Optimal Rational Approximation Research Articles

Gradient-based structural change detection for nonstationary time series M-estimation

We consider structural change testing for a wide class of time series M-estimation with nonstationary predictors and errors. Flexible predictor-error relationships, including exogenous, state-heteroscedastic and autoregressive regressions and their mixtures, are allowed. New uniform Bahadur representations are established with nearly optimal approximation rates. A CUSUM-type test statistic based on the gradient vectors of the regression is considered. In this paper, a simple bootstrap method is proposed and is proved to be consistent for M-estimation structural change detection under both abrupt and smooth nonstationarity and temporal dependence. Our bootstrap procedure is shown to have certain asymptotically optimal properties in terms of accuracy and power. A public health time series dataset is used to illustrate our methodology, and asymmetry of structural changes in high and low quantiles is found.

Boundary elements with mesh refinements for the wave equation

The solution of the wave equation in a polyhedral domain in $\mathbb{R}^3$ admits an asymptotic singular expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as equivalent boundary integral equations in time domain, study the regularity properties of their solutions and the numerical approximation. Guided by the theory for elliptic equations, graded meshes are shown to recover the optimal approximation rates known for smooth solutions. Numerical experiments illustrate the theory for screen problems. In particular, we discuss the Dirichlet and Neumann problems, as well as the Dirichlet-to-Neumann operator and applications to the sound emission of tires.

Optimal Rational Approximations by the Modified Fourier Basis

We consider convergence acceleration of the modified Fourier expansions by rational trigonometric corrections which lead to modified-trigonometric-rational approximations. The rational corrections contain some unknown parameters and determination of their optimal values for improved pointwise convergence is the main goal of this paper. The goal was accomplished by deriving the exact constants of the asymptotic errors of the approximations with further elimination of the corresponding main terms by appropriate selection of those parameters. Numerical experiments outline the convergence improvement of the optimal rational approximations compared to the expansions by the modified Fourier basis.

Study on A Combined Optimal Rational Approximation Method of Fractional-order Calculus Operator and Its Application

Background: Because the fractional-order calculus operator is an irrational function of the complex variable, it is required to deal with rational approximation. Objective: It is difficult to directly implement the fractional-order calculus operator in the numerical simulation and practical application. Methods: So the minimum amplitude frequency approximation error and phase frequency approximation error of rational approximation function are comprehensively considered in the same rational function class. And the rational approximation function construction method and combined optimal rational approximation method of fractional-order calculus operator based on the function approximation theory and logarithmic frequency characteristic asymptote idea are described in detail. Then, an approximation method of fractional-order calculus operator based on the combined optimal rational approximation is proposed in this paper. The approximation effectiveness of fractional-order calculus operator is analyzed by using s-0.6 . And the combined optimal rational approximation method is applied to the design and digital realization of fractional-order controller. In order to simplify the controller design, the controller structure of parallel connection mode is used and the simplified rational approximation expression and transform of fractionalorder integral operator in the different order is obtained. Results: The analyzed results show that the combined optimal rational approximation method of fractional-order calculus operator is consistent with the ideal c in amplitude frequency domain and has smallest error for the phase frequency characteristic. And the applied results show that the proposed combined optimal rational approximation method can conveniently control and simulate the practical computer control system by analyzing the simplified rational approximation expression and z transform of fractional-order integral operator. Conclusion: So the combined optimal rational approximation method can provide better engineering applications of fractional-order controller in the practical control system. Keywords: Fractional-order calculus, rational approximation of operator, combined optimal rational approximation, discretization, fractional-order PIα Dβ controller, parallel connection structure.

Approximation of Integral Operators Using Product-Convolution Expansions

We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computationally intensive problem necessary for many practical problems. We analyze a technique called product-convolution expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices.

