Showing papers on "Trigonometric interpolation published in 2014"

High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

Abstract: We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE's, we have shown in Chkifa et al. (Model. Math. Anal. Numer. 47(1):253---280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE's.

Fast Discrete Fourier Transform on Generalized Sparse Grids

Abstract: In this paper, we present an algorithm for trigonometric interpolation of multivariate functions on generalized sparse grids and study its application for the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. In particular, we derive estimates for the error and the cost. We construct interpolants with a computational cost complexity which is substantially lower than for the standard full grid case. The associated generalized sparse grid interpolants have the same approximation order as the standard full grid interpolants, provided that certain additional regularity assumptions on the considered functions are fulfilled. Numerical results validate our theoretical findings.

A G2 Continuous Path-smoothing Algorithm Using Modified Quadratic Polynomial Interpolation

Abstract: Path searching algorithm is one of the main topics in studies on path planning. These algorithms are used to avoiding obstacles and find paths from starting point to target point. There are dynamic problems that must be addressed when these paths are applied in real environments. In order to be applicable in actual situations, the path must be a smooth path. A smooth path is a path that maintains continuity. Continuity is decided by the differential values of the path. In order to be G2 continuous, the secondary differential values of the path must be connected throughout the path. In this paper, the interpolation method is used to construct continuous paths. The quadratic polynomial interpolation is a simple method for obtaining continuous paths about three points. The proposed algorithm makes a connection of three points with curves and the proposed path is rotated using the parametric method in order to make the path optimal and smooth. The polynomials expand to the next three points and they merge int...

Kötter interpolation in skew polynomial rings

Abstract: Skew polynomials are a noncommutative generalization of ordinary polynomials that, in recent years, have found applications in coding theory and cryptography. Viewed as functions, skew polynomials have a well-defined evaluation map; however, little is known about skew-polynomial interpolation. In this work, we apply Kotter's interpolation framework to free modules over skew polynomial rings. As a special case, we introduce a simple interpolation algorithm akin to Newton interpolation for ordinary polynomials.

Interpolation functions and?the Lions-Peetre?interpolation construction

Abstract: The generalization of the Lions-Peetre interpolation method of means considered in the present survey is less general than the generalizations known since the 1970s. However, our level of generalization is sufficient to encompass spaces that are most natural from the point of view of applications, like the Lorentz spaces, Orlicz spaces, and their analogues. The spaces considered here have three parameters: two positive numerical parameters and? of equal standing, and a function parameter?. For these spaces can be regarded as analogues of Orlicz spaces under the real interpolation method. Embedding criteria are established for the family of spaces , together with optimal interpolation theorems that refine all the known interpolation theorems for operators acting on couples of weighted spaces and that extend these theorems beyond scales of spaces. The main specific feature is that the function parameter can be an arbitrary natural functional parameter in the interpolation. Bibliography: 43 titles.

Fractal Interpolation Functions: A Short Survey

Abstract: The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various approximation techniques, the article is interspersed with comparison of fractal interpolation functions and diverse conventional interpolation schemes. There are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric and splines to name a few. But it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. One of the goals of the paper is the definition of interpolants which are not smooth, and likely they are nowhere differentiable. They are defined by means of an appropriate iterated function system. Their appearance fills the gap of non-smooth methods in the field of approximation. Another interesting topic is that, if one chooses the elements of the iterated function system suitably, the resulting fractal curve may be close to classical mathematical functions like polynomials, exponentials, etc. The authors review many results obtained in this field so far, although the article does not claim any completeness. Theory as well as applications concerning this new topic published in the last decade are discussed. The one dimensional case is only considered.

