Piecewise Polynomial Functions Research Articles

Analytic Expressions in Stochastic Max-Plus-Linear Algebra and their Application in Model Predictive Control

The class of max-plus-linear systems can model discrete event systems with synchronization but no choice. Model mismatch and/or disturbances can be characterized as stochastic uncertainties. In stochastic max-plus-linear systems one often needs to compute the expectation of a max-plus-scaling (MPS) function or the chance constraint of a MPS function. The algorithms available in literature are either computationally too expensive or only give an approximation. In this article, we derive an analytic expression for both the expectation and the chance constraint of a MPS function. Both can be written in the form of a piecewise polynomial function in the components of the control variables. The analytic function can be derived offline and can be evaluated online in a quick and efficient way. We also show how the expressions can be used in a model predictive control setting and show the efficiency of the proposed approach with a worked example.

Mathematical programming formulations for piecewise polynomial functions

This paper studies mathematical programming formulations for solving optimization problems with piecewise polynomial (PWP) constraints. We elaborate on suitable polynomial bases as a means of efficiently representing PWPs in mathematical programs, comparing and drawing connections between the monomial basis, the Bernstein basis, and B-splines. The theory is presented for both continuous and semi-continuous PWPs. Using a disjunctive formulation, we then exploit the characteristic of common polynomial basis functions to significantly reduce the number of nonlinearities, and to suggest a bound-tightening technique for PWP constraints. We derive several extensions using Bernstein cuts, an expanded Bernstein basis, and an expanded monomial basis, which upon a standard big-M reformulation yield a set of new MINLP models. The formulations are compared by globally solving six test sets of MINLPs and a realistic petroleum production optimization problem. The proposed framework shows promising numerical performance and facilitates the solution of PWP-constrained optimization problems using standard MINLP software.

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

О ГЛАДКОЙ ИНТЕРПОЛЯЦИИ ТРИАНГУЛИРОВАННОЙ ПОВЕРХНОСТИ

A method and algorithm for rebuilding a surface triangulation in three-dimensional space defined by an STL file is proposed. An initial surface in 3D space (STL file) is represented as a polyhedron composed of triangular faces. The method is based on the analytical representation of the surface as a piecewise polynomial function. This function is built on a polyhedral surface composed of triangles and satisfies the following requirements: 1) within one face, the function is an algebraic polynomial of the third degree; 2) the function is continuous on the entire surface and preserves the continuity of the first partial derivatives; 3) the surface determined by the function passes through the vertices of the initial triangulated surface. The restructuring of computational meshes is required in cases of distortion of the shape of cells when solving problems of mathematical physics using mesh methods (finite-difference, FEM, etc.). Cell distortion can be due to various reasons. These can be large distortions of moving Lagrangian meshes in the calculations in the current configuration, with instability of the hourglass type, with distortion of the faces of the interface between interacting gaseous, liquid and elastoplastic bodies. The rebuilding of the mesh reduces to solving the problem of constructing a smooth surface passing through the nodes of an existing triangulated surface or part of it. Later the nodes of the new mesh are placed on the constructed smooth surface with existing requirements for the size and shape of the cells. The construction of a smooth piecewise polynomial surface is based on the ideas of spline approximation and reduces to the building of a cubic polynomial on each triangular face, taking into account the smooth conjugation of polynomial pieces of the surface constructed on adjacent faces. The proposed method for rebuilding surface triangulation can be useful for calculating the motion of deformable bodies when solving problems of the dynamics of continuous media on immovable Euler grids.

Min-Sum Soft Symbol Estimation for Iterative MMSE Detection

Soft symbol estimation (SSE) is the first process in soft interference cancellation minimum mean squared error (SIC-MMSE) detection for multiple-input multiple-output (MIMO) systems. SSE requires the sum of exhaustive multiplications of probability that occupies a non-negligible amount of the entire SIC-MMSE complexity. This paper proposes two approaches to reduce the complexity of SSE. The first is to find the approximation of SSE by investigating the estimation in the log-domain, which leads to a simple min-sum operation. The second is to approximate the hyperbolic tangent with low-order piecewise polynomial functions. These two approaches are then integrated. The simulation results show that the proposed methods do not degrade the performance in terms of bit error rate (BER), and that they can greatly reduce the number of multiplications, from $O(K2^{K})$ to $O(K)$ .

