Biorthogonal System Research Articles

Optimal Hessian recovery using a biorthogonal system with an application to adaptive refinement

We present a method for recovering the Hessian from a linear finite element approach to achieve a higher rate of convergence. This method uses an \(L^{2}\)-based projection as well as boundary modification to achieve and improve the convergence rate. The projection uses a biorthogonal system to make the computation more numerically efficient. We present numerical examples to illustrate the efficiency and optimality of our approach on different meshes. The performance of our approach on adaptively refined meshes is briefly explored.

Neumann Problem for Second-Order Differential Equation with Fractional Derivative in the Analysis and Modeling of Structures Made of Viscoelastic Elements

This article addresses a second-order differential equation containing a Gerasimov-Kaputo fractional differentiation operator of order less than two. The Neumann problem is formulated for this equation. A system of eigenfunctions and eigenvalues for the considered homogeneous boundary problem of the second kind is found. A conjugate boundary problem for the Gerasimov-Kaputo fractional derivative is introduced. A biorthogonal system is obtained that is orthogonal to the found system of eigenfunctions. Visualizations of the eigenfunction system, biorthogonal system, and an example of eigenvalue distribution on the real axis are provided.

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.

A mixed finite element method using a biorthogonal system for optimal control problems governed by a biharmonic equation

In this article, we consider an optimal control problem governed by a biharmonic equation with clamped boundary conditions. We use the Ciarlet--Raviart formulation combined with a biorthogonal system to obtain an efficient numerical scheme. We discuss the a priori error analysis and present results of the numerical experiments that validate the theoretical estimates.

Sturm-Liouville Problem with Mixed Boundary Conditions for a Differential Equation with a Fractional Derivative and Its Application in Viscoelasticity Models

In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem of a second-order differential equation containing a fractional derivative operator. The fractional differentiation operator was considered according to two definitions: Gerasimov-Caputo and Riemann-Liouville-Visualizations of the system of eigenfunctions, the biorthogonal system, and the distribution of eigenvalues on the real axis were presented. The numerical behavior of eigenvalues was studied depending on the order of the fractional derivative for both definitions of the fractional derivative.

Biorthogonal decomposition of the disturbance flow field generated by particle impingement on a hypersonic boundary layer

The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only.

A mixed finite element discretisation of linear and nonlinear multivariate splines using the Laplacian penalty based on biorthogonal systems

We consider a mixed finite element method for a linear multivariate spline using the Laplacian penalty. Our discretisation is based on biorthogonal systems leading to a very simple and efficient finite element scheme. We also extend our approach to a nonlinear case and describe a split Bregman iteration scheme for the resulting nonlinear equations. We apply our numerical schemes to remove the mixture of Gaussian and impulsive noise for some test images.•This paper presents a method of discretising a multivariate spline using a finite element method.•The method uses a biorthogonal system to achieve an efficient finite element method.•The method is extended to cover a discretisation scheme for a nonlinear case, including an adaptation of the split Bregman method for the nonlinear case.

Rigged Hilbert space approach for non-Hermitian systems with positive definite metric

We investigate Dirac’s bra–ket formalism based on a rigged Hilbert space for a non-Hermitian quantum system with a positive-definite metric. First, the rigged Hilbert space, characterized by a positive-definite metric, is established. With the aid of the nuclear spectral theorem for the obtained rigged Hilbert space, spectral expansions are shown for the bra–kets by the generalized eigenvectors of a quasi-Hermitian operator. The spectral expansions are utilized to endow the complete bi-orthogonal system and the transformation theory between the Hermitian and non-Hermitian systems. As an example of application, we show a specific description of our rigged Hilbert space treatment for some parity-time symmetrical quantum systems.

On Biorthogonalization of a Dirichlet System Over a Finite Interval

Ultimately aiming to estimate Dirichlet polynomials, a representation problem for special biorthogonal systems of exponentials is explored in $L^2(0,a)$. If $a=+\infty$, a method of construction of such systems through suitable Blaschke products is known, but the method ceases to operate when $a$ is finite.
 It turns out that the Blaschke product cannot be even adjusted to maintain the old method for the new situation. The biorthogonal system is then represented by a single determinant of a modified Gram matrix of the original system. Bernstein-type inequalities for Dirichlet polynomials and their higher order derivatives are established. The best constants and extremal polynomials are obtained in terms of the Gram matrix.

Smooth and Polyhedral Norms via Fundamental Biorthogonal Systems

Abstract Let $\mathcal {X}$ be a Banach space with a fundamental biorthogonal system, and let $\mathcal {Y}$ be the dense subspace spanned by the vectors of the system. We prove that $\mathcal {Y}$ admits a $C^\infty $-smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that $\mathcal {Y}$ admits locally finite, $\sigma $-uniformly discrete $C^\infty $-smooth and LFC partitions of unity and a $C^1$-smooth locally uniformly rotund norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every weakly Lindelöf determined Banach space (hence, all reflexive ones), $L_1(\mu )$ for every measure $\mu $, $\ell _\infty (\Gamma )$ spaces for every set $\Gamma $, $C(K)$ spaces where $K$ is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum MM, all Banach spaces of density $\omega _1$ are covered by our result.

