Uniform Rational Approximation Research Articles

Uniform Rational Approximation of Even and Odd Continuations of Functions

Uniform Rational Approximation of Even and Odd Continuations of Functions

Uniform Rational Approximation of Even and Odd Continuations of Functions

Исследуется поведение наилучших рациональных приближений нечетного продолжения функции. Показано, что без дополнительных условий на гладкость функции оценить наилучшее рациональное приближение нечетного продолжения функции на $[-1,1]$ через наилучшее рациональное приближение исходной функции на $[0,1]$ невозможно. Найдена точная верхняя оценка для наилучших рациональных приближений четного (нечетного) продолжения функции через нечетное (четное) продолжение и экстремальное произведение Бляшке. Библиография: 12 названий.

A Newton method for best uniform rational approximation

A Newton method for best uniform rational approximation

Application of the real Hardy – Sobolev space on the line to study the order of uniform rational approximations of functions

The real space of Hardy – Sobolev on a straight line is considered and some sufficient conditions for belonging to functions to this space are described. Estimates of the norm of functions from this space are also obtained. Various examples of functions from the Hardy – Sobolev space are given and the order of their best uniform rational approximations are investigated. Estimates of the best rational approximations for even and odd continuations of functions with monotonous derivatives are obtained. The order of the best rational approximations of the even and odd continuations of functions in the general case have also been studied. Estimates are given both considering the continuity module and without it. The obtained results are also used to study the best rational approximations of functions with a kink, introduced by A. A. Gonchar.

Computation of generalized matrix functions with rational Krylov methods

We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special case. By exploiting the quasiseparable structure of the projected matrices, we show that the basis vectors can be updated using a short recurrence, which can be seen as a generalization to the rational case of the Golub-Kahan bidiagonalization. We also prove error bounds that relate the error of these methods to uniform rational approximation. The effectiveness of the algorithms and the accuracy of the bounds is illustrated with numerical experiments.

Non-Overlapping Domain Decomposition via BURA Preconditioning of the Schur Complement

A new class of high-performance preconditioned iterative solution methods for large-scale finite element method (FEM) elliptic systems is proposed and analyzed. The non-overlapping domain decomposition (DD) naturally introduces coupling operator at the interface γ. In general, γ is a manifold of lower dimensions. At the operator level, a key property is that the energy norm associated with the Steklov-Poincaré operator is spectrally equivalent to the Sobolev norm of index 1/2. We define the new multiplicative non-overlapping DD preconditioner by approximating the Schur complement using the best uniform rational approximation (BURA) of Lγ1/2. Here, Lγ1/2 denotes the discrete Laplacian over the interface γ. The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the BURA-based non-overlapping DD preconditioner has optimal computational complexity O(n), where n is the number of unknowns (degrees of freedom) of the FEM linear system. All theoretical estimates are robust, with respect to the geometry of the interface γ. Results of systematic numerical experiments are given at the end to illustrate the convergence properties of the new method, as well as the choice of the involved parameters.

On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type

The main aim of this article is to propose a general framework for the theoretical analysis of discrete schemes used to solve multi-dimensional parabolic problems with fractional power elliptic operators. This analysis is split into three parts. The first part is based on techniques well developed for the solution of nonlocal elliptic problems. The obtained discrete elliptic operators are used to formulate semi-discrete approximations. Next, the fully discrete schemes are constructed by applying the classical and robust approximations of time derivatives. The existing stability and convergence results are directly included in the new framework. In the third part, approximations of transfer operators are constructed by using uniform and the best uniform rational approximations. The stability and accuracy of the obtained local discrete schemes are investigated. The results of computational experiments are presented and analyzed. A three-dimensional test problem is solved. The rational approximations are constructed by using the BRASIL algorithm.