Sparse Hierarchical Preconditioners Using Piecewise Smooth Approximations of Eigenvectors

When solving linear systems arising from PDE discretizations, iterative methods (such as conjugate gradient (CG), GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. One approach to preconditioning is provided by the hierarchical approximate factorization methods. However, to guarantee sufficient accuracy on the eigenvectors corresponding to the smallest eigenvalues, these methods typically have to be performed at very stringent accuracies, making the preconditioner expensive to apply. On the other hand, for a large class of problems, including many elliptic equations, the eigenvectors corresponding to small eigenvalues are smooth functions of the PDE grid. In this paper, we describe a hierarchical approximate factorization approach which focuses on improving accuracy on the smooth eigenvectors. The improved accuracy is achieved by preserving the action of the factorized matrix on piecewise polynomial functions of the grid. Based on the factorization, we propose a family of sparse preconditioners with $\mathcal{O}({n})$ or $\mathcal{O}({n \log{n}})$ construction complexities. Our methods exhibit rapid convergence of CG in benchmarks run on large elliptic problems, arising for example in flow or mechanical simulations. In the case of the linear elasticity equation the preconditioners are exact on the near-kernel rigid body modes.

Algorithm 999

Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines with similar properties to standard B-splines. These basis functions are called multi-degree B-splines (or MDB-splines ). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A M ATLAB implementation is provided to illustrate the computation and use of MDB-splines.

Analysis of Electromagnetic Wave Propagation in Unmagnetized Hypersonic Vehicle Plasma Sheath Using the Variational Method

The propagation of electromagnetic waves in the plasma sheath is investigated theoretically and numerically using the variational method, which handles the inhomogeneity of the plasma sheath. We derive the variation of the action integral, which can be numerically represented by expanding the velocity and electromagnetic fields to the piecewise polynomial function space. We analyze the transmissivity of electromagnetic waves propagating through the plasma sheath with the barrier and the barrier-parabolic electron number density profile. We give the frequency-dependent transmissivity of electromagnetic wave propagation in different inhomogeneous plasma sheaths. The electromagnetic energy density derived from the Lagrangian density is also given, and the result shows that the collision between electrons and neutral particles can be used to enhance the transmission of electromagnetic waves in the plasma sheath.

Evaluation of the Dawson function and its antiderivative needed for the Gaussian broadening of piecewise polynomial functions

The broadening of a sharp (unbroadened) dielectric function is a fruitful approach to the construction of models of dielectric response of materials. It naturally includes structural disorder or finite state lifetime and allows parameterization of such effects. The unbroadened function is often taken as a piecewise polynomial. Broadening it with the Lorentzian then leads to relatively simple analytical formulae. The Gaussian broadening, however, requires evaluation of several special functions, including the antiderivative of the Dawson function which is not generally available in mathematical libraries. Recently, the authors described the simple recurrent formulae for the construction of a Gaussian-broadened piecewise polynomial model of a complex dielectric function using three special functions, the error function, the Dawson function, and its antiderivative. In this paper, for the Dawson function and its antiderivative an efficient evaluation method is developed enabling the utilization of this model in optical spectra fitting. The effectiveness of this approach is illustrated using elementary and real-world examples of complex dielectric function models.

Evaluation of the Dawson function and its antiderivative needed for the Gaussian broadening of piecewise polynomial functions

The broadening of a sharp (unbroadened) dielectric function is a fruitful approach to the construction of models of dielectric response of materials. It naturally includes structural disorder or finite state lifetime and allows parameterization of such effects. The unbroadened function is often taken as a piecewise polynomial. Broadening it with the Lorentzian then leads to relatively simple analytical formulae. The Gaussian broadening, however, requires evaluation of several special functions, including the antiderivative of the Dawson function which is not generally available in mathematical libraries. Recently, the authors described the simple recurrent formulae for the construction of a Gaussian-broadened piecewise polynomial model of a complex dielectric function using three special functions, the error function, the Dawson function, and its antiderivative. In this paper, for the Dawson function and its antiderivative an efficient evaluation method is developed enabling the utilization of this model in optical spectra fitting. The effectiveness of this approach is illustrated using elementary and real-world examples of complex dielectric function models.

Transmissivity of electromagnetic wave propagating in magnetized plasma sheath using variational method

A variational method is introduced to analyze the transmissivity of an electromagnetic wave propagating in the magnetized plasma sheath. The plasma density is modeled by two parabolic inhomogeneous regions separated by one homogeneous region. The Lagrangian density of the system is constructed based on the fluid energy density and the electromagnetic energy density. The total variation of the Lagrangian density is derived. The fluid and electromagnetic fields are numerically solved by expansion in piecewise polynomial function space. We investigate the effect of an external magnetic field on the transmissivity of the electromagnetic wave. It is found that the transmissivity is increased when an external magnetic field is applied. The dependence of transmissivity on the collision frequency between the electrons and the neutral particles has also been studied. We also show that the external magnetic field causes a shift in the critical frequency of the plasma sheath.

