hinder the convergence of these series. It is proved that, depending on the size of the domain, the set of non-zero solutions of the expression in the denominator is not empty. And also, it is shown that if this denominator is equal to zero, then this problem will have a non-trivial solution under homogeneous conditions. It is also proved that the solution of the inverse problem is stable in the norms of the spaces L2​,W2n​ and C(Ω±) with respect to changes in the input data.

Evaluating the performance of different mother wavelet functions for down-sampling of earthquake records

Application of discrete wavelet transform for down-sampling earthquake records has been started since 2002. Although the mother wavelet function plays an important role in the accuracy of wavelet transform, there is no study to investigate the effect of this function on dynamic analysis error. In addition, most studies have used a limited number of earthquake records for down-sampling application of wavelet transform. Therefore, in this paper, a comprehensive study is performed on 36 different mother wavelet functions and 100 earthquake records. The linear and nonlinear response spectra of the main records and their down-sampled ones have been used to evaluate the performance of different mother wavelet functions. Approximate waves are obtained for three levels using wavelet transform. Also, dynamic analyses are done for 101 single-degree-of-freedom systems (with a period between 0 and 5 s) and 5 behavior coefficients consisting of more than 7 million analyses. Results show that the Daubechies family of wavelet functions used in previous studies for down-sampling earthquake records often cause errors greater than 15%. But the Biorthogonal wavelet functions which have not been used for down-sampling before, have an error of less than 2.5% and are the best wavelet functions for down-sampling earthquake records. In addition, a method called wavelet functions combination is presented to ensure minimum errors at different levels of down-sampling. In the next step, to evaluate the effect of the wavelet function on the accuracy of dynamic analysis of real structures, 3, 9, and 20 story structures are analyzed using the IDA method and finally, an irregular structure is also analyzed. The results obtained from the nonlinear analysis of these structures show that Biorthogonal wavelet functions can reduce the wavelet error from 27% to 6%.

A Legendre dual-Petrov-Galerkin spectral element method for the Kawahara-type equations

An efficient and accurate Legendre dual-Petrov-Galerkin spectral element method for solving the Kawahara-type equations is proposed. The Sobolev bi-orthogonal basis functions are constructed, which reduce the number of nonzero elements in the matrix and improve the computational efficiency. The generalized stability of the fully discrete spectral element scheme is analyzed. For applications, we consider the Kawahara, KdV-Kawahara and modified Kawahara equations, and discuss numerically the motion of single solitary wave, conservation laws, the phenomena of wave generation and interaction of two and three solitons. Numerical results illustrate the effectiveness of the suggested approach.

Bispectrality and biorthogonality of the rational functions of q-Hahn type

We introduce families of rational functions that are biorthogonal with respect to the q-hypergeometric distribution. A triplet of q-difference operators X, Y, Z is shown to play a role analogous to the pair of bispectral operators of orthogonal polynomials. The recurrence relation and difference equation take the form of generalized eigenvalue problems involving the three operators. The algebra generated by X, Y, Z is akin to the algebras of Askey–Wilson type in the case of orthogonal polynomials. The actions of these operators in three different bases are presented. Connections with Wilson's ϕ910 biorthogonal rational functions are also discussed.

Band structures under non-Hermitian periodic potentials: Connecting nearly-free and bi-orthogonal tight-binding models

We explore band structures of one-dimensional open systems described by periodic non-Hermitian operators, based on continuum models and tight-binding models. We show that imaginary scalar potentials do not open band gaps but instead lead to the formation of exceptional points as long as the strength of the potential exceeds a threshold value, which is contrast to closed systems where real potentials open a gap with infinitesimally small strength. The imaginary vector potentials hinder the separation of low energy bands because of the lifting of degeneracy in the free system. In addition, we construct tight-binding models through bi-orthogonal Wannier functions based on Bloch wavefunctions of the non-Hermitian operator and its Hermitian conjugate. We show that the bi-orthogonal tight-binding model well reproduces the dispersion relations of the continuum model when the complex scalar potential is sufficiently large.