Mean convergence of an extended Lagrange interpolation process on [0,+∞)

Abstract: We study some extended Lagrange interpolation processes based on the zeros of the generalized Laguerre polynomials. We give necessary and sufficient conditions such that the convergence of these processes, in suitable Lp weighted spaces on the real semiaxis, is assured for 1

Optimal Point Sets for Total Degree Polynomial Interpolation in Moderate Dimensions

Abstract: This paper is concerned with Lagrange interpolation by total degree polynomials in moderate dimensions. In particular, we are interested in characterising the optimal choice of points for the interpolation problem, where we define the optimal interpolation points as those which minimise the Lebesgue constant. We give a novel algorithm for numerically computing the location of the optimal points, which is independent of the shape of the domain and does not require computations with Vandermonde matrices. We perform a numerical study of the growth of the minimal Lebesgue constant with respect to the degree of the polynomials and the dimension, and report the lowest values known as yet of the Lebesgue constant in the unit cube and the unit ball in up to 10 dimensions.

Multilevel Fast Multipole Algorithm With Global and Local Interpolators

Abstract: In this paper a new version of the multilevel fast multipole algorithm (MLFMA), based on an interpolator that utilizes trigonometric polynomial expansion, is presented. This novel version allows in levels, where the cube size is large, the use of the local interpolator that utilizes the Lagrange interpolating polynomial. The accuracy of aggregation and disaggregation is improved by choosing the sample rates for each cube according to the distribution of the basis functions inside the cube in question. Additionally an algorithm for an efficient evaluation of the translator is presented. Further some optimizations and good compromises, both in terms of memory and CPU-time, are suggested. Hierarchical matrices are employed to reduce the memory requirements of the sparse block system matrix. The provided numerical results demonstrate that the presented version is able to maintain good accuracy provided that the local interpolator is only utilized in levels where cube size is large enough.

Iterative sine-wave frequency estimation by generalized Fourier interpolation algorithms

Abstract: In this paper two iterative interpolation algorithms proposed in the scientific literature for estimating the frequency of complex-valued sine-waves are generalized to a generic Maximum Sidelobe Decay (MSD) window in order to achieve highly accurate estimates even when real-valued pure or harmonically distorted sine-waves are analyzed. The analytical expressions for the frequency estimations formulas are derived. Moreover the accuracy achieved when pure, noisy, and noisy and harmonically distorted sine-waves are analyzed is compared with those provided by the classical Interpolated Discrete Fourier Transform (IpDFT) and three-point IpDFT algorithms through computer simulations. The performed comparison allows us to determine in which situations the proposed algorithms can be advantageously used.

p-adic interpolation of automorphic periods for GL(2)

Abstract: We give a new and representation theoretic construction of $p$-adic interpolation series for central values of self-dual Rankin-Selberg $L$-functions for $\operatorname{GL}_2$ in dihedral towers of CM fields, using expressions of these central values as automorphic periods The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local We give the construction here for a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real field corresponding to a $\mathfrak{p}$-ordinary Hilbert modular forms of parallel weight two and trivial character, although a similar approach can be taken in any setting where the underlying $\operatorname{GL}_2$-representation can be chosen to take values in a discrete valuation ring A certain choice of vectors allows us to establish a precise interpolation formula thanks to theorems of Martin-Whitehouse and File-Martin-Pitale Such interpolation formulae had been conjectured by Bertolini-Darmon in antecedent works Our construction also gives a conceptual framework for the nonvanishing theorems of Cornut-Vatsal in that it describes the underlying theta elements To highlight this latter point, we describe how the construction extends in the parallel weight two setting to give a $p$-adic interpolation series for central derivative values when the root number is generically equal to $-1$, in which case the formula of Yuan-Zhang-Zhang can be used to give an interpolation formula in terms of heights of CM points on quaternionic Shimura curves

Local Polynomial and Penalized Trigonometric Series Regression

Abstract: We investigate the connections between local polynomial regression, mixed models, and penalized trigonometric series regression. Expressing local polynomial regression in a projection framework, we derive equivalent kernels for both the interior and boundary points. For interior points, it is shown that the asymptotic bias decreases as the order of polynomial increases. Then we show that, under some conditions, the local polynomial projection approach admits an equivalent mixed model formulation where the fixed effects part includes the polynomial functions. The random effects part in the representation is shown to be the trigonometric series asymptotically. The connections are extended to partial linear models and additive models. These results suggest a new smoothing approach using a combination of unpenalized polynomials and penalized trigonometric functions. We illustrate the potential usefulness of the new approach with real data analysis.