Rational Approximations in Robust Preconditioning of Multiphysics Problems

Multiphysics or multiscale problems naturally involve coupling at interfaces which are manifolds of lower dimensions. The block-diagonal preconditioning of the related saddle-point systems is among the most efficient approaches for numerically solving large-scale problems in this class. At the operator level, the interface blocks of the preconditioners are fractional Laplacians. At the discrete level, we propose to replace the inverse of the fractional Laplacian with its best uniform rational approximation (BURA). The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the proposed preconditioners have optimal computational complexity O(N), where N is the number of unknowns (degrees of freedom) of the coupled discrete problem. The main theoretical contribution is the condition number estimates of the BURA-based preconditioners. It is important to note that the obtained estimates are completely analogous for both positive and negative fractional powers. At the end, the analysis of the behavior of the relative condition numbers is aimed at characterizing the practical requirements for minimal BURA orders for the considered Darcy–Stokes and 3D–1D examples of coupled problems.

Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved.

On geometry of numbers and uniform rational approximation to the Veronese curve

Consider the classical problem of rational simultaneous approximation to a point in \({\mathbb {R}}^{n}\). The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided information on the best approximating rational vectors to the points where the gap is close to this minimal value. Combining the latter result with parametric geometry of numbers, we effectively bound the dual linear form exponents in the described situation. As an application, we slightly improve the upper bound for the classical exponent of uniform Diophantine approximation \({\widehat{\lambda }}_{n}(\xi )\), for even \(n\ge 4\). Unfortunately our improvements are small, for \(n=4\) only in the fifth decimal digit. However, the underlying method in principle can be improved with more effort to provide better bounds. We indeed establish reasonably stronger results for numbers which almost satisfy equality in the estimate by Marnat and Moshchevitin. We conclude with consequences on the classical problem of approximation to real numbers by algebraic numbers/integers of uniformly bounded degree.

Projective view at optimization problem for multiband filter

The best uniform rational approximation of the \emph{sign} function on two intervals separated by zero was explicitly solved by E.I. Zolotarev in 1877. This optimization problem is the initial step in the staircase of the so called approximation problems for multiband filters which are of great importance for electrical engineering. We show that known in the literature optimality criterion for this problem may be contradictory since it does not take into account the projective invariance of the problem. We propose a new consistently projective formulation of this problem and give a constructive optimality criterion for it.

An algorithm for best rational approximation based on barycentric rational interpolation

We present a novel algorithm for computing best uniform rational approximations to real scalar functions in the setting of zero defect. The method, dubbed BRASIL (best rational approximation by successive interval length adjustment), is based on the observation that the best rational approximation r to a function f must interpolate f at a certain number of interpolation nodes (xj). Furthermore, the sequence of local maximum errors per interval (xj− 1,xj) must equioscillate. The proposed algorithm iteratively rescales the lengths of the intervals with the goal of equilibrating the local errors. The required rational interpolants are computed in a stable way using the barycentric rational formula. The BRASIL algorithm may be viewed as a fixed-point iteration for the interpolation nodes and converges linearly. We demonstrate that a suitably designed rescaled and restarted Anderson acceleration (RAA) method significantly improves its convergence rate. The new algorithm exhibits excellent numerical stability and computes best rational approximations of high degree to many functions in a few seconds, using only standard IEEE double-precision arithmetic. A free and open-source implementation in Python is provided. We validate the algorithm by comparing to results from the literature. We also demonstrate that it converges quickly in some situations where the current state-of-the-art method, the minimax function from the Chebfun package which implements a barycentric variant of the Remez algorithm, fails.

Rational approximation to real points on quadratic hypersurfaces

Let $Z$ be a quadratic hypersurface of $\mathbb{P}^n(\mathbb{R})$ defined over $\mathbb{Q}$ containing points whose coordinates are linearly independent over $\mathbb{Q}$. We show that, among these points, the largest exponent of uniform rational approximation is the inverse $1/\rho$ of an explicit Pisot number $\rho<2$ depending only on $n$ if the Witt index (over $\mathbb{Q}$) of the quadratic form $q$ defining $Z$ is at most $1$, and that it is equal to $1$ otherwise. Furthermore there are points of $Z$ which realize this maximum. They constitute a countably infinite set in the first case, and an uncountable set in the second case. The proof for the upper bound $1/\rho$ uses a recent transference inequality of Marnat and Moshchevitin. In the case $n=3$, we recover results of the second author while for $n>3$, this completes recent work of Kleinbock and Moshchevitin.