Weakly monotone sets and continuous selection in asymmetric spaces

Sets admitting a continuous selection from the set of near best approximations are studied. Applications of the geometric theory of approximation to the existence of continuous selections for the sets of -link piecewise linear functions, -link piecewise polynomial functions and generalizations thereof are also discussed. Bibliography: 23 titles.

A comparison of piecewise cubic Hermite interpolating polynomials, cubic splines and piecewise linear functions for the approximation of projectile aerodynamics

Abstract Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions. An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics. The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions: they are continuous, differentiable at least once, and have a relatively low degree. The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools, and then compares Piecewise Cubic Hermite Interpolating Polynomial (PCHIP), cubic splines, and piecewise linear functions, and their variant, as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile. A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics, and its evaluation against a set of criteria. Finally, the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.

Size-density trajectories for even-aged sessile oak (Quercus petraea (Matt.) Liebl.) and common beech (Fagus sylvatica L.) stands revealing similarities and differences in the mortality process

We studied the size-density trajectories of pure even-aged unthinned experimental sessile oak ( Quercus petraea (Matt.) Liebl.) stands in the ranges of 994–135,555 trees per hectare initial densities, observed from the ages of 5 to 38. We compared them to unthinned beech ( Fagus sylvatica L.) stands from the same experimental area. An original piecewise polynomial function was fitted to the trajectories, giving way to various applications. For each species, the initial number of trees per hectare ( N 0 ) and the mean girth at breast height at the onset of mortality (Cg 0 ) were parameters of the trajectory model, in addition to the parameters of the maximum size-density lines. The two former parameters (Cg 0, N 0 ) were tied by a linear relationship, which allowed the prediction of trajectories for initial densities not included in the study data. For oak and beech, mortality onset occurred at a constant relative density (RDI), for all initial stand densities, respectively, 0.35 and 0.29. The comparison of the size-density trajectories of oak and beech allowed to establish that oak needs more space than beech for comparable mean girth, and then is less efficient than beech in its space requirements. This paper models the size-density trajectories of pure even-aged sessile oak stands, including the early development stage. It compares the oak results with those on common beech on the same site from a previous study. A novel approach to size-density trajectories, with an original polynomial piecewise function previously used for beech stands on the same site, was satisfactorily used again as a mortality model to provide references to managers of oak forests. A 38-year-old oak spacing trial, re-measured from year 5 to year 38, provided the opportunity to study the size-density trajectories of unthinned stands of this species. The fit of the piecewise polynomial function allowed us to estimate the parameters of the size-density trajectories of all stands, which were the initial number of trees per hectare (N0) and the mean girth at breast height at the onset of mortality (Cg0), in addition to the intercept (a) and slope (b) of the maximum size-density line. A linear relationship between Ln(N0) and Ln(Cg0) (where Ln is the Neperian logarithm) allowed us to reduce the number of parameters needed to fit the trajectories and made it possible to predict a size-density trajectory from any initial density not observed in the experimental stands. Moreover, this later line appeared to be parallel to the maximum size-density line, and new data allowed to establish that this was also the case for the beech stands on the same site. This parallelism feature translates to the onset of mortality occurring at the same relative density for stands of every initial density that is 0.35 for oak and 0.29 for beech. Given the parameters of the maximum size-density line, a single-parameter function family could be used to predict the size-density trajectories of oak stands. The predicted trajectories have various applications in oak silviculture and growth simulators. The oak data and new data for beech stands on the same site allowed to compare the two species and draw conclusions on similitudes and differences concerning mortality and space requirements of both species.

On the strong separation conjecture

This paper contains a partial result on the Pierce–Birkhoff conjecture on piece-wise polynomial functions defined by a finite collection {f 1 ,. .. , f r } of polynomials. In the nineteen eighties, generalizing the problem from the polynomial ring to an artibtrary ring Σ, J. Madden proved that the Pierce–Birkhoff conjecture for Σ is equivalent to a statement about an arbitrary pair of points α, β ∈ Sper Σ and their separating ideal ; we refer to this statement as the local Pierce-Birkhoff conjecture at α, β. In [8] we introduced a slightly stronger conjecture, also stated for a pair of points α, β ∈ Sper Σ and the separating ideal , called the Connectedness conjecture, about a finite collection of elements {f 1 , . . . , fr} ⊂ Σ. In the paper [10] we introduced a new conjecture, called the Strong Connectedness conjecture, and proved that the Strong Connectedness conjecture in dimension n−1 implies the Strong Connectedness conjecture in dimension n in the case when ht( ) ≤ n − 1. The Pierce-Birkhoff Conjecture for r = 2 is equivalent to the Connectedness Conjecture for r = 1; this conjecture is called the Separation Conjecture. The Strong Connectedness Conjecture for r = 1 is called the Strong Separation Conjecture. In the present paper, we fix a polynomial f ∈ R[x, z] where R is a real closed field and x = (x1, . . . , xn), z are n + 1 independent variables. We define the notion of two points α, β ∈ Sper R[x, z] being in good position with respect to f. The main result of this paper is a proof of the Strong Separation Conjecture in the case when α and β are in good position with respect to f.

