Multivariate Polynomial Approximation Research Articles

Construction of Multivariate Polynomial Approximation Kernels via Semidefinite Programming

In this paper we construct a hierarchy of multivariate polynomial approximation kernels for uniformly continuous functions on the hypercube via semidefinite programming. We give details on the implementation of the semidefinite programs defining the kernels. Finally, we show how symmetry reduction may be performed to increase numerical tractability.

Comparison of Artificial Neural Network and Polynomial Approximation Models for Reflectance Spectra Reconstruction

Knowledge of surface reflection of an object is essential in many technological fields, including graphics and cultural heritage. Compared to direct multi- or hyper-spectral capturing approaches, commercial RGB cameras allow for a high resolution and fast acquisition, so the idea of mapping this information into a reflectance spectrum (RS) is promising. This study compared two modelling approaches based on a training set of RGB-reflectance pairs, one implementing artificial neural networks (ANN) and the other one using multivariate polynomial approximation (PA). The effect of various parameters was investigated: the ANN learning algorithm-standard backpropagation (BP) or Levenberg-Marquardt (LM), the number of hidden layers (HLs) and neurons, the degree of multivariate polynomials in PA, the number of inputs, and the training set size on both models. In the two-layer ANN with significantly fewer inputs than outputs, a better MSE performance was found where the number of neurons in the first HL was smaller than in the second one. For ANNs with one and two HLs with the same number of neurons in the first layer, the RS reconstruction performance depends on the choice of BP or LM learning algorithm. RS reconstruction methods based on ANN and PA are comparable, but the ANN models' better fine-tuning capabilities enable, under realistic constraints, finding ANNs that outperform PA models. A profiling approach was proposed to determine the initial number of neurons in HLs-the search centre of ANN models for different training set sizes.

Multivariate approximation by polynomial and generalized rational functions

In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables.

Correction/Addendum to “The extremal function for the complex ball for generalized notions of degree and multivariate polynomial approximation” (Ann. Polon. Math. 123 (2019), 171–195)

Correction/Addendum to “The extremal function for the complex ball for generalized notions of degree and multivariate polynomial approximation” (Ann. Polon. Math. 123 (2019), 171–195)

Chebyshev Multivariate Polynomial Approximation and Point Reduction Procedure

We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case.

Market Adaptive Control Function Optimization in Continuous Cover Forest Management

Economically optimal management of a continuous cover forest is considered here. Initially, there is a large number of trees of different sizes and the forest may contain several species. We want to optimize the harvest decisions over time, using continuous cover forestry, which is denoted by CCF. We maximize our objective function, the expected present value, with consideration of stochastic prices, timber quality variations and dynamically changing spatial competition. The problem is solved using an adaptive control function. The parameters of the control function are optimized via the first order optimum conditions based on a multivariate polynomial approximation of the objective function. The second order maximum conditions are investigated and used to determine relevant parameter ranges. The procedure is described and optimal results are derived for a general function multi-species CCF forest. Concrete examples from Germany, with beech, and from Sweden, with Norwegian spruce, are used to illustrate the methodology and typical numerical results. It is important to make market adapted harvest decisions. If the stochastic price variations are not considered when the harvest decisions are taken, the expected present value is reduced by 23%.

Weierstrass type approximation by weighted polynomials in [formula omitted

In this paper we consider weighted polynomial approximation on unbounded multidimensional domains in the spirit of the weighted version of the Weierstrass trigonometric theorem according to which every continuous function on the real line with equal finite limits at ±∞ is a uniform limit on R of weighted algebraic polynomials of degree 2n with varying weights (1+t2)−n. We will verify a similar statement in the multivariate setting for a general class of convex weights.We also consider the similar problem of multivariate polynomial approximation with varying weights for some typical non convex weights. In case of non convex weights of the form wα(x)≔(1+|x1|α+...+|xd|α)1α,0<α<1 in order for weighted polynomial approximation to hold for a given continuous function it is necessary that the function vanishes on a certain exceptional set consisting of all coordinate hyperplanes and ∞. Moreover, in case of rational α this condition is also sufficient for weighted polynomial approximation to hold.