Generalised rational approximation and its application to improve deep learning classifiers

A rational approximation (that is, approximation by a ratio of two polynomials) is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non-Lipschitz functions, where polynomial approximations are not efficient. We prove that the optimisation problems appearing in the best uniform rational approximation and its generalisation to a ratio of linear combinations of basis functions are quasiconvex even when the basis functions are not restricted to monomials. Then we show how this fact can be used in the development of computational methods. This paper presents a theoretical study of the arising optimisation problems and provides results of several numerical experiments. We apply our approximation as a preprocessing step to deep learning classifiers and demonstrate that the classification accuracy is significantly improved compared to the classification of the raw signals.

Neumann fractional diffusion problems: BURA solution methods and algorithms

Let us consider the non-local problem −Lαu=f, α∈(0,1), L is a second order self-adjoint elliptic operator in Ω⊂Rd with Neumann boundary conditions on ∂Ω. The problem is discretized by finite difference or finite element method, thus obtaining the linear system Aαu=f, A is sparse symmetric and positive semidefinite matrix. The proposed method is based on best uniform rational approximations (BURA) of degree k, rα,k, of the scalar function tα, t∈[0,1]. Then, the approximate solution ur of the fractional power linear system is defined as u≈ur=λ2−αrα,k(λ2A†)f, where λ2 is the first positive eigenvalue of A, and A† stands for the Moore–Penrose pseudo inverse of A. The BURA method reduces the non-local problem to solution of k linear systems with matrices A+diI, di>0, i=1,…,k. An exponential convergence rate with respect to k is proven. The error estimates are robust with respect to the condition number of A in the subspace orthogonal to the constant vectors. The algorithm has almost optimal computational complexity assuming that optimal iterative solvers are applied to the auxiliary sparse linear systems. The first group of numerical tests illustrates in detail the obtained theoretical results. Finite difference discretization of a model 2D problem is used for this purpose. The second part of numerical tests demonstrates the applicability of BURA methods for 3D problems in domains with general geometry. Linear finite elements on unstructured tetrahedral meshes with local mesh refinement are used in the presented large-scale experiments, confirming also the almost optimal computational complexity.

Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation

Here we study theoretically and compare experimentally with the methods developed in [1,2] an efficient method for solving systems of algebraic equations A˜αu˜h=f˜h, 0<α<1, where A˜ is an N×N matrix coming from the discretization of a fractional diffusion operator. More specifically, we focus on matrices obtained from finite difference or finite element approximation of second order elliptic problems in Rd, d=1,2,3. The proposed methods are based on the best uniform rational approximation (BURA) rα,k(t) of tα on [0,1]. Here rα,k is a rational function of t involving numerator and denominator polynomials of degree at most k.The approximation of u˜h=A˜−αf˜h is then w˜h=λ1−αrα,k(λ1A˜−1)f˜h, where λ1 is the smallest eigenvalue of A˜. We show that the proposed method is exponentially convergent with respect to k and has some attractive properties. First, it reduces the solution of the nonlocal system to solution of k systems with matrix (A˜+cjI˜) and cj>0, j=1,2,…,k. Thus, good computational complexity can be achieved if fast solvers are available for such systems. Second, the original problem and its rational approximation in the finite difference case are positivity preserving. In the finite element case, this valid for schemes obtained by mass lumping under certain mild conditions on the mesh. Further, we prove that the lumped mass schemes still have the expected rate of convergence, at times assuming additional regularity on the right hand side. Finally, we present comprehensive numerical experiments on a number of model problems for various α in one and two spatial dimensions. These illustrate the computational behavior of the proposed method and compare its accuracy and efficiency with that of other methods developed by Harizanov et al. [1] and Bonito and Pasciak [2].

