Multivariate Polynomial Interpolation Research Articles

Dimensional reduction for multivariate Lagrange polynomial interpolation problems

Dimensional reduction for multivariate Lagrange polynomial interpolation problems

Surrogate model based on hierarchical sparse polynomial interpolation for the phosphate ore dissolution

This paper deals with the development of accurate surrogate models for first-principles models constructed for the dissolution of phosphate ore in a phosphoric acid solution. The surrogate models are based on sparse multivariate polynomial interpolation. The main goal is to reduce the computational time of the first-principles model while preserving its properties, namely the outputs’ monotonicity and positivity. The temporal profiles of the concentrations of the different components involved in the liquid phase and in the ore particles, the particle size, and the thickness of the liquid film surrounding the particles are approximated by the surrogate models. The inputs of the models are the particle size distribution, the initial acid concentration, and the hydrodynamic conditions. A design of experiments method is used to produce the required sampling points for the surrogate models and several simulations are conducted in the MATLAB environment. A final comparison of the proposed surrogate models’ performance demonstrates high efficiency, both in terms of accuracy as well as computation time, and highlights the strength of the introduced surrogate modeling methodology.

Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach

The problems of polynomial interpolation with several variables present more difficulties than those of one-dimensional interpolation. The first problem is to study the regularity of the interpolation schemes. In fact, it is well-known that, in contrast to the univariate case, there is no universal space of polynomials which admits unique Lagrange interpolation for all point sets of a given cardinality, and so the interpolation space will depend on the set Z of interpolation points. Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain schemes. In this work, we propose a random algorithm for computing several interpolating multivariate Lagrange polynomials, called RLMVPIA (Random Lagrange Multivariate Polynomial Interpolation Algorithm), for any finite interpolation set. We will use a Newton-type polynomials basis, and we will introduce a new concept called Z , z -partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage.

Numerical differentiation on scattered data through multivariate polynomial interpolation

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

Finite difference method based on polynomial interpolation for solving Helmholtz equation

The generalized finite difference formula plays an important role in the meshless method for solving differential equations. The main purpose of this paper is to find the numerical solution of Helmholtz equation, an elliptic partial differential equation describing electromagnetic waves in physics, by using the generalized finite difference formula. Firstly, this paper introduces a simple and practical nodal distribution, which not only guarantees the uniqueness of multivariate polynomial interpolation, but also makes the matrix triangular, so that the constructed basic polynomials can be transformed into Lagrange basis polynomials. Secondly, the generalized finite difference formula is created by polynomial interpolation. Finally, a numerical example of Helmholtz equation under general boundary conditions is given.

The Representation of D-Invariant Polynomial Subspaces Based on Symmetric Cartesian Tensors

Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).

Polynomial-division-based algorithms for computing linear recurrence relations

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence.Several algorithms solve this problem. The so-called Berlekamp–Massey–Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process.We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp–Massey–Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations.A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial.As an addition from the paper published at the ISSAC conference, we give an adaptive variant of this algorithm taking into account the shape of the final Gröbner basis gradually as it is discovered. The main advantage of this algorithm is that its complexity in terms of operations and sequence queries only depends on the output Gröbner basis.All these algorithms have been implemented in Maple and we report on our comparisons.

First order polarization tensor approximation using multivariate polynomial interpolation method via least square minimization technique

This paper proposes a new numerical approach useful in dealing with nearly singular integrals, specifically, the integral of the first order polarization tensor (PT). Polarization tensor represents the integral equations in an asymptotic series, and it can also define the boundary value problem of a partial differential equation (PDE). Since PT has been widely used and implemented in many engineering areas, particularly electric and magnetic field areas, it is crucial to estimate the first order PT solutions accurately. In this regard, the computation of PT for different geometry types is basically from the quadratic interpolation and the multivariate polynomial fitting using the least square method. The numerical calculation of the integral of the singular integral operator, ?? ∗ which is one of the primary integral processes before we obtained the solution of PT uses the multivariate polynomial fitting. This paper aims to provide an accurate numerical solution for first order PT for different geometry types, particularly sphere and ellipsoid geometry. The numerical results of the proposed method are shown together with the comparison of its analytical solutions. From the results obtained, the numerical solution of first order PT shows higher accuracy and higher convergence as the number of surface elements increases. The numerical and the analytical solution of first order PT for a sphere is discussed and represented in graphical form. The utilization of two different software types throughout this study is Netgen Mesh Generator and MATLAB to aid the numerical computation process. The simulation and the numerical examples verify the effectiveness and efficiency of the proposed method.