Control function optimization for stochastic continuous cover forest management

Economically optimal management of continuous cover forests can be obtained via a new approach to adaptive control function optimization We maximize our objective function the expected present value with consideration of stochastic prices timber quality variations and dynamically changing spatial competition The parameters of the control function are optimized via the first order optimum conditions based on a multivariate polynomial approximation of the objective function The second order maximum conditions are investigated and used to determine relevant parameter ranges The procedure is described and optimal results are derived for general function multi species CCF forests A numerically specified model with empirical data showed that if the stochastic price variations are not considered when the harvest decisions are taken the expected present value is reduced by

The extremal function for the complex ball for generalized notions of degree and multivariate polynomial approximation

We discuss the Siciak–Zakharyuta extremal function of pluripotential theory for the unit ball in ${\mathbb {C}}^d$ for spaces of polynomials with the notion of degree determined by a convex body $P.$ We then use it to analyze the approximation properties

Receiver Techniques for Diffusion-Based Molecular Nano Communications Using an Adaptive Neuro-Fuzzy-Based Multivariate Polynomial Approximation

Molecular communication (MC) nanonetworks, the interconnection of biological nanomachines (Bio-NMs), are envisaged to significantly expand the applications of nano-technology in the areas of the biomedical, material science, and electrical engineering. In this paper, we consider the scenario of a diffusion-based molecular communication (DMC) system that is a promising bio-inspired approach to implementing nanonetwork systems. Considering the limited computation capability, ultra-low energy assumption, and expensive information exchange cost among Bio-NMs, this paper proposes a pulse-based demodulation scheme in three biologically inspired techniques for the detection of the molecular pulses: 1) a multivariate polynomial approximation (MPA) scheme; 2) an adaptive fuzzy threshold-based detection (AFTD) scheme; and 3) an adaptive neuro-fuzzy-based MPA (ANF-MPA) scheme. These methods are suitable for binary ON- OFF keying (BOOK signaling), where the Rx Bio-NM adapts the 1/0-bit detection threshold based on the previously received bits that help to alleviate the inter-symbol interference (ISI) problem resulting from residual (tail) diffusion molecules arriving at the receiver due to past bit transmissions and reception noise. To evaluate the performance of DMC systems, the essential communication metrics in each detection scheme are identified, and the numerical results are obtained and validated by MATLAB simulation. Moreover, the complexity of the proposed schemes is calculated, and the performance evaluation in various noisy channel sources shows a promising improvement in the un-coded bit error rate (BER) performance compared with the threshold detection schemes in the literature. These results provide insights that may guide the implementation of future DMC nanonetworks.

Computation of induced orthogonal polynomial distributions

We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad class of measures, encompassing those associated to classical orthogonal polynomial families, which is stable for polynomial degrees up to at least 1000. Paired with other standard tools such as a numerical root-finding algorithm and inverse transform sampling, this provides a methodology for generating random samples from an induced orthogonal polynomial measure. Generating samples from this measure is one ingredient in optimal numerical methods for certain types of multivariate polynomial approximation. For example, sampling from induced distributions for weighted discrete least-squares approximation has recently been shown to yield convergence guarantees with a minimal number of samples. We also provide publicly-available code that implements the algorithms in this paper for sampling from induced distributions.

Multivariate polynomial approximation in the hypercube

A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the s s -dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial, but by the Euclidean degree, defined in terms of the 2-norm rather than the 1-norm of the exponent vector k \mathbf {k} of a monomial x 1 k 1 ⋯ x s k s x_1^{k_1}\cdots \kern .8pt x_s^{k_s} .