A unified view of some numerical methods for fractional diffusion

In recent years, a number of numerical methods for the solution of fractional Laplace and, more generally, fractional diffusion problems have been proposed. The approaches are quite diverse and include, among others, the use of best uniform rational approximations, quadrature for Dunford–Taylor-like integrals, finite element approaches for a localized elliptic extension into a space of increased dimensions, and time stepping methods for a parabolic reformulation of the fractional differential equation.A systematic comparison, both theoretical and experimental, of these approaches has thus far been lacking. A main contribution of the present work is the observation that all approaches mentioned above can, in fact, be interpreted as realizing different rational approximations of a univariate function over the spectrum of the original (non-fractional) diffusion operator. While this is obvious for some of the methods, it is a new result in particular for extension-based and time stepping approaches.This observation allows us to cast all described methods into a unified theoretical and computational framework, which has a number of benefits. Theoretically, it enables us to develop new convergence proofs for several of the studied methods, clarifies similarities and differences between the approaches, suggests how to design new and improved methods, and allows a direct comparison of the relative performance of the various methods. Practically, it provides a single, simple to implement, efficient and fully parallel algorithm for the realization of all studied methods; for instance, this immediately yields a fast and memory-efficient way of realizing all tensor product extension methods and lets us parallelize the otherwise inherently sequential time stepping approach.Finally, we present a detailed numerical study comparing all investigated methods for various fractional exponents and draw conclusions from the results. The comparison is made fair by the central insight that the computational effort of all these methods depends only on a single parameter, the degree of the underlying rational approximation. As a point of comparison, we also test a simple rational approximation method based on a black-box direct rational approximation algorithm which performs very well in practice.

Numerical solution of fractional diffusion–reaction problems based on BURA

The paper is devoted to the numerical solution of algebraic systems of the type (Aα+qI)u=f, 0<α<1, q>0, u,f∈RN, where A is a symmetric and positive definite matrix. We assume that A is obtained by finite difference approximation of a second order diffusion problem in Ω⊂Rd, d=1,2 so that Aα+qI approximates the related fractional diffusion–reaction operator or could be a result of a time-stepping procedure in solving time-dependent sub-diffusion problems. We also assume that a method of optimal complexity for solving linear systems with matrices A+cI, c≥0 is available. We analyze and study numerically a class of solution methods based on the best uniform rational approximation (BURA) of a certain scalar function in the unit interval.The first such method, originally proposed in Harizanov et al. (2018) for numerical solution of fractional-in-space diffusion problems, was based on the BURA rα(ξ) of ξ1−α in [0,1] through scaling of the matrix A by its largest eigenvalue. Then the BURA of t−α in [1,∞) is given by t−1rα(t) and correspondingly, A−1rα(A) is used as an approximation of A−α. Further, this method was improved in Harizanov et al. (2019) using the same concept but by scaling the matrix A by its smallest eigenvalue. In this paper we consider the BURA rα(ξ) of 1∕(ξ−α+q) for ξ∈(0,1]. Then we define the approximation of (Aα+qI)−1 as rα(A−α). We also propose an alternative method that uses BURA of ξα to produce certain uniform rational approximation (URA) of 1∕(ξ−α+q). Comprehensive numerical experiments are used to demonstrate the computational efficiency and robustness of the new BURA and URA methods.

Criteria for the individual -approximability of functions on compact subsets of by solutions of second-order homogeneous elliptic equations

Criteria for the individual approximability of functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients in the norms of Whitney-type -spaces on compact subsets of , , are obtained for . These results, which are analogues of Vitushkin's celebrated criteria for uniform rational approximation, were previously established by Mazalov for harmonic approximations (for and ) and by Mazalov and Paramonov for bi-analytic approximation. Bibliography: 11 titles.

Optimal solvers for linear systems with fractional powers of sparse SPD matrices

SummaryIn this paper, we consider efficient algorithms for solving the algebraic equation , 0&lt;α&lt;1, where is a properly scaled symmetric and positive definite matrix obtained from finite difference or finite element approximations of second‐order elliptic problems in , d=1,2,3. This solution is then written as with with β positive integer. The approximate solution method we propose and study is based on the best uniform rational approximation of the function tβ−α for 0&lt;t≤1 and on the assumption that one has at hand an efficient method (e.g., multigrid, multilevel, or other fast algorithms) for solving equations such as , c≥0. The provided numerical experiments confirm the efficiency of the proposed algorithms.