Adaptive Interpolation Algorithm Based on a kd-Tree for the Problems of Chemical Kinetics with Interval Parameters

The questions of simulating chemical processes with uncertainty in their parameters are considered. A new approach consisting in the construction of a dynamic structured grid based on a kd-tree over the space formed by the interval parameters of a problem is proposed. When the algorithm is executed, a piecewise polynomial function interpolating the dependence of a solution on certain values of interval parameters is built at every step of integrating the initial system of ordinary differential equations. The results of testing the algorithm on the problems of chemical kinetics, including the combustion processes, demonstrate its efficiency and broad range of applications.

Error indicators and adaptive refinement of the discrete thin plate spline smoother

The discrete thin plate spline is a data fitting and smoothing technique for large datasets. Current research only uses uniform grids for this discrete smoother, which may require a fine grid to achieve a certain accuracy. This leads to a large system of equations and high computational costs. Adaptive refinement adapts the precision of the solution to reduce computational costs by refining only in sensitive regions. The error indicator is an essential part of the adaptive refinement as it identifies whether certain regions should be refined. Error indicators are well researched in the finite element method, but they might not work for the discrete smoother as data may be perturbed by noise and not uniformly distributed. Two error indicators are presented: one computes errors by solving an auxiliary problem and the other uses the bounds of the finite element error. Their performances are evaluated and compared with 2D model problems.
 
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The Approximation of the Fourth-Order Partial Differential Equations in the Class of the Piecewise Polynomial Functions on the Triangular Grid

The present work determines the deviation of the piecewise cubic almost-solution of the biharmonic equation and derives the general formula (3) for its calculation. Based on this concept, we obtained an approximation of the equation. A number of numerical calculations were carried out in order to confirm the obtained formula (see pictures 1 and 2) experimentally. In general, for all selected biharmonic functions, the deviation value "ΔΔ(𝑢) turned out to be, as expected, rather small even with a relatively small number of triangulation nodes. On average, with 25 ≤ 𝑁 ≤ 35, the absolute error was about 0, 0001, which gives an approximately asymptotic estimate of 𝑂(ℎ2) when the partitioning step is ℎ → 0.

Dual-Channel Multiplier for Piecewise-Polynomial Function Evaluation for Low-Power 3-D Graphics

A dual-channel multiplier (DCM) for energy efficient second-order piecewise-polynomial function evaluation for 3-D graphics applications is presented in this paper. The performance of the evaluation process is highly dependent on the design of the multiplication and squaring structure. A novel hardware implementation for polynomial evaluation is presented. The proposed approach compensates the complex multipliers by using DCM which reduces the hardware complexity. The DCM scheme performs complex functions with power-efficient and area-efficient approach. The multiplier reduces the hardware computational effort in the piecewise polynomial approximation with uniform or nonuniform segmentation. For large operand input size, a multiplier adder converter and a dedicated radix-4 squaring unit are also proposed. These units achieve the least power consumption compared to previous approaches with large input word size. Comparison with general purpose multiplication has shown reduction in power, and delay by up to 36%, and 50%, respectively. The proposed technique exhibits up to 93% saving in power consumption compared to the current traditional schemes.

Data analysis and forecasting of tourism development in Ukraine

The article contains a thorough study of tourist flows in Ukraine during the last 18 years. The tendencies of the development of international tourism during the last 20 years and their influence on the development of tourism in Ukraine have been explored. Particular attention is paid to the analysis of inbound tourist flows. The negative dynamics of tourist arrivals in Ukraine led to the construction of a forecast model for the development of this component of tourism activity with the aim of an objective assessment of future tourist arrivals and the adoption of effective management decisions on improving the situation of tourism in Ukraine. Tourist destinations today are rigorously competing for consumer interest in tourism products. In view of this, tourist facilities operating in the tourist market place particular emphasis on the elements of tourism marketing (product, price, distribution, advertising, human capital, actual data, processes).Modern forecasting systems and methods have been used to build a forecast of tourism development in Ukraine and, in particular, arrivals of foreign tourists to Ukraine. An analytical forecasting model was built by the system of CurveExpert in the form of a polynomial function, analytical forecasting models were built by the system of computer algebra Maple in the form of piecewise linear and piecewise polynomial functions. Numerical prediction models in the MathCAD system using different types of spline-interpolation and predictive functions have been constructed. A comparative analysis of the results of forecasting in different systems was carried out. The results of the comparative analysis give confidence in the development of inbound tourism in Ukraine.