Polynomials Of Arbitrary Degree Research Articles

A C^1 Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence

<p style='text-indent:20px;'>In this paper, we present and study <inline-formula><tex-math id="M2">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M4">\begin{document}$ \ge 3 $\end{document}</tex-math></inline-formula>) for one-dimen\-sional elliptic equations. We prove that, the solution and its derivative approximations converge with rate <inline-formula><tex-math id="M5">\begin{document}$ 2k-2 $\end{document}</tex-math></inline-formula> at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree <inline-formula><tex-math id="M6">\begin{document}$ k+1 $\end{document}</tex-math></inline-formula> in each element, the first-order derivative approximation is superconvergent at all interior <inline-formula><tex-math id="M7">\begin{document}$ k-2 $\end{document}</tex-math></inline-formula> Lobatto points, and the second-order derivative approximation is superconvergent at <inline-formula><tex-math id="M8">\begin{document}$ k-1 $\end{document}</tex-math></inline-formula> Gauss points, with an order of <inline-formula><tex-math id="M9">\begin{document}$ k+2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ k+1 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M11">\begin{document}$ k $\end{document}</tex-math></inline-formula>, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in <inline-formula><tex-math id="M12">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M14">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> norms. All theoretical findings are confirmed by numerical experiments.

A virtual element formulation for general element shapes

The virtual element method is a lively field of research, in which considerable progress has been made during the last decade and applied to many problems in physics and engineering. The method allows ansatz function of arbitrary polynomial degree. However, one of the prerequisite of the formulation is that the element edges have to be straight. In the literature there are several new formulations that introduce curved element edges. These virtual elements allow for specific geometrical forms of the course of the curve at the edges. In this contribution a new methodology is proposed that allows to use general mappings for virtual elements which can model arbitrary geometries.

Rational Bézier curves approximated by Bernstein–Jacobi hybrid polynomial curves

In this paper, we propose a linear method for $C^{(r,s)}$ approximation of rational B\'{e}zier curve with arbitrary degree polynomial curve. Based on weighted least-squares, the problem be converted to an approximation between two polynomial curves. Then applying Bernstein-Jacobi hybrid polynomials, we obtain the resulting curve. In order to reduce error, degree reduction method for B\'{e}zier curve is used. A error bound between rational B\'{e}zier curve and B\'{e}zier curve is presented. Finally, some examples and figures were offered to demonstrate the efficiency, simplicity, and stability of our methods.

On the zeros of certain composite polynomials and an operator preserving inequalities

If all the zeros of nth degree polynomials f(z) and $$g(z) = \sum _{k=0}^{n}\lambda _k\left( {\begin{array}{c}n\\ k\end{array}}\right) z^k$$ respectively lie in the cricular regions $$|z|\le r$$ and $$|z| \le s|z-\sigma |$$ , $$s>0$$ , then it was proved by Marden (Geometry of polynomials, Math Surveys, No. 3, American Mathematical Society, Providence, 1949, p. 86) that all the zeros of the polynomial $$h(z)= \sum _{k=0}^{n}\lambda _k f^{(k)}(z) \frac{(\sigma z)^k}{k!}$$ lie in the circle $$|z| \le r ~ \max (1,s)$$ . In this paper, we relax the condition that f(z) and g(z) are of the same degree and instead assume that f(z) and g(z) are polynomials of arbitrary degree n and m, respectively, $$m\le n,$$ and obtain a generalization of this result. As an application, we also introduce a linear operator which preserves Bernstein type polynomial inequalities.

Finite Element Simulation of Ionic Electrodiffusion in Cellular Geometries.

Mathematical models for excitable cells are commonly based on cable theory, which considers a homogenized domain and spatially constant ionic concentrations. Although such models provide valuable insight, the effect of altered ion concentrations or detailed cell morphology on the electrical potentials cannot be captured. In this paper, we discuss an alternative approach to detailed modeling of electrodiffusion in neural tissue. The mathematical model describes the distribution and evolution of ion concentrations in a geometrically-explicit representation of the intra- and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the Extracellular-Membrane-Intracellular (EMI) framework, we refer to this model as the KNP-EMI model. Here, we introduce and numerically evaluate a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree. Moreover, we compare the electrical potentials predicted by the KNP-EMI and EMI models. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the KNP-EMI framework can give new insights in this setting.