RMVPIA: a new algorithm for computing the Lagrange multivariate polynomial interpolation

The problems of univariate polynomial interpolation of Lagrange or Hermite have been treated by several recent researches (Gasca and Lopez-Carmona J. Approx. Theory. 34 361–374 1982; Messaoudi et al. Numer. Algorithms J 80, 253–278 2019; Messaoudi and Sadok Numer. Algorithms J 76, 675–694 2017; Muhlbach Numer. Math. 31, 97–110 1978). The study of the multivariate polynomial interpolation is more difficult and the approaches are less obvious (Gasca and Lopez-Carmona J. Approx. Theory. 34, 361–374 1982; Gasca and Sauer 2000; Lorentz 2000; Muhlbach Numer. Math. 31, 97–110 1978; Neidinger Siam Rev. 61, 361–381 2019). In Gasca and Sauer (2000), there are a large number of interesting theoretical ideas developed around the theme in the last years of the last century. The numerical schemes proposed are based on the Newton formulas. Recently in (Siam Rev. 61, 361–381 2019), R.D Neidinger has studied the multivariate polynomial interpolation problem using the techniques of Newton’s polynomial interpolation and the divided difference. In this work, we propose another approach to study the problem of the Lagrange multivariate polynomial interpolation in a particular case where the set of the interpolation nodes is a grid. Indeed, to solve this problem, we will use the Schur complement (Brezinski J. Comput. Appl. Math. 9, 369–376 1983; Brezinski Linear Algebra Appl. 111, 231–247 1988; Cottle Linear Algebra Appl. 8, 189–211 1974; Ouellette Linear Algebra Appl. 36, 187–295 1981; Schur J. Reine. Angew. Math. 147, 205–232 1917) and we will give a new algorithm for computing the interpolating polynomial which will be called the Recursive MultiVariate Polynomial Interpolation Algorithm: RMVPIA. A simplified version and some properties of this algorithm will be also studied and some examples will be given.

Robust computation methods for sparse interpolation of multivariate polynomials

For the problem of sparse interpolation of multivariate polynomials, we propose robust computation methods based on the modified numerical Ben-Or/Tiwari algorithm by M. Giesbrecht, G. Labahn, and W.-s. Lee.

Sliding mode control for a grid connected PV-system using interpolation polynomial MPPT approach

In this paper, two controllers are proposed to optimize the photovoltaic (PV) energy injected into three phase grid. The first controller is used to predict the DC voltage that allows the three-phase inverter to track the maximum power point of photovoltaic generator under unbalanced climatic conditions and variable load. This new controller uses a multivariate polynomial interpolation based on the Lagrange’s theory. The second controller is based on the robust Sliding Mode Control (SMC). It is used to control the active and reactive powers injected into the network in order to achieve the maximum active power into grid with high efficiency and minimum total harmonic distortion using the same three-phase inverter. The control strategies efficiency is confirmed by simulations and experimental results. In addition, in order to validate the choice of control strategies, a comparative study was carried out.

A Meshless Finite Difference Method Based on Polynomial Interpolation

The finite difference (FD) formula plays an important role in the meshless methods for the numerical solution of partial differential equations. It can be created by polynomial interpolation, however, this idea has not been widely used due to the complexity of multivariate polynomial interpolation. Instead, radial basis functions interpolation is widely used to generate the FD formula. In this paper, we first propose a simple and practicable node distribution, which makes it convenient for one to choose the interpolation node set that guarantee the unique solvability of multivariate polynomial interpolation. The greatest advantage of the interpolation node set is that we can only face triangular matrix in order to obtain the Lagrange basis polynomials by the constructed basis polynomials. We then use Taylor’s formula to establish error estimates of the FD formula based on polynomial interpolation. We finally give some numerical experiments for the numerical solutions of the Poisson equation and the heat equation.

Robust algorithms for sparse interpolation of multivariate polynomials

We consider the problem of symbolic-numeric sparse interpolation of multivariate polynomials. The problem is to find the coefficients and the exponents of a given black-box polynomial [EQUATION] by evaluating the value of f ( x 1 ,..., x n ) at any point in C n in floating-point arithmetic and by using the conditions of the input.

Multivariate Polynomial Interpolation in Newton Forms

Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain regions. While this idea, known as the tensor product case, has been around for over a century, this exposition uses modern multi-index notation to provide generality, concise algorithms, and rigor, appropriate as a topic in introductory numerical analysis. Node configurations where the tensor product case applies are called lower sets, and they include $n$-dimensional rectangles (boxes) and triangles (corners), the latter having unique interpolation over polynomials of fixed degree. Arbitrary distinct nodes do not ensure unique interpolating polynomials but, when possible, a different basis generalization, which we call Newton--Sauer, results in a nice triangular linear system for the coefficients. For the right number of distinct nodes, an algorithm based on Gaussian elimination produces either this Newton--Sauer basis or a polynomial that is zero on all nodes, showing that unique interpolation is impossible. The algorithm may also be used on any distinct nodes to produce a polynomial subspace of minimal degree where unique interpolation is possible. A variation of this algorithm using matrix block operations produces another basis generalization that we call Newton--Olver, which exactly agrees with the classic one for a corner of nodes and computes an interpolating polynomial using formulas similar to divided differences.