Fractional Calculus of Fractal Interpolation Function on

Abstract: The paper researches the continuity of fractal interpolation function’s fractional order integral on and judges whether fractional order integral of fractal interpolation function is still a fractal interpolation function on or not. Relevant theorems of iterated function system and Riemann-Liouville fractional order calculus are used to prove the above researched content. The conclusion indicates that fractional order integral of fractal interpolation function is a continuous function on and fractional order integral of fractal interpolation is still a fractal interpolation function on the interval .

A New (Proposed) Formula for Interpolation and Comparison with Existing Formula of Interpolation

Abstract: The word “interpolation” originates from the Latin verb interpolare, a contraction of “inter,” meaning “between,” and “polare,” meaning “to polish.” That is to say, to smooth in between given pieces of information. A number of different methods have been developed to construct useful interpolation formulas for evenly and unevenly spaced points. The aim of this paper is to develop a central difference interpolation formula which is derived from Gauss’s Backward Formula and another formula in which we retreat the subscripts in Gauss’s Forward Formula by one unit and replacing by . Also, we make the comparisons of the developed interpolation formula with the existing interpolation formulas based on differences. Results show that the new formula is very efficient and posses good accuracy for evaluating functional values between given data. Keywords: Interpolation, Central Difference, Gauss’s Formula.

Lagrange interpolation for signal reconstruction in event-based GPC

Abstract: This work presents the application of Lagrange interpolation method for a signal reconstruction in event-based Generalized Predictive Control (GPC). The event-based control system is governed by level crossing sampling techniques, which monitors the controlled variable. The Lagrange interpolation method is used to reconstruct signal values between two consecutive events. The interpolated signal values are used to unify the obtained values to a base signal, which is resampled with fixed frequency. The developed event-based GPC with Lagrange interpolation is verified through a simulation study, considering several process models commonly used in industrial applications. The obtained results show a proper operation of the event-based controllers, due to good interpolation accuracy.

Application of new family of atomic functions ch a, n to solution of boundary value problems

Abstract: Atomic functions present an effective mathematical apparatus for interpolation of functions. Given function is represented as a sum of scaled shifts of compactly supported atomic function. Fundamental property of such interpolation is that derivatives of given function are simultaneously interpolated with corresponding derivatives of interpolation. This property allows application of atomic function to numerical solution of differential equations.

Application of integrodifferential splines to solving an interpolation problem

Abstract: This paper deals with cases when the values of derivatives of a function are given at grid nodes or the values of integrals of a function over grid intervals are known. Polynomial and trigonometric integrodifferential splines for computing the value of a function from given values of its nodal derivatives and/or from its integrals over grid intervals are constructed. Error estimates are obtained, and numerical results are presented.

Influence of the choice of Lagrange interpolation nodes on the exact and approximate values of the Lebesgue constants

Abstract: The behavior of the Lebesgue constants corresponding to two classical Lagrange interpolation polynomials is studied in dependence on the number of interpolation nodes uniformly distributed on the period divided into three classes. We obtain new exact and approximate formulas for the constants corresponding to each of these classes: the errors are estimated uniformly in the degree of a polynomial. Two standing problems are solved in interpolation theory that are connected with asymptotic equalities for Lebesgue constants.