A Robust Discretization of the Reissner-Mindlin Plate with Arbitrary Polynomial Degree

A Robust Discretization of the Reissner-Mindlin Plate with Arbitrary Polynomial Degree

On the development of symmetry-preserving finite element schemes for ordinary differential equations

In this paper we introduce a procedure, based on the method of equivariant moving frames, for formulating continuous Galerkin finite element schemes that preserve the Lie point symmetries of initial value problems for ordinary differential equations. Our methodology applies to projectable and non-projectable symmetry group actions, to ordinary differential equations of arbitrary order, and finite element approximations of arbitrary polynomial degree. Several examples are included to illustrate various features of the symmetry-preserving process. We summarise extensive numerical experiments showing that symmetry-preserving finite element schemes may provide better long term accuracy than their non-invariant counterparts and can be implemented on larger elements.

Protecting data privacy in publicly verifiable delegation of matrix and polynomial functions

Outsourcing computation has gained significant attention in recent years in particular due to the prevalence of cloud computing. There are two main security concerns in outsourcing computation: guaranteeing that the server performs the computation correctly, and protecting the privacy of the client's data. The verifiable computation of Gennaro, Gentry and Parno addresses both concerns for outsourcing the computation of a function f on an input x to the cloud. The GGP scheme is privately delegatable, privately verifiable, and based on the expensive cryptographic primitives such as fully homomorphic encryption (FHE). In this paper we consider the problem of outsourcing matrix-vector multiplications of the form Fxwhere F is a matrix and xis a column vector, and construct publicly delegatable and publicly verifiable schemes. Our schemes are either input private or function private, highly efficient, and provably secure under the well-established assumptions such as the discrete-logarithm assumption. We decompose a polynomial computation, such as computing a univariate polynomial of arbitrary degree, a bivariate polynomial of arbitrary degree, a quadratic multivariate polynomial, and in general any multivariate polynomial, into a two-step computation in which the computaionally expensive step is a matrix-vector multiplication. We use the matrix schemes to outsource the computation of high-degree polynomials and obtain the first high-degree polynomial outsourcing schemes that simultaneously have public delegation, public verification and input privacy/function privacy.

A hybrid high-order method for a coupled Stokes-Darcy problem on general meshes

In this work, a hybrid high-order (HHO) method on general meshes is presented to solve a coupled Stokes-Darcy problem with the Beavers-Joseph-Saffman interface condition. Constructed on polynomials of arbitrary degree k≥0, the numerical method is established in terms of discrete unknowns attached to mesh faces and cells (or elements). The unified discrete scheme for Stokes equation and Darcy equation is given by the continuity condition of the interface. The unique solvability of the discrete scheme is proved. Moreover, the energy error estimate for the velocity and L2-error estimate for pressure of order (k+1) are derived. Finally, a series of numerical experiments are reported to illustrate the accuracy, mass conservation and robustness of our method.

A posteriori error analysis for the mixed Laplace eigenvalue problem: investigations for the BDM‐element

AbstractA posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem can be derived using a reconstruction in the standard ‐conforming space for the primal variable of the mixed problem. In the case of Raviart–Thomas finite elements of arbitrary polynomial degree the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. This paper shows that the extension for the BDM‐element is not straightforward.

Asymptotically Exact A Posteriori Error Analysis for the Mixed Laplace Eigenvalue Problem

Abstract This paper derives a posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem. We discuss a reconstruction in the standard H 0 1 {H_{0}^{1}} -conforming space for the primal variable of the mixed Laplace eigenvalue problem and compare it with analogous approaches present in the literature for the corresponding source problem. In the case of Raviart–Thomas finite elements of arbitrary polynomial degree, the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. Our reconstruction is performed locally on a set of vertex patches.