Polynomial approximation and quadrature on geographic rectangles

Using some recent results on subperiodic trigonometric interpolation and quadrature, and the theory of admissible meshes for multivariate polynomial approximation, we study product Gaussian quadrature, hyperinterpolation and interpolation on some regions of Sd,d ≥ 2. Such regions include caps, zones, slices and more generally spherical rectangles defined on S2 by longitude and (co)latitude (geographic rectangles). We provide the corresponding Matlab codes and discuss several numerical examples on S2.

Performance modelling for wind turbines operating in harsh conditions

This paper presents a new method of online wind turbine performance modelling (recursive parameter estimation) that addresses the nonlinearity associated with wind turbine performance characteristics. A sliding linearization algorithm is implemented to track changes in the turbine operating environment. A multivariate polynomial approximation of the turbine power coefficient is developed to produce a linear process model approximating the operating wind turbine. The estimated model parameters are recursively calculated to compensate for changes in both the turbine operating environment and the condition of the wind turbine. The algorithm models both the steady-state and dynamic wind turbine performance throughout the entire operating range, producing a continuously valid turbine linearization with applications in gain scheduling and turbine performance optimization. Copyright © 2016 John Wiley & Sons, Ltd.

Compressed sampling inequalities by Tchakaloff's theorem

We show that a discrete version of Tchakaloff’s theorem on the existence of positive algebraic cubature formulas, entails that the information required for multivariate polynomial approximation can be suitably compressed.

Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

We study the accuracy of the discrete least-squares approximation on a finite-dimensional space of a real-valued target function from noisy pointwise evaluations at independent random points distributed according to a given sampling probability measure. The convergence estimates are given in mean-square sense with respect to the sampling measure. The noise may be correlated with the location of the evaluation and may have nonzero mean (offset). We consider both cases of bounded or square-integrable noise/offset. We prove conditions between the number of sampling points and the dimension of the underlying approximation space that ensure a stable and accurate approximation. Particular focus is on deriving estimates in probability within a given confidence level. We analyze how the best approximation error and the noise terms affect the convergence rate and the overall confidence level achieved by the convergence estimate. The proofs of our convergence estimates in probability use arguments from the theory of large deviations to bound the noise term. Finally we address the particular case of multivariate polynomial approximation spaces with any density in the beta family, including uniform and Chebyshev.