Lattice QCD with N f = 2 + 1 + 1 domain-wall quarks

We perform hybrid Monte Carlo simulation of (2+1+1)-flavors lattice QCD with the optimal domain-wall fermion (which has the effective 4D Dirac operator exactly equal to the Zolotarev optimal rational approximation of the overlap Dirac operator). The gauge ensemble is generated on the $32^3 \times 64 $ lattice with the extent $ N_s = 16 $ in the fifth dimension, and with the plaquette gauge action at $ \beta = 6/g^2 = 6.20 $. The lattice spacing ($ a \simeq 0.063 $ fm) is determined by the Wilson flow, using the value $ \sqrt{t_0} = 0.1416(8) $ fm obtained by the MILC Collaboration for the $(2+1+1)$-flavors QCD. The masses of $s$ and $c$ quarks are fixed by the masses of the vector mesons $ \phi(1020) $ and $ J/\psi(3097) $ respectively; while the mass of the $u/d$ quarks is heavier than their physical values, with the unitary pion mass $ M_\pi \simeq 280$ MeV (and $ M_\pi L \simeq 3 $). We compute the point-to-point quark propagators, and measure the time-correlation functions of meson and baryon interpolators. Our results of the mass spectra of the lowest-lying hadrons containing $ s $ and $ c $ quarks are in good agreement with the high energy experimental values, together with the predictions of the charmed baryons which have not been observed in experiments.

Optimal Approximation of Skorohod Integrals

In this manuscript, we determine the optimal approximation rate for Skorohod integrals of sufficiently regular integrands. This generalizes the optimal approximation results for Ito integrals. However, without adaptedness and the Ito isometry, new proof techniques are required. The main tools are a characterization via S-transform and a reformulation of the Wiener chaos decomposition in terms of Wick-analytic functionals.

Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data

Brenier and Grenier [SIAM J. Numer. Anal., 1998] proved that sticky particle dynamics with a large number of particles allow to approximate the entropy solution to scalar one-dimensional conservation laws with monotonic initial data. In [arXiv:1501.01498], we introduced a multitype version of this dynamics and proved that the associated empirical cumulative distribution functions converge to the viscosity solution, in the sense of Bianchini and Bres-san [Ann. of Math. (2), 2005], of one-dimensional diagonal hyperbolic systems with monotonic initial data of arbitrary finite variation. In the present paper, we analyse the L 1 error of this approximation procedure, by splitting it into the discretisation error of the initial data and the non-entropicity error induced by the evolution of the particle system. We prove that the error at time t is bounded from above by a term of order (1 + t)/n, where n denotes the number of particles, and give an example showing that this rate is optimal. We last analyse the additional error introduced when replacing the multitype sticky particle dynamics by an iterative scheme based on the typewise sticky particle dynamics, and illustrate the convergence of this scheme by numerical simulations.

Multivariate [formula omitted]-molecules

The suboptimal performance of wavelets with regard to the approximation of multivariate data gave rise to new representation systems, specifically designed for data with anisotropic features. Some prominent examples of these are given by ridgelets, curvelets, and shearlets, to name a few.The great variety of such so-called directional systems motivated the search for a common framework, which unites many under one roof and enables a simultaneous analysis, for example with respect to approximation properties. Building on the concept of parabolic molecules, the recently introduced framework of α-molecules does in fact include the previous mentioned systems. Until now however it is confined to the bivariate setting, whereas nowadays one often deals with higher dimensional data. This motivates the extension of this unifying theory to dimensions larger than 2, put forward in this work. In particular, we generalize the central result that the cross-Gramian of any two systems of α-molecules will to some extent be localized.As an exemplary application, we investigate the sparse approximation of video signals, which are instances of 3D data. The multivariate theory allows us to derive almost optimal approximation rates for a large class of representation systems.

Quadrature-based IRKA for optimal H2 model reduction

Iterative Rational Krylov Algorithm (irka) of Gugercin et al. [2008] is an effective tool for optimal H2 rational approximation. Beattie and Gugercin [2012] has recently developed a new formulation of irka that only uses transfer function evaluations, without requiring any particular realization; thus extending irka to H2 approximation of irrational, infinitedimensional dynamical systems. The main computational cost of this new framework is the need for repeated transfer function evaluations. In this paper, based on the reproducing kernel formulation of the underlying Hilbert space, we introduce a quadrature-based formulation of irka where a number of transfer function evaluation are computed only at the beginning (the offline phase) and the irka steps (the online phase) never revisit the original transfer function performing all the computations in the reduced order dimension.