On the Turán Inequality for Quaternionic Polynomials

In this paper we prove that the Turan inequality in quaternionic setting holds for all polynomials of degree $$n\le 2$$ and for some particular subclasses of polynomials of arbitrary degree $$n\ge 3$$ . It is important to note that the proofs of Turan’s inequality in the complex case do not work in the quaternionic setting for $$n\ge 3$$ , however we are lead to make the intriguing conjecture that the Turan inequality still holds for all quaternionic polynomials of arbitrary degree $$n\ge 3$$ .

An HDG formulation for incompressible and immiscible two-phase porous media flow problems

ABSTRACTWe develop a high-order hybridisable discontinuous Galerkin (HDG) formulation to solve the immiscible and incompressible two-phase flow problem in a heterogeneous porous media. The HDG method is locally conservative, has fewer degrees of freedom than other discontinuous Galerkin methods due to the hybridisation procedure, provides built-in stabilisation for arbitrary polynomial degrees and, if the error of the temporal discretisation is low enough, the pressure, the saturation and their fluxes converge with order P+1 in -norm, being P the polynomial degree. In addition, an element-wise post-process can be applied to obtain a convergence rate of P+2 in -norm for the scalar variables. All of these advantages make the HDG method suitable for solving multiphase flow trough porous media. We show numerical evidence of the convergences rates. Finally, to assess the capabilities of the proposed formulation, we apply it to several cases of water-flooding technique for oil recovery.

МОДУЛЬ ОБРАБОТКИ ЭКСПЕРИМЕНТАЛЬНЫХ ТЕПЛОВЫХ ХАРАКТЕРИСТИК СТАНКОВ В РЕЖИМЕ РЕАЛЬНОГО ВРЕМЕНИ С ПРИМЕНЕНИЕМ ПРОИЗВОДНЫХ ВЫСОКОГО ПОРЯДКА

The article presents processing module of experimental thermal characteristics of machine tools in real time using high order derivatives. It is developed in a high level programming language. This module is a part of automated system of tests and diagnostics of machines thermal state. The module is equipped with a graphical user interface; it works in real time, allows performing procedures for smoothing experimental thermal characteristics and determining their prognostic values. This allows building curves of speed, acceleration and sharpness of temperature change over the time. The developed software module implements a new algorithm based on the Horner’s method. The practice of developing algorithmic software shows that the chosen method is very convenient and effective for machine implementation, due to the absence of a division operation at each computational step. It is experimentally confirmed that the total costs of performing the calculations of high order derivatives of an arbitrary degree polynomial using the Horner’s method according to the described algorithm are acceptable for real-time calculations on a standard personal computer. Data temperature and temperature movements of the machine working bodies can be processed using the developed algorithms. In addition, this algorithm allows calculating the value of the approximating polynomial and simultaneously obtaining the values of all its derivatives at a given point. This allows to solve the problem of "shift in time by step" of the position of characteristic points.

The images of multilinear polynomials on strictly upper triangular matrices

The purpose of this paper is to describe the images of multilinear polynomials of arbitrary degree on the strictly upper triangular matrix algebra.

On the minimal ranks of matrix pencils and the existence of a best approximate block-term tensor decomposition

Under the action of the general linear group with tensor structure, the ranks of matrices A and B forming an m×n pencil A+λB can change, but in a restricted manner. Specifically, with every pencil one can associate a pair of minimal ranks, which is unique up to a permutation. This notion can be defined for matrix pencils and, more generally, also for matrix polynomials of arbitrary degree. In this paper, we provide a formal definition of the minimal ranks, discuss its properties and the natural hierarchy it induces in a pencil space. Then, we show how the minimal ranks of a pencil can be determined from its Kronecker canonical form. For illustration, we classify the orbits according to their minimal ranks (under the action of the general linear group) in the case of real pencils with m,n≤4. Subsequently, we show that real regular 2k×2k pencils having only complex-valued eigenvalues, which form an open positive-volume set, do not admit a best approximation (in the norm topology) on the set of real pencils whose minimal ranks are bounded by 2k−1. Our results can be interpreted from a tensor viewpoint, where the minimal ranks of a degree-(d−1) matrix polynomial characterize the minimal ranks of matrices constituting a block-term decomposition of an m×n×d tensor into a sum of matrix-vector tensor products.

