Optimal Rational Approximation Research Articles

Open mushrooms: stickiness revisited

We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system ρ, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit quasi-regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well-known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe for the first time a zero measure set of control parameter values ρ ∊ (0, 1) for which all MUPOs are destroyed and therefore the system is less sticky, leading to a different power law exponent for the Poincaré recurrence time distribution statistics. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited due to MUPOs and exact leading order expressions for the algebraic decay of the survival probability function are calculated for mushrooms with triangular and rectangular stems. Numerical simulations are also performed which confirm our predictions for both sticky and less sticky mushrooms.

Instability of nondiscrete free subgroups in lie groups

We study finitely-generated nondiscrete free subgroups in Lie groups. We address the following question first raised by Etienne Ghys: is it always possible to make arbitrarily small perturbations of the generators of the free subgroup in such a way that the new group formed by the perturbed generators be not free? In other words, is it possible to approximate generators of a free subgroup by elements satisfying a nontrivial relation? We prove that the answer to Ghys’ question is positive and generalize this result to certain nonfree subgroups. We also consider the question on the best approximation rate in terms of the minimal length of relation in the approximating group. We give an upper bound on the optimal approximation rate as $$ {e^{ - c{l^\kappa }}} $$ , where c > 0 is a constant, l the minimal length of relation and 0.19 < κ < 0.2.

Greedy approximation of characteristic functions

We discuss the problem of sparse representation of domains in ℝd. We demonstrate how the recently developed general theory of greedy approximation in Banach spaces can be used in this problem. The use of greedy approximation has two important advantages: (1) it works for an arbitrary dictionary of sets used for sparse representation and (2) the method of approximation does not depend on smoothness properties of the domains and automatically provides a near optimal rate of approximation for domains with different smoothness properties. We also give some lower estimates of the approximation error and discuss a specific greedy algorithm for approximation of convex domains in ℝ2.

Trigonometric Approximation of SO(N) Loops

This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Holder class Lip α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})$ for n≥k, where k=k(Q) is determined by topological properties of the loop and e>0 is arbitrarily small. The convergence rate is therefore e-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.

The lower estimate for the linear combinations of Bernstein–Kantorovich operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein–Kantorovich operators, which gives the optimal approximation rate. On the basis of this inequality, we further obtain the lower estimate for these operators.

Rate of Convergence in Approximating the Spectral Factor of Regular Stochastic Sequences

Common methods for the calculation of the spectral factorization rely on an approximation of the given spectral density by a polynomial and a subsequent factorization of this polynomial. It is known that the regularity of the stochastic sequence determines the achievable approximation rate of its spectrum. However, since the approximative polynomial should be factorized, it has to be positive. It is shown that this restriction on the approximation polynomial implies a limitation on the approximation rate for linear methods whereas for nonlinear methods the optimal approximation rate can still be achieved. This has also consequences for the rate of convergence of the spectral factor, which is investigated in the second part. There, a lower and an upper bound for the error in the spectral factor is derived, which shows the dependency on the approximation degree and on the regularity of the stochastic sequence. Finally, if the spectral density is given only on a finite set of sampling points, no linear approximation method exists such that the error in the spectral factor can be controlled by the approximation degree.

Representation and Compression of Multidimensional Piecewise Functions Using Surflets

We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M-1)-dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal rate-distortion performance; for piecewise smooth functions, our coder has near-optimal rate-distortion performance. Our coder for piecewise constant functions employs surflets, a new multiscale geometric tiling consisting of M-dimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints, which wed surflets to wavelets for piecewise smooth approximation. Both of these schemes achieve the optimal asymptotic approximation performance. Key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity. We also extend our results to the corresponding discrete function spaces for sampled data. We provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities. For approximation of discrete data, we propose a new scale-adaptive dictionary that contains few elements at coarse and fine scales, but many elements at medium scales. Simulation results on synthetic signals provide a comparison between surflet-based coders and previously studied approximation schemes based on wedgelets and wavelets.

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an ex- panded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu 0 s work (SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777) on stan- dard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h = O(H (4k+1)/(k+1) ). AMS subject classifications: 65N30, 65N15, 65M12

Compensated optimal grids for elliptic boundary-value problems

A method is proposed which allows to efficiently treat elliptic problems on unbounded domains in two and three spatial dimensions in which one is only interested in obtaining accurate solutions at the domain boundary. The method is an extension of the optimal grid approach for elliptic problems, based on optimal rational approximation of the associated Neumann-to-Dirichlet map in Fourier space. It is shown that, using certain types of boundary discretization, one can go from second-order accurate schemes to essentially spectrally accurate schemes in two-dimensional problems, and to fourth-order accurate schemes in three-dimensional problems without any increase in the computational complexity. The main idea of the method is to modify the impedance function being approximated to compensate for the numerical dispersion introduced by a small finite-difference stencil discretizing the differential operator on the boundary. We illustrate how the method can be efficiently applied to nonlinear problems arising in modeling of cell communication.