Exact pseudofermion action for Monte Carlo simulation of domain-wall fermion

We present an exact pseudofermion action for hybrid Monte Carlo simulation (HMC) of one-flavor domain-wall fermion (DWF), with the effective 4-dimensional Dirac operator equal to the optimal rational approximation of the overlap-Dirac operator with kernel H=cHw(1+dγ5Hw)−1, where c and d are constants. Using this exact pseudofermion action, we perform HMC of one-flavor QCD, and compare its characteristics with the widely used rational hybrid Monte Carlo algorithm (RHMC). Moreover, to demonstrate the practicality of the exact one-flavor algorithm (EOFA), we perform the first dynamical simulation of the (1+1)-flavors QCD with DWF.

Cascadic multilevel algorithms for symmetric saddle point systems

In this paper, we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems. The algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive definite systems. On each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a Schur complement is implemented. The level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error. We prove new estimates that relate the iteration error and the residual for the constraint equation. The new estimates are the key ingredients in imposing an efficient level change criterion. The first iteration on each new level uses information about the best approximation of the discrete solution from the previous level. The theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non-increasing number of iterations on each level. Even though numerical results supporting the efficiency of the algorithms are presented for the Stokes system, the algorithms can be applied to a large class of boundary value problems, including first order systems that can be reformulated at the continuous level as symmetric saddle point problems, such as the Maxwell equations.

Geometric multiscale decompositions of dynamic low-rank matrices

The present paper is concerned with the study of manifold-valued multiscale transforms with a focus on the Stiefel manifold. For this specific geometry we derive several formulas and algorithms for the computation of geometric means which will later enable us to construct multiscale transforms of wavelet type. As an application we study compression of piecewise smooth families of low-rank matrices both for synthetic data and also real-world data arising in hyperspectral imaging. As a main theoretical contribution we show that the manifold-valued wavelet transforms can achieve an optimal N-term approximation rate for piecewise smooth functions with possible discontinuities. This latter result is valid for arbitrary manifolds.

Strong type of Steckin inequality for the linear combination of Bernstein operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein operators, which gives the optimal approximation rate. Moreover, a method to prove lower estimates for linear operators is introduced. As a result the lower estimate for the linear combinations of Bernstein operators is obtained by using the Ditzian–Totik modulus of smoothness.

Rational approximations for tomographic reconstructions

We use optimal rational approximations of projection data collected in x-ray tomography to improve image resolution. Under the assumption that the object of interest is described by functions with jump discontinuities, for each projection we construct its rational approximation with a small (near optimal) number of terms for a given accuracy threshold. This allows us to augment the measured data, i.e., double the number of available samples in each projection or, equivalently, extend (double) the domain of their Fourier transform. We also develop a new, fast, polar coordinate Fourier domain algorithm which uses our nonlinear approximation of projection data in a natural way. Using augmented projections of the Shepp–Logan phantom, we provide a comparison between the new algorithm and the standard filtered back-projection algorithm. We demonstrate that the reconstructed image has improved resolution without additional artifacts near sharp transitions in the image.

A Krylov Stability-Corrected Coordinate-Stretching Method to Simulate Wave Propagation in Unbounded Domains

The Krylov subspace projection approach is a well-established tool for the reduced-order modeling of dynamical systems in the time domain. In this paper, we address the main issues obstructing the application of this powerful approach to the time-domain solution of exterior wave problems. We use frequency-independent perfectly matched layers to simulate the extension to infinity. Pure imaginary stretching functions based on Zolotarev's optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete spectral interval of interest. A new Krylov-based solution method via stability-corrected operator exponents is presented which allows us to construct reduced-order models (ROMs) that respect the delicate spectral properties of the original scattering problem. The ROMs are unconditionally stable and are based on a renormalized bi-Lanczos algorithm. We give a theoretical foundation of our method and illustrate its performance through a number of numerical examples in which we simulate two-dimensional electromagnetic wave propagation in unbounded domains, including a photonic waveguide example. The new algorithm outperforms the conventional finite-difference time-domain method for problems on large time intervals.

Near optimal rational approximations of large data sets

We introduce a new computationally efficient algorithm for constructing near optimal rational approximations of large (one-dimensional) data sets. In contrast to wavelet-type approximations, these new approximations are effectively shift invariant. We note that the complexity of current algorithms for computing near optimal rational approximations prevents their use for large data sets.In order to obtain a near optimal rational approximation of a large data set, we first construct its B-spline representation. Then, by using a new rational approximation of B-splines, we arrive at a suboptimal rational approximation of the data set. We then use a recently developed fast and accurate reduction algorithm for obtaining a near optimal rational approximation from a suboptimal one. Our approach requires first splitting the data into large segments, which may later be merged together, if needed. We also describe a fast algorithm for evaluating these rational approximations. In particular, this allows us to interpolate the original data to any grid.One of the practical applications of our algorithm is the compression of audio signals. To demonstrate the potential competitiveness of our approach, we construct a near optimal rational approximation of a piano recording.