Local projection stabilisation for convection-diffusion-reaction equations using a biorthogonal system and adaptive refinement

We consider a local projection stabilisation based on biorthogonal systems and adaptive refinement for convection-diffusion-reaction differential equations. The local projection stabilisation and adaptive finite element method are both based on a biorthogonal system. We investigate the numerical efficiency of the approach when compared to the standard finite element method. Numerical examples are presented to demonstrate the performance of the approach.

Basis properties of Fučík eigenfunctions for the Neumann Laplacian

We investigate the basis properties of sequences of Fučík eigenfunctions of the one-dimensional Neumann Laplacian. We show that any such sequence is complete in L2(0,π) and a Riesz basis in the subspace of functions with zero mean. Moreover, we provide sufficient assumptions on Fučík eigenvalues which guarantee that the corresponding Fučík eigenfunctions form a Riesz basis in L2(0,π) and we explicitly describe the corresponding biorthogonal system.

Generalized eigenvectors of linear operators and biorthogonal systems

The notions of a set of generalized eigenvalues and a set of generalized eigenvectors of a linear operator in Euclidean space are introduced. In addition, we provide a method to find a biorthogonal system of a subsystem of eigenvectors of some linear operators in a Hilbert space whose systems of canonical eigenvectors are over-complete. Related to our problem, we will show an example of a linear differential operator that is formally adjoint to Bessel-type differential operators. We also investigate basis properties (completeness, minimality, basicity) of the systems of generalized eigenvectors of this differential operator.

Research on Towed Linear Array Shape Measurement Method Based on Biorthogonal Signal

Systematic error and random error are main factors affecting the positioning of biorthogonal signal underwater positioning system. Based on the operation theory of USBL, this paper quantitatively analyzes a theoretical analysis and simulation calculations on influence of different error sources on the positioning accuracy and qualitatively analyzes the effectiveness of the towed linear array shape measurement method based on biorthogonal signal through practical experiments. By comparing the measurement effect after correction by using the measurement array with the measurement effect based on the linear array, the effectiveness of the method is verified.

AN INVERSE SOURCE PROBLEM FOR A GENERALIZED TIME FRACTIONAL DIFFUSION EQUATION

Abstract This paper is devoted to the study of the inverse problem of finding the time- dependent coefficient of a generalized time fractional diffusion equation, in the case of non- local boundary and integral overdetermination conditions. The existence and uniqueness of the solution of the considered inverse problem are obtained by a method based on the expan- sion of the solution by using a bi-orthogonal system of functions and the fractional calculus. Moreover, we show its continuous dependence on the data. At the end, two examples are presented to illustrate the obtained results.

A generalized trigonometric moment problem in a weighted L2(−∞,0) space

In this paper we consider a generalized trigonometric moment problem in a weighted L2(−∞,0) space. We obtain a solution which is in the closed span of some exponential system EΛ={tkeiλnt} in the weighted space. Our motivation is a classical trigonometric moment problem in L2(−π,π). Based on an entire function, a lower bound for the distance between exponential functions and the closed span of the remaining elements of the system EΛ is derived. An upper bound is then obtained for the norm of the elements of a biorthogonal system to EΛ. The proof of our result utilizes the notions of Bessel sequences and Riesz-Fischer sequences.

Receptivity and stability of hypersonic leading-edge sweep flows around a blunt body

Abstract

A local projection stabilisation finite element method for the Stokes equations using biorthogonal systems

We present a stabilised finite element method for the Stokes equations. The stabilisation is based on a biorthogonal system, which preserves the locality of the approach. We present a priori error estimates of the presented scheme and demonstrate some numerical results.

Finitely determined functions

We study the notion of finitely determined functions defined on a topological vector space E equipped with a biorthogonal system. We prove that, for real-valued convex functions defined on a Banach space with a Schauder basis, the notion of finitely determined function coincides with the classical continuity but outside the convex case there are many finitely determined nowhere continuous functions. This notion will be used to obtain a necessary and sufficient condition for a convex function to attain a minimum at some point. An application to the Karush–Kuhn–Tucker theorem will be given.

Greedy approximation for biorthogonal systems in quasi-Banach spaces

The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems in quasi-Banach spaces from a functional-analytic point of view. If (Formula Presented) is a biorthogonal system in X then for each x ∈ X we have a formal expansion (Formula Presented). The thresholding greedy algorithm (with threshold e > 0) applied to x is formally defined as (Formula Presented). The properties of this operator give rise to the different classes of greedy-type bases. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (non-trivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations among them are carefully discussed.