Image Approximation by Rectangular Wavelet Transform

We study image approximation by a separable wavelet basis $$ \{\psi(2^{k_1}x-i)\psi(2^{k_2}y-j), \phi(x-i)\psi(2^{k_2}y-j), \psi(2^{k_1}(x-i)\phi(y-j), \phi(x-i)\phi(y-i)\},$ where $k_1, k_2 \in \mathbb{Z}_+; i,j\in\mathbb{Z}; $$ and ?,? are elements of a standard biorthogonal wavelet basis in L2(?). Because k1? k2, the supports of the basis elements are rectangles, and the corresponding transform is known as the rectangular wavelet transform. We provide a self-contained proof that if one-dimensional wavelet basis has M dual vanishing moments then the rate of approximation by N coefficients of rectangular wavelet transform is $$ \mathcal{O}(N^{-M}) $$ for functions with mixed derivative of order M in each direction. These results are consistent with optimal approximation rates for such functions. The square wavelet transform yields the approximation rate is $$ \mathcal{O}(N^{-M/2}) $$ for functions with all derivatives of the total order M. Thus, the rectangular wavelet transform can outperform the square one if an image has a mixed derivative. We provide experimental comparison of image approximation which shows that rectangular wavelet transform outperform the square one.

EDGE ELEMENT COMPUTATION OF MAXWELL'S EIGENVALUES ON GENERAL QUADRILATERAL MESHES

Recent results prove that Nédélec edge elements do not achieve optimal rate of approximation on general quadrilateral meshes. In particular, lowest order edge elements provide stable but non-convergent approximation of Maxwell's eigenvalues. In this paper we analyze a modification of standard edge element that restores the optimality of the convergence. This modification is based on a projection technique that can be interpreted as a reduced integration procedure.

Is the Approximation Rate for European Pay-offs in the Black–Scholes Model Always 1/ $$\sqrt{n}$$

For a Borel-function \(f:\mathbb{R} \to \mathbb{R}\), we consider the approximation of a random variable f(W1) with \(\mathbb{E}f^{2}(W_{1})<\infty\) by stochastic integrals with respect to the Brownian motion \(W = (W_{t})_{t \in[0, 1]}\) and the geometric Brownian motion, where the integrands are piecewise constant within certain deterministic time intervals. In earlier papers it has been shown that under certain regularity conditions the optimal approximation rate is 1/ \(\sqrt{n}\), if one optimizes over deterministic time-nets of cardinality n. We will show the existence of random variables f(W1) such that the approximation error tends as slowly to zero as one wishes.

The contourlet transform: an efficient directional multiresolution image representation

The limitations of commonly used separable extensions of one-dimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a "true" two-dimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discrete-domain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and, thus, it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for N-pixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuous-domain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications. Index Terms-Contourlets, contours, filter banks, geometric image processing, multidirection, multiresolution, sparse representation, wavelets.

Sparse geometric image representations with bandelets

This paper introduces a new class of bases, called bandelet bases, which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates directions in which the image gray levels have regular variations. The image decomposition in a bandelet basis is implemented with a fast subband-filtering algorithm. Bandelet bases lead to optimal approximation rates for geometrically regular images. For image compression and noise removal applications, the geometric flow is optimized with fast algorithms so that the resulting bandelet basis produces minimum distortion. Comparisons are made with wavelet image compression and noise-removal algorithms.

Bi-framelet systems with few vanishing moments characterize Besov spaces

We study the approximation properties of wavelet bi-frame systems in L p( R d) . For wavelet bi-frame systems the approximation spaces associated with best m-term approximation are completely characterized for a certain range of smoothness parameters limited by the number of vanishing moments of the generators of the dual frame. The approximation spaces turn out to be essentially Besov spaces, just as in the classical orthonormal wavelet case. We also prove that for smooth functions, the canonical expansion in the wavelet bi-frame system is sparse and one can reach the optimal rate of approximation by simply thresholding the canonical expansion. For twice oversampled MRA based wavelet frames, a characterization of the associated approximation space is obtained without any restrictions given by the number of vanishing moments, but at a price of replacing the canonical expansion by another linear expansion.

