1,319 research outputs found
Chebyshev Interpolation Polynomial-based Tools for Rigorous Computing
17 pagesInternational audiencePerforming numerical computations, yet being able to provide rigorous mathematical statements about the obtained result, is required in many domains like global optimization, ODE solving or integration. Taylor models, which associate to a function a pair made of a Taylor approximation polynomial and a rigorous remainder bound, are a widely used rigorous computation tool. This approach benefits from the advantages of numerical methods, but also gives the ability to make reliable statements about the approximated function. Despite the fact that approximation polynomials based on interpolation at Chebyshev nodes offer a quasi-optimal approximation to a function, together with several other useful features, an analogous to Taylor models, based on such polynomials, has not been yet well-established in the field of validated numerics. This paper presents a preliminary work for obtaining such interpolation polynomials together with validated interval bounds for approximating univariate functions. We propose two methods that make practical the use of this: one is based on a representation in Newton basis and the other uses Chebyshev polynomial basis. We compare the quality of the obtained remainders and the performance of the approaches to the ones provided by Taylor models
Chebyshev model arithmetic for factorable functions
This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden
Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
We analyze the convergence of compressive sensing based sampling techniques
for the efficient evaluation of functionals of solutions for a class of
high-dimensional, affine-parametric, linear operator equations which depend on
possibly infinitely many parameters. The proposed algorithms are based on
so-called "non-intrusive" sampling of the high-dimensional parameter space,
reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a
functional of the parametric solution is then computed via compressive sensing
methods from samples of functionals of the solution. A key ingredient in our
analysis of independent interest consists in a generalization of recent results
on the approximate sparsity of generalized polynomial chaos representations
(gpc) of the parametric solution families, in terms of the gpc series with
respect to tensorized Chebyshev polynomials. In particular, we establish
sufficient conditions on the parametric inputs to the parametric operator
equation such that the Chebyshev coefficients of the gpc expansion are
contained in certain weighted -spaces for . Based on this we
show that reconstructions of the parametric solutions computed from the sampled
problems converge, with high probability, at the , resp.
convergence rates afforded by best -term approximations of the parametric
solution up to logarithmic factors.Comment: revised version, 27 page
On the resolution power of Fourier extensions for oscillatory functions
Functions that are smooth but non-periodic on a certain interval possess
Fourier series that lack uniform convergence and suffer from the Gibbs
phenomenon. However, they can be represented accurately by a Fourier series
that is periodic on a larger interval. This is commonly called a Fourier
extension. When constructed in a particular manner, Fourier extensions share
many of the same features of a standard Fourier series. In particular, one can
compute Fourier extensions which converge spectrally fast whenever the function
is smooth, and exponentially fast if the function is analytic, much the same as
the Fourier series of a smooth/analytic and periodic function.
With this in mind, the purpose of this paper is to describe, analyze and
explain the observation that Fourier extensions, much like classical Fourier
series, also have excellent resolution properties for representing oscillatory
functions. The resolution power, or required number of degrees of freedom per
wavelength, depends on a user-controlled parameter and, as we show, it varies
between 2 and \pi. The former value is optimal and is achieved by classical
Fourier series for periodic functions, for example. The latter value is the
resolution power of algebraic polynomial approximations. Thus, Fourier
extensions with an appropriate choice of parameter are eminently suitable for
problems with moderate to high degrees of oscillation.Comment: Revised versio
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