Binary correspondences and the inverse problem of chemical kinetics

We show how binary correspondences can be used for simple formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions. In particular, formalization of the following notions is presented: condition, data, unknowns, and solutions of a problem, solvability and unique solvability, inverse problem, composition and restriction of problems, isomorphism between problems. We also consider topological problems and the related notions of stability and correctness. A connection is indicated between the stability and continuity of a uniquely solvable topological problem. The definition of parametrized set is given. The notions are introduced of parametrized problem, the problem of reconstruction of an object by the values of parameters, as well as the notions of locally free set of parameters and stability with respect to a set of parameters. As an illustration, we consider a singularly perturbed system of ordinary differential equations which describe a process in chemical kinetics and burning. Direct and inverse problems are stated for such a system. We extend the class of problems under study by considering polynomials of arbitrary degree as the right-hand sides of the differential equations. It is shown how the inverse problem of chemical kinetics can be corrected and made more practical by means of the composition with a simple auxiliary problem which represents the relation between functions and finite sets of numerical characteristics being measured. For the corrected inverse problem, formulas for the solution are presented and the conditions of unique solvability are indicated. Within the study of solvability, a criterion is established for linear independence of functions in terms of finite sets of their values. With the help of the criterion, realizability is clarified of the condition for unique solvability of the inverse problem of chemical kinetics.

New robust LMI synthesis conditions for mixed gain‐scheduled reduced‐order DOF control of discrete‐time LPV systems

SummaryThis paper investigates the problems of stabilization and mixed reduced‐order dynamic output‐feedback control of discrete‐time linear systems. The synthesis conditions are formulated in terms of parameterdependent linear matrix inequalities (LMIs) combined with scalar parameters, dealing with state‐space models where the matrices depend polynomially on time‐varying parameters and are affected by norm‐bounded uncertainties. The motivation to handle these models comes from the context of networked control systems, particularly when a continuous‐time plant is controlled by a digitally implemented controller. The main technical contribution is a distinct LMI‐based condition for the dynamic output‐feedback problem, allowing an arbitrary structure (polynomial of arbitrary degree) for the measured output matrix. Additionally, an innovative heuristic is proposed to reduce the conservativeness of the stabilization problem. Numerical examples are provided to illustrate the potentialities of the approach to cope with several classes of discrete‐time linear systems (time‐invariant and time‐varying) and the efficiency of the proposed design conditions when compared with other methods available in the literature.

Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework

This paper develops a general framework for a posteriori error estimates in numerical approximations of the Laplace eigenvalue problem, applicable to all standard numerical methods. Guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given, as well as on the energy error in the approximation of the associated eigenvector. The bounds are valid under the sole condition that the approximate i-th eigenvalue lies between the exact $$(i-1)$$ -th and $$(i+1)$$ -th eigenvalue, where the relative gaps are sufficiently large. We give a practical way how to check this; the accuracy of the resulting estimates depends on these relative gaps. Our bounds feature no unknown (solution-, regularity-, or polynomial-degree-dependent) constant, are optimally convergent (efficient), and polynomial-degree robust. Under a further explicit, a posteriori, minimal resolution condition, the multiplicative constant in our estimates can be reduced by a fixed factor; moreover, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, this multiplicative constant can be brought to the optimal value of 1 with mesh refinement. Applications of our framework to nonconforming, discontinuous Galerkin, and mixed finite element approximations of arbitrary polynomial degree are provided, along with numerical illustrations. Our key ingredients are equivalences between the i-th eigenvalue error, the associated eigenvector energy error, and the dual norm of the residual. We extend them in an appendix to the generic class of bounded-below self-adjoint operators with compact resolvent.

Recursive computation of gravitational field of a right rectangular parallelepiped with density varying vertically by following an arbitrary degree polynomial

Recursive computation of gravitational field of a right rectangular parallelepiped with density varying vertically by following an arbitrary degree polynomial