Porting a sphere optimization program from LAPACK to ScaLAPACK

The sphere optimization program sphopt was originally written as a sequential program using LAPACK, and was converted to use ScaLAPACK, primarily to overcome memory limitations. The conversion was relatively straightforward, using a small number of organizing principles which are widely applicable to the ScaLAPACK parallelization of serial code. The main innovation is the use of a compressed block cyclic storage scheme to store two symmetric matrices in little more than the storage needed for one matrix. The resulting sphopt program scales at least as well as the ScaLAPACK Cholesky decomposition routine which it uses. References E. Anderson, Z. Bai, C. Bischof, L. S. Blackford, J. Demmel, J. E. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users' Guide (third ed.). SIAM, Philadelphia, 1999. http://www.netlib.org/lapack/lug/ G. E. Andrews, R. Askey, and R. Roy. Special Functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2000. L. S. Blackford, J. Choi, A. Cleary, E. D'Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley. ScaLAPACK Users' Guide. SIAM, Philadelphia, 1997. http://www.netlib.org/scalapack/slug/ J. Choi, J. J. Dongarra, L. S. Ostrouchov, A. P. Petitet, D. W. Walker, and R. C. Whaley. The design and implementation of the ScaLAPACK LU, QR, and Cholesky factorization routines. Technical Report 80, LAPACK Working Note, Knoxville, September 1994. http://www.netlib.org/lapack/lawnspdf/lawn80.pdf J. Choi, J. J. Dongarra, R. Pozo, and D. W. Walker. ScaLAPACK: A scalable linear algebra for distributed memory concurrent computers. Technical Report 55, LAPACK Working Note, Knoxville, November 1992. http://www.netlib.org/lapack/lawnspdf/lawn55.pdf E. F. D'Azevedo and J. J. Dongarra. Packed storage extensions for ScaLAPACK. Technical Report 135, LAPACK Working Note, Knoxville, April 1998. http://www.netlib.org/lapack/lawnspdf/lawn135.pdf J. J. Dongarra, R. A. Vandegeijn, and D. W. Walker. Scalability issues affecting the design of a dense linear algebra library, Journal of Parallel and Distributed Computing 22 (3):523--537, September 1994. doi:10.1006/jpdc.1994.1108 J. J. Dongarra and R. C. Whaley. A user's guide to the BLACS v1.1. Technical Report 94, LAPACK Working Note, Knoxville, May 1997. originally released March 1995. http://www.netlib.org/lapack/lawnspdf/lawn94.pdf P. Leopardi. Converting a sphere optimization program from LAPACK to ScaLAPACK, 2004. http://wwwmaths.anu.edu.au/ leopardi/conversion-LAPACK-ScaLAPACK.pdf D. C. Liu and J. Nocedal. On the limited memory BFGS method for large scale optimization. Math. Programming, 45(3, (Ser. B)):503--528, 1989. doi:10.1007/BF01589116 M. Reimer. Multivariate Polynomial Approximation, volume 144 of International Series of Numerical Mathematics. {Birkhauser} Verlag, Basel, 2003. I. H. Sloan and R. S. Womersley. Extremal systems of points and numerical integration on the sphere. Advances in Computational Mathematics, 21:107--125, 2004. doi:10.1023/B:ACOM.0000016428.25905.da D. W. Walker and J. J. Dongarra. MPI: a standard Message Passing Interface. Supercomputer, 12(1):56--68, 1996. http://users.cs.cf.ac.uk/David.W.Walker/MPI/supercomputer96.html C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal. Algorithm 778 L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization. ACM Transactions on Mathematical Software (TOMS), 23 (4): 550--560, December 1997. doi:10.1145/279232.279236

Multivariate Polynomial Approximation of Powers of the Euclidean Distance Function

The polynomial approximation behaviour of the class of functions $$F_s: {\bf R}^2 \backslash \{ (x_0, y_0) \} \to {\bf R}, \quad F_s(x,y) = ( (x-x_0)^2 + (y-y_0)^2 )^{-s},\quad s \in (0, \infty),$$ is studied in [Bra01]. There it is claimed that the obtained results can be embedded in a more general setting. This conjecture will be confirmed and complemented by a different approach than in [Bra01]. The key is to connect the approximation rate of Fs with its holomorphic continuability for which the classical Bernstein approximation theorem is linked with the convexity of best approximants. Approximation results of this kind also play a vital role in the numerical treatment of elliptic differential equations [Sau].

Transient security assessment using multivariate polynomial approximation

This paper presents an efficient method for transient security assessment. The proposed method exploits the computational efficiency of linear estimation methods to determine an accurate nonlinear mapping function. The novelty of the proposed method is that a nonlinear transformation is applied to the original variables, voltage magnitudes and phase angles, so that the mapping function is approximately linear in terms of the transformed variables. The linear function obtained using the transformed variables is indeed nonlinear in terms of original variables. The attractiveness of this method is that the estimated function is not a linearized approximation although a linear estimation method is used. The potential of the proposed method is demonstrated using the New England 39-bus system and a larger power system with 470 buses.

Jackson-type theorems on some transcendental curves in [formula omitted

“Efficient curves” in the sense of the best rate of multivariate polynomial approximation to contractive functions on these curves, were first introduced by D.J. Newman and L. Raymon in 1969. They proved that algebraic curves are efficient, but claimed that the exponential curve γ : = { ( t , e t ) : 0 ⩽ t ⩽ 1 } is not. We prove to the contrary that this exponential curve and its generalization to higher dimensions are indeed efficient. We also investigate helical curves in R d and show that they too are efficient. Transcendental curves of the form { ( t , t λ ) : δ ⩽ t ⩽ 1 } are shown to be efficient for δ > 0 , contradicting another claim of Newman and Raymon.