Multivariate Polynomial Interpolation Based Indoor Fingerprinting Localization Using Bluetooth

Comparing with mature outdoor positioning technology, the demand for indoor positioning is also becoming increasingly extensive. Fingerprinting-based localization using Bluetooth is a promising method for indoor localization. The localization accuracy mainly depends on the quality of fingerprints database, and we have to collect a larger number of training sites in order to achieve finer-grained positioning. However, the site survey for constructing a fine-grained fingerprints database is very time consuming and labor intensive. In order to improve localization accuracy under the condition of without additional fingerprints, we use multivariate polynomial function to construct the fine-grained interpolated fingerprints database derived from the coarse-grained collected fingerprints. Extensive simulation results verify that the proposed multivariate polynomial function interpolation based fingerprinting localization method can improve localization accuracy effectively.

Correspondence of Trap Spaces in Different Models of Bioregulatory Networks

Mathematical models for bioregulatory networks can be based on different formalisms, depending on the quality of available data and the research question to be answered. Discrete Boolean models can be constructed based on qualitative data, which are frequently available. On the other hand, continuous models in terms of ordinary differential equations (ODEs) can incorporate time-series data and give more detailed insight into the dynamics of the underlying system. A few years ago, a method based on multivariate polynomial interpolation and Hill functions was developed for an automatic conversion of Boolean models to systems of ODEs. This method is frequently used by modelers in systems biology today, but there are only a few results available about the conservation of mathematical structures and properties across the formalisms. Here, we consider subsets of the phase space where some components stay fixed, called trap spaces, and demonstrate how Boolean trap spaces can be linked to invariant sets in the continuous state space. This knowledge is of practical relevance since finding trap spaces in the Boolean setting, which is relatively easy, allows for the construction of reduced ODE models.

Quantum algorithm for multivariate polynomial interpolation.

How many quantum queries are required to determine the coefficients of a degree-d polynomial in n variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields [Formula: see text], [Formula: see text] and [Formula: see text]. We show that [Formula: see text] and [Formula: see text] queries suffice to achieve probability 1 for [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] except for d=2 and four other special cases. For [Formula: see text], we show that ⌈(d/(n+d))(n+dd) ⌉ queries suffice to achieve probability approaching 1 for large field order q. The classical query complexity of this problem is (n+dd) , so our result provides a speed-up by a factor of n+1, (n+1)/2 and (n+d)/d for [Formula: see text], [Formula: see text] and [Formula: see text], respectively. Thus, we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of 2. For the case of [Formula: see text], we conjecture that [Formula: see text] queries also suffice to achieve probability approaching 1 for large field order q, although we leave this as an open problem.

Computing Sparse GCD of Multivariate Polynomials via Polynomial Interpolation

The problem of computing the greatest common divisor (GCD) of multivariate polynomials, as one of the most important tasks of computer algebra and symbolic computation in more general scope, has been studied extensively since the beginning of the interdisciplinary of mathematics with computer science. For many real applications such as digital image restoration and enhancement, robust control theory of nonlinear systems, L1-norm convex optimization in compressed sensing techniques, as well as algebraic decoding of Reed-Solomon and BCH codes, the concept of sparse GCD plays a core role where only the greatest common divisors with much fewer terms than the original polynomials are of interest due to the nature of problems or data structures. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of Zippel’s method and Ben-Or/Tiwari algorithm, respectively. To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate GCD computation and univariate polynomial interpolation. The authors demonstrate the practical performance of our algorithms on a significant body of examples. The implemented experiment illustrates that our algorithms are efficient for a quite wide range of input.

Multivariate polynomial interpolation using even and odd polynomials

The Lebesgue constant is a measure for the stability of the Lagrange interpolation. The decomposition of the Lagrange interpolation operator in their even and odd parts with respect to the last variable can be used to find a relation between the Lebesgue constant for a space of polynomials and the corresponding Lebesgue constants for subspaces of even and odd polynomials. It is shown that such a decomposition preserves the stability properties of the Lagrange interpolation operator. We use the Lebesgue functions to provide pointwise quantitative measures of the stability properties and illustrate with examples the behaviour in simple cases.

A case of multivariate Birkhoff interpolation using high order derivatives

We consider a specific scheme of multivariate Birkhoff polynomial interpolation. Our samples are derivatives of various orders kj at fixed points vj along fixed straight lines through vj in directions uj, under the following assumption: the total number of sampled derivatives of order k,k=0,1,… is equal to the dimension of the space homogeneous polynomials of degree k. We show that this scheme is regular for general directions. Specifically this scheme is regular independent of the position of the interpolation nodes. In the planar case, we show that this scheme is regular for distinct directions.Next we prove a “Birkhoff-Remez” inequality for our sampling scheme extended to larger sampling sets. It bounds the norm of the interpolation polynomial through the norm of the samples, in terms of the geometry of the sampling set.