Subsymmetric bases have the factorization property

Let $(e_j)_{j=1}^\infty $ denote a Schauder basis for a Banach space $X$, and let $(e_j^*)_{j=1}^\infty $ denote the biorthogonal functionals. We say that $(e_j^*)_{j=1}^\infty $ has the factorization property if the identity operator $I_{X^*}$ on $X^

Exact Solution for a Rectangle with Rigidly Clamped Horizontal Sides

In this paper, in the form of expansions in Papkovich–Fadle eigenfunctions, an exact solution to the boundary value problem of the theory of elasticity is constructed for a rectangle whose horizontal sides are rigidly clamped. The expansion coefficients are determined with the help of biorthogonal functions by simple formulas.

The rational Heun operator and Wilson biorthogonal functions

We consider the rational Heun operator defined as the most general second-order q-difference operator which sends any rational function of type \([(n-1)/n]\) to a rational function of type \([n/(n+1)]\). We shall take the poles to be located on the Askey–Wilson grid. It is shown that this operator is related to the one-dimensional degeneration of the Ruijsenaars–van Diejen Hamiltonians. The Wilson biorthogonal functions of type \({_{10}}\Phi _9\) are found to be solutions of a generalized eigenvalue problem involving rational Heun operators of the special “classical” kind.

Generation of orthogonal rational functions by procedures for structured matrices

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

A generalized inverse eigenvalue problem and m-functions

In this manuscript, a generalized inverse eigenvalue problem is considered that involves a linear pencil (zJ[0,n]−H[0,n]) of (n+1) by (n+1) matrices arising in the theory of rational interpolation and biorthogonal rational functions. In addition to the reconstruction of the Hermitian matrix H[0,n], characterizations of the rational functions that are components of the prescribed eigenvectors are given. A condition concerning the positive-definiteness of J[0,n] which is often an assumption in the direct problem is also isolated. Further, the reconstruction of H[0,n] is viewed through the inverse of the pencil (zJ[0,n]−H[0,n]) which involves the concept of m-functions.

Modeling Skew by Single- and Multi-Slice 2-D Machine Models

Skew is taken into account in single- and multi-slice 2-D finite-element machine models by skew interface conditions applied at a cylinder hull in the center of the air gap. A set of biorthogonal basis functions is constructed to discretize the tangential magnetic field strength at the interface and to weakly enforce the tangential continuity of the magnetic vector potential. The harmonic filtering as a result of skew is already observed in a single-slice model. Multi-slice models with different numbers of stator and rotor slices increase the modeling accuracy.

Biorthogonal Wavelet on a Logarithm Curve ℂ

According to the length-preserving projection and Euler discretization method, biorthogonal wavelet function on a smooth curve C is constructed in this paper, such as a logarithm curve. The properties of biorthogonal wavelet filters on a smooth curve C are discussed, such as induced refinable equation and symmetry. Moreover, an example is given for discussing the biorthogonal scaling function and its dual on a logarithm curve C . Finally, a numerical application is given for dealing with financial data.

Two-Dimensional Quantum Walk with Non-Hermitian Skin EffectsSupported by the National Natural Science Foundation of China (Grant Nos. 11974331, 11674306, and 61590932), and the National Key R&D Program (Grant Nos. 2016YFA0301700 and 2017YFA0304100).

We construct a two-dimensional, discrete-time quantum walk, exhibiting non-Hermitian skin effects under open-boundary conditions. As a confirmation of the non-Hermitian bulk-boundary correspondence, we show that the emergence of topological edge states is consistent with the Floquet winding number, calculated using a non-Bloch band theory, invoking time-dependent generalized Brillouin zones. Further, the non-Bloch topological invariants associated with quasienergy bands are captured by a non-Hermitian local Chern marker in real space, defined via the local biorthogonal eigenwave functions of a non-unitary Floquet operator. Our work aims to stimulate further studies of non-Hermitian Floquet topological phases where skin effects play a key role.