Finite‐difference modeling of viscoelastic materials with quality factors of arbitrary magnitude

To minimize acoustic noise, designers of sonic logging tools often consider coatings of viscoelastic materials with very high attenuation properties. Efficient finite‐difference modeling of viscoelastic materials is a topic of current research. To model viscoelastic materials in the time domain through finite differences efficiently, one needs to replace the time convolution, which enters in the stress–strain relations, by a set of first‐order differential equations. This procedure is equivalent to computing a rational approximation of a certain form to the frequency‐dependent complex modulus of viscoelasticity. Known schemes for computing such approximations are designed to treat materials with low attenuation, such as underground formations, but fail to produce accurate or even physically meaningful results for highly attenuative materials. We propose a novel scheme that allows one to construct, for a given frequency range, a uniformly optimal rational approximation for the most widely used model of materials with constant quality (Q‐) factors of arbitrary magnitude. We present the proof of convergence and demonstrate it on numerical finite‐difference examples. These examples also demonstrate the effective transparency of a simple tool modeled as a pipe of highly viscoelastic material. For frequency‐dependent quality factors we present a modified numerical scheme to compute a nearly optimal rational approximation of the viscoelastic modulus.

On Approximation with Spline Generated Framelets

We characterize the approximation spaces associated with the best n-term approximation in L p (R) by elements from a tight wavelet frame associated with a spline scaling function. The approximation spaces are shown to be interpolation spaces between L p and classical Besov spaces, and the result coincides with the result for nonlinear approximation with an orthonormal wavelet with the same smoothness as the spline scaling function. We also show that, under certain conditions, the Besov smoothness can be measured in terms of the sparsity of expansions in the wavelet frame, just like the nonredundant wavelet case. However, the characterization now holds even for wavelet frame systems that do not have the usually required number of vanishing moments, e.g., for systems built through the Unitary Extension Principle, which can have no more than one vanishing moment. Using these results, we describe a fast algorithm that takes as input any function and provides a near sparsest expansion of it in the framelet system as well as approximants that reach the optimal rate of nonlinear approximation. Together with the existence of a fast algorithm, the absence of the need for vanishing moments may have an important qualitative impact for applications to signal compression, as high vanishing moments usually introduce a Gibbs-type phenomenon (or “ringing” artifacts)in the approximants.

Note on the Zolotarev optimal rational approximation for the overlap Dirac operator

We discuss the salient features of Zolotarev optimal rational approximation for the inverse square root function, in particular, for its applications in lattice QCD with overlap Dirac quark. The theoretical error bound for the matrix-vector multiplication $ H_w (H_w^2)^{-1/2}Y $ is derived. We check that the error bound is always satisfied amply, for any QCD gauge configurations we have tested. An empirical formula for the error bound is determined, together with its numerical values (by evaluating elliptic functions) listed in Table 2 as well as plotted in Figure 3. Our results suggest that with Zolotarev approximation to $ (H_w^2)^{-1/2} $, one can practically preserve the exact chiral symmetry of the overlap Dirac operator to very high precision, for any gauge configurations on a finite lattice.

Taylor Approximations on Sierpinski Gasket Type Fractals

For a class of fractals that includes the familiar Sierpinski gasket, there is now a theory involving Laplacians, Dirichlet forms, normal derivatives, Green's functions, and the Gauss–Green integration formula, analogous to the theory of analysis on manifolds. This theory was originally developed as a by-product of the construction of stochastic processes analogous to Brownian motion, but has been given by a direct analytic construction in the work of Kigami. Until now, this theory has not provided anything analogous to the gradient of a function, or a local Taylor approximation. In this paper we construct a family of derivatives, which includes the known normal derivative, at vertex points in the graphs that approximate the fractal, and obtain Taylor approximations at these points. We show that a function in the domain of Δn can be locally well approximated by an n-harmonic function (solution of Δnu=0). One novel feature of this result is that it requires several different estimates to describe the optimal rate of approximation.

Hybrid Monte Carlo algorithm for lattice QCD with two flavors of dynamical Ginsparg-Wilson quarks

We study aspects concerning numerical simulations of lattice QCD with two flavors of dynamical Ginsparg-Wilson quarks with degenerate masses. A hybrid Monte Carlo algorithm is described and a formula for the fermionic force is derived for two specific implementations. The implementation with the optimal rational approximation method is favored in both CPU time and memory consumption.