Polynomials Of Fixed Degree Research Articles

Higher-Order Finite Element Methods for the Nonlinear Helmholtz Equation

In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and error estimates of the finite element solution under a resolution condition between the wave number k, the mesh size h and the polynomial degree p of the form “k(kh)p+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k(kh)^{p+1}$$\\end{document} sufficiently small” and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in h from the case p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document} in Wu and Zou (SIAM J Numer Anal 56(3):1338–1359, 2018) can be removed for p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document}. We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data.

Piecewise interpolation solution of the Cauchy problem for the transport equation with iterative refinement

This article deals with the method of approximate solution of the Cauchy problem for the transport equation based on the Newton interpolation polynomial of two variables with iterative refinement. In each element of the division of a rectangular domain into subdomains, a polynomial approximation of the partial time derivative is constructed. The interpolation polynomial is transformed into the form of an algebraic polynomial with numerical coefficients. In the integrated form, the polynomial is substituted for the dependent variable in the right side of the expression of the partial time derivative. Iterative renewal of the process at a fixed polynomial degree leads to the analogue of Picard’s method of successive approximations.

Bernstein-Bézier weight-adjusted discontinuous Galerkin methods for wave propagation in heterogeneous media

This paper presents an efficient discontinuous Galerkin method to simulate wave propagation in heterogeneous media with sub-cell variations. This method is based on a weight-adjusted discontinuous Galerkin method (WADG), which achieves high order accuracy for arbitrary heterogeneous media [1]. However, the computational cost of WADG grows rapidly with the order of approximation. In this work, we propose a Bernstein-Bézier weight-adjusted discontinuous Galerkin method (BBWADG) to address this cost. By approximating sub-cell heterogeneities by a fixed degree polynomial, the main steps of WADG can be expressed as polynomial multiplication and L2 projection, which we carry out using fast Bernstein algorithms. The proposed approach reduces the overall computational complexity from O(N2d) to O(Nd+1) in d dimensions. Numerical experiments illustrate the accuracy of the proposed approach, and computational experiments for a GPU implementation of BBWADG verify that this theoretical complexity is achieved in practice.

Comparison of Moving Least Squares and RBF+poly for Interpolation and Derivative Approximation

The combination of polyharmonic splines (PHS) with high degree polynomials (PHS+poly) has recently opened new opportunities for radial basis function generated finite difference approximations. The PHS+poly formulation, which relies on a polynomial least squares fitting to enforce the local polynomial reproduction property, resembles somehow the so-called moving least squares (MLS) method. Although these two meshfree approaches are increasingly used nowadays, no direct comparison has been done yet. The present study aims to fill this gap, focusing on scattered data interpolation and derivative approximation. We first review the MLS approach and show that under some mild assumptions PHS+poly can be formulated analogously. Based on heuristic perspectives and numerical demonstrations, we then compare their performances in 1-D and 2-D. One key result is that, as previously found for PHS+poly, MLS can also overcome the edge oscillations (Runge’s phenomenon) by simply increasing the stencil size for a fixed polynomial degree. This is, however, controlled by a weighted least squares fitting which fails for high polynomial degrees. Overall, PHS+poly is found to perform superior in terms of accuracy and robustness.

An insight into RBF-FD approximations augmented with polynomials

Radial basis function-generated finite differences (RBF-FD) based on the combination of polyharmonic splines (PHS) with high degree polynomials have recently emerged as a powerful and robust numerical approach for the local interpolation and derivative approximation of functions over scattered node layouts. Among the key features, (i) high orders of accuracy can be achieved without the need of selecting a shape parameter or the issues related to numerical ill-conditioning, and (ii) the harmful edge effects associated to the use of high order polynomials (better known as Runge’s phenomenon) can be overcome by simply increasing the stencil size for a fixed polynomial degree. The present study complements our previous results, providing an analytical insight into RBF-FD approximations augmented with polynomials. It is based on a closed-form expression for the interpolant, which reveals the mechanisms underlying these features, including the role of polynomials and RBFs in the interpolant, the approximation error, and the behavior of the cardinal functions near boundaries. Numerical examples are included for illustration.

Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form

It is shown that the h-adaptive mixed finite element method for the discretization of eigenvalue clusters of the Laplace operator produces optimal convergence rates in terms of nonlinear approximation classes. The results are valid for the typical mixed spaces of Raviart-Thomas or Brezzi-Douglas-Marini type with arbitrary fixed polynomial degree in two and three space dimensions.

An alternative proof of a PTAS for fixed-degree polynomial optimization over the simplex

The problem of minimizing a polynomial over the standard simplex is one of the basic NP-hard nonlinear optimization problems—it contains the maximum clique problem in graphs as a special case. It is known that the problem allows a polynomial-time approximation scheme (PTAS) for polynomials of fixed degree, which is based on polynomial evaluations at the points of a sequence of regular grids. In this paper, we provide an alternative proof of the PTAS property. The proof relies on the properties of Bernstein approximation on the simplex. We also refine a known error bound for the scheme for polynomials of degree three. The main contribution of the paper is to provide new insight into the PTAS by establishing precise links with Bernstein approximation and the multinomial distribution.