Solving Burgersʼ equation using optimal rational approximations

We solve viscous Burgersʼ equation using a fast and accurate algorithm—referred to here as the reduction algorithm—for computing near optimal rational approximations.Given a proper rational function with n poles, the reduction algorithm computes (for a desired L∞-approximation error) a rational approximation of the same form, but with a (near) optimally small number m≪n of poles. Although it is well known that (nonlinear) optimal rational approximations are much more efficient than linear representations of functions via a fixed basis (e.g. wavelets), their use in numerical computations has been limited by a lack of efficient, robust, and accurate algorithms. The reduction algorithm presented here computes reliably (near) optimal rational approximations with high accuracy (e.g., ≈10−14) and a complexity that is essentially linear in the number n of original poles. A key tool is a recently developed algorithm for computing small con-eigenvalues of Cauchy matrices with high relative accuracy, an impossible task for standard algorithms without extended precision.Using the reduction algorithm, we develop a numerical calculus for rational representations of functions. Indeed, while operations such as multiplication and convolution increase the number of poles in the representation, we use the reduction algorithm to maintain an optimally small number of poles.To demonstrate the efficiency, robustness, and accuracy of our approach, we solve Burgersʼ equation with small viscosity ν. It is well known that its solutions exhibit moving transition regions of width O(ν), so that this equation provides a stringent test for adaptive PDE solvers. We show that optimal rational approximations capture the solutions with high accuracy using a small number of poles. In particular, we solve the equation with local accuracy ϵ=10−9 for viscosity as small as ν=10−5.

Fast and Accurate Con-Eigenvalue Algorithm for Optimal Rational Approximations

The need to compute small con-eigenvalues and the associated con-eigenvectors of positive-definite Cauchy matrices naturally arises when constructing rational approximations with a (near) optimally small $L^{\infty}$ error. Specifically, given a rational function with $n$ poles in the unit disk, a rational approximation with $m\ll n$ poles in the unit disk may be obtained from the $m$th con-eigenvector of an $n\times n$ Cauchy matrix, where the associated con-eigenvalue $\lambda_{m}>0$ gives the approximation error in the $L^{\infty}$ norm. Unfortunately, standard algorithms do not accurately compute small con-eigenvalues (and the associated con-eigenvectors) and, in particular, yield few or no correct digits for con-eigenvalues smaller than the machine roundoff. We develop a fast and accurate algorithm for computing con-eigenvalues and con-eigenvectors of positive-definite Cauchy matrices, yielding even the tiniest con-eigenvalues with high relative accuracy. The algorithm computes the $m$th con-eigenvalue in $\mathcal{O}\left(m^{2}n\right)$ operations and, since the con-eigenvalues of positive-definite Cauchy matrices decay exponentially fast, we obtain (near) optimal rational approximations in $\mathcal{O}(n\left(\log\delta^{-1}\right)^{2})$ operations, where $\delta$ is the approximation error in the $L^{\infty}$ norm. We provide error bounds demonstrating high relative accuracy of the computed con-eigenvalues and the high accuracy of the unit con-eigenvectors. We also provide examples of using the algorithm to compute (near) optimal rational approximations of functions with singularities and sharp transitions, where approximation errors close to machine roundoff are obtained. Finally, we present numerical tests on random (complex-valued) Cauchy matrices to show that the algorithm computes all the con-eigenvalues and con-eigenvectors with nearly full precision.

Essential rate for approximation by spherical neural networks

We consider the optimal rate of approximation by single hidden feed-forward neural networks on the unit sphere. It is proved that there exists a neural network with n neurons, and an analytic, strictly increasing, sigmoidal activation function such that the deviation of a Sobolev class W²(2r)(S(d)) from the class of neural networks Φ(n)(ϕ), behaves asymptotically as n(-2r/d-1). Namely, we prove that the essential rate of approximation by spherical neural networks is n(-2r/d-1).