Three-dimensional modeling method based on vector closure

Abstract: The invention discloses a three-dimensional modeling method based on vector closure The three-dimensional modeling method comprises the specific steps of obtaining seed points to form a region network, fitting a fault curved surface by use of a fault Delaunay triangulation network interpolation algorithm and fitting a position curved surface by use of a position Delaunay trigonometric interpolation algorithm and a complementary interpolation algorithm The three-dimensional modeling method based on vector closure has the advantages that the Delaunay triangulation network is introduced to preprocess seed data so that Kriging interpolation can be applied at higher efficiency, meanwhile, a method of directly obtaining points of intersection is adopted so that the complex steps of manually drawing fault intersecting lines are avoided, the modeling accuracy is improved and the vector closure of the obtained model can be realized

On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation

Abstract: We consider optimal Lagrange interpolation with polynomials of degree at most two on the unit interval [−1, 1]. In a largely unknown paper, Schurer (1974, Stud. Sci. Math. Hung. 9, 77-79) has analytically described the infinitely many zero-symmetric and zero-asymmetric extremal node systems −1 ≤ x1 < x2 < x3 ≤ 1 which all lead to the minimal Lebesgue constant 1.25 that had already been determined by Bernstein (1931, Izv. Akad. Nauk SSSR 7, 1025-1050). As Schurer’s proof is not given in full detail, we formally verify it by providing two new and sound proofs of his theorem with the aid of symbolic computation using quantifier elimination. Additionally, we provide an alternative, but equivalent, parameterized description of the extremal node systems for quadratic Lagrange interpolation which seems to be novel. It is our purpose to bring the computer-assisted solution of the first nontrivial case of optimal Lagrange interpolation to wider attention and to stimulate research of the higher-degree cases. This is why our style of writing is expository.

On a Pointwise Convergence of Quasi-Periodic-Rational Trigonometric Interpolation

Abstract: We introduce a procedure for convergence acceleration of the quasi-periodic trigonometric interpolation by application of rational corrections which leads to quasi-periodic-rational trigonometric interpolation. Rational corrections contain unknown parameters whose determination is important for realization of interpolation. We investigate the pointwise convergence of the resultant interpolation for special choice of the unknown parameters and derive the exact constants of the main terms of asymptotic errors.

Hermite Interpolation on the Unit Circle Considering up to the Second Derivative

Abstract: The paper is devoted to study the Hermite interpolation problem on the unit circle. The interpolation conditions prefix the values of the polynomial and its first two derivatives at the nodal points and the nodal system is constituted by complex numbers equally spaced on the unit circle. We solve the problem in the space of Laurent polynomials by giving two different expressions for the interpolation polynomial. The first one is given in terms of the natural basis of Laurent polynomials and the remarkable fact is that the coefficients can be computed in an easy and efficient way by means of the Fast Fourier Transform (FFT). The second expression is a barycentric formula, which is very suitable for computational purposes.

Optimal boundary interpolation and one-block optimal control problem with invariant zeros on the imaginary axis and infinity

Abstract: We formulate a matrix interpolation problem with existing interpolation points on the imaginary axis and infinity and existing equal left and right interpolation points, using the concept of parametrisation of stabilising controllers. Then, we solve the problem of obtaining all its solutions. If interpolation points at infinity are absent, we show that the introduced problem is equivalent to the existing one. We apply this result to solve the problem of optimal interpolation with existent interpolation points on the imaginary axis and infinity. We show by an example that the solution of optimal interpolation is directly applicable to the one-block optimal control with existent invariant zeros on the imaginary axis and infinity. It is seen from the example that not only the transfer matrix of the closed-loop system is constrained on the extended imaginary axis, but also its derivatives.

Positivity Preserving Interpolation of Positive Data by Rational Quadratic Trigonometric Spline

Abstract: In Computer Aided Geometric Design it is often needed to produce a positivity preserving curve according to the given positive data. The main focus of this work is to visualize the positive data in such a way that its display looks smooth and pleasant. A rational quadratic trigonometric spline function with three shape parameters has been developed. In the description of the rational quadratic trigonometric spline interpolant, positivity is preserved everywhere. Constraints are derived for shape parameters to preserve the positivity thorough positive data. The curves scheme under discussion is attained C 1 continuity.

Interpolation between Hilbert spaces

Abstract: This note comprises a synthesis of certain results in the theory of exact interpolation between Hilbert spaces. In particular, we examine various characterizations of interpolation spaces and their relations to a number of results in operator-theory and in function-theory.