A Refined Error Analysis for Fixed-Degree Polynomial Optimization over the Simplex

We consider the problem of minimizing a fixed-degree polynomial over the standard simplex. This problem is well known to be NP-hard, since it contains the maximum stable set problem in combinatorial optimization as a special case. In this paper, we revisit a known upper bound obtained by taking the minimum value on a regular grid, and a known lower bound based on Polya’s representation theorem. More precisely, we consider the difference between these two bounds and we provide upper bounds for this difference in terms of the range of function values. Our results refine the known upper bounds in the quadratic and cubic cases, and they asymptotically refine the known upper bound in the general case.

Evolvability of Real Functions

We formulate a notion of evolvability for functions with domain and range that are real-valued vectors, a compelling way of expressing many natural biological processes. We show that linear and fixed-degree polynomial functions are evolvable in the following dually-robust sense: There is a single evolution algorithm that, for all convex loss functions, converges for all distributions. It is possible that such dually-robust results can be achieved by simpler and more-natural evolution algorithms. Towards this end, we introduce a simple and natural algorithm that we call wide-scale random noise and prove a corresponding result for the L 2 metric. We conjecture that the algorithm works for a more general class of metrics.

Approximation classes for adaptive higher order finite element approximation

We provide an almost characterization of the approximation classes appearing when using adaptive finite elements of Lagrange type of any fixed polynomial degree. The characterization is stated in terms of Besov regularity, and requires the approximation within spaces with integrability indices below one. This article generalizes to higher order finite elements the results presented for linear finite elements by Binev et. al. [BDDP 2002].

An Adaptive Finite Element Eigenvalue Solver of Asymptotic Quasi-Optimal Computational Complexity

This paper presents a combined adaptive finite element method with an iterative algebraic eigenvalue solver for a symmetric eigenvalue problem of asymptotic quasi-optimal computational complexity. The analysis is based on a direct approach for eigenvalue problems and allows the use of higher-order conforming finite element spaces with fixed polynomial degree. The asymptotic quasi-optimal adaptive finite element eigenvalue solver (AFEMES) involves a proper termination criterion for the algebraic eigenvalue solver and does not need any coarsening. Numerical evidence illustrates the asymptotic quasi-optimal computational complexity in 2 and 3 dimensions.

Overlapping Schwarz Methods for Isogeometric Analysis

We construct and analyze an overlapping Schwarz preconditioner for elliptic problems discretized with isogeometric analysis. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric problems on these subdomains, and solving an additional coarse isogeometric problem associated with the subdomain mesh. We develop an $h$-analysis of the preconditioner, showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and „overlap sizes” for fixed polynomial degree $p$ and regularity $k$ of the basis functions. Numerical results in two- and three-dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters $h, p, k$, number of subdomains $N$, overlap size, and also jumps in the coefficients of the elliptic operator.

Divergence (Runge Phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock–Chebyshev subset interpolation

Runge showed more than a century ago that polynomial interpolation of a function f( x), using points evenly spaced on x ∈ [ - 1 , 1 ] , could diverge on parts of this interval even if f( x) was analytic everywhere on the interval. Least-squares fitting of a polynomial of degree N to an evenly spaced grid with P points should improve accuracy if P ≫ N . We show through numerical experiments that such an overdetermined fit reduces but does not eliminate the Runge Phenomenon. More precisely, define β ≡ ( N + 1 ) / P . The least-squares fit will fail to converge everywhere on [−1, 1] as N → ∞ for fixed β if f( x) has a singularity inside a curve D ( β ) in the complex-plane. The width of the region enclosed by the convergence boundary D shrinks as β diminishes (i. e., more points for a fixed polynomial degree), but shrinks to the interval [−1, 1] only when β → 0 . We also show that the Runge Phenomenon can be completely defeated by interpolation on a “mock–Chebyshev” grid: a subset of ( N + 1 ) points from an equispaced grid with O ( N 2 ) points chosen to mimic the non-uniform N + 1 -point Chebyshev–Lobatto grid.

Adaptive-degree polynomial filter for voltammetric signals

The adaptive-degree polynomial filter (ADPF) for data smoothing is further improvement of the least-squares regression formalism introduced to the experimental data processing by Savitzky and Golay. The adaptive approach bases on the statistical testing of the fitting quality of polynomial function to the experimental data in the smoothing window sliding along the curve. It allows the automatic choice of the proper degree of the polynomial function in the various parts of the experimental curve. The sets of the simulated according to the Feldberg model voltammetric curves reflecting calibration process are considered. The normal random noises of different level were added to simulated curves and next processed with ADPF and fixed-degree polynomial filter. The performance of smoothing methods were tested by means of the sum of squared errors, signal to noise ratio and improvement of the analytical parameters such as: detection limit, sensitivity, linearity and other.

Improving Error Bounds for Multipole-Based Treecodes

Rapid evaluation of potentials in particle systems is an important, time-consuming step in many physical simulations. Over the past decade, the development of treecodes, such as the fast multipole method and Barnes--Hut method, has enabled large scale simulations in astrophysics, molecular dynamics, material science, etc. These methods use fixed-degree polynomial approximations of the potential at a point, due to a set of particles in a hierarchical representation of the particle system. In this paper, we present analysis and experiments to illustrate that fixed-degree multipole approximations can lead to large aggregate errors. We describe an alternate strategy based on careful selection of the multipole degree that leads to asymptotically lower errors while incurring minimal computation overhead. First, we estimate the error associated with each particle-cluster interaction and the aggregate error for each particle. Then we describe a technique for computing the multipole degree of each interaction with a view to reducing aggregate error, and establish the computational complexity of the new method. Following this, we discuss numerical experiments that demonstrate favorable error properties of the new method at the expense of marginal increase in computation. Finally, we report a parallel implementation on the SGI Origin 2000 that achieves excellent speedup for problems from domains such as astrophysics and boundary element solvers.

Cramér-von Mises Variance Estimators for Simulations

We study estimators for the variance parameter σ2 of a stationary process. The estimators are based on weighted Cramér-von Mises statistics, and certain weightings yield estimators that are “first-order unbiased” for σ2. We derive an expression for the asymptotic variance of the new estimators; this expression is then used to obtain the first-order unbiased estimator having the smallest variance among fixed-degree polynomial weighting functions. Our work is based on asymptotic theory; however, we present exact and empirical examples to demonstrate the new estimators' small-sample robustness. We use a single batch of observations to derive the estimators' asymptotic properties, and then we compare the new estimators among one another. In real-life applications, one would use more than one batch; we indicate how this generalization can be carried out.

Adaptive hp-finite element for transient analysis

Even though a tremendous effort has been devoted to the area of transient analysis, many of these models fall short of achieving the overall desired objective due to oversights and rigid simplifying assumptions. Some of those shortfalls are summarized as follows: 1. (i) The grid used is too fine or too coarse and is usually fixed in size; 2. (ii) No error estimates are being used to trigger a grid size change; 3. (iii) A fixed polynomial degree is being used to interpolate the approximate solution of the differential equation at hand (finite element approach); 4. (iv) An approximate differential equation is used to model the physical phenomena rather than the exact one (finite difference approach). In our study, a self-adaptive grid refinement technique coupled with the p-version of the finite element method is investigated. This type of approach is called the hp-version of the finite element. In the modeling process, the approximate solution to the exact differential equation achieves convergence by applying two distinct solution enrichment strategies. One strategy is to solve the problem using a very coarse grid and to enrich the quality of the approximation by increasing the degree p of the interpolating polynomial shape functions for those elements where it is needed. The other strategy involves local grid h refinements for those elements where it is needed. Both strategies would interact synergistically to produce an optimal computational model that produces an accurate simulation of the transient phenomena addressed with minimal computational effort. The triggering for the increase or decrease in polynomial degree p and placement or removal of local mesh h refinements is based on the calculation of a reliable a posteriori error estimate over each element at the end of each time-step. Acase study of particular interest is the simulation of wave propagation in a semi-infinite media.

Efficiency of local polynomials in contour mapping

On the basis of samples taken from a known topographic surface, the parameters of two types of linear models are estimated. The first category is defined by polynomials or trigonometric functions, whose parameters are simultaneously computed from available data. In the second category a set of local centers is defined, and in the neighborhood of each center a fixed-degree polynomial is developed. An approximative resemblance index is calculated, and contour maps corresponding to various models are compared with the topographic map. It is found that with an increasing number of grid points, maps of local polynomials are converging both in continuity and in resemblance. For a sufficient number of grid points, this resemblance is always higher than those produced by models of the first category.

Computer Generation of Optimized Subroutines

Ideally, subroutines used for function evaluation should be both flexible (suitable for use over arbitrary intervals as well as accurate to any desired order) and fast. Often these characteristics conflict, and, as an alternative, a number of subroutines for the same function may be kept in the Subroutine Library, with the programmer selecting the one closest to his needs. Among the fastest subroutines are those using fixed-degree polynomial approximations, where the fixed number of terms has been selected so that the error does not exceed a certain limit throughout the interval. This paper describes a subroutine generator program which, for a number of functions, produces a fixed-degree polynomial approximation subroutine which is adapted to the particular interval and accuracy needed by the programmer (within wide limits). The generation of the polynomial is based on the use of Chebyshev polynomials, and therefore the approximation is optimum in the sense of the Chebyshev polynomial approximations (for a given degree polynomial, very nearly minimum maximum error over the interval).