We consider the integration of one-dimensional highly oscillatory functions. Based on analytic continuation, rapidly converging quadrature rules are derived for a fairly general class of oscillatory integrals with an analytic integrand. The accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency, and for the case of an increasing number of points and fixed frequency. These results are then used to obtain quadrature rules for more general oscillatory integrals, i.e., for functions that exhibit some smoothness but that are not analytic. The approach described in this paper is related to the Steepest Descent method, but it does not employ asymptotic expansions. It can be used for small or moderate frequencies as well as for very high frequencies. The approach is compared with the oscillatory integration techniques recently developed by Iserles and Nørsett.
In a previous paper [4] we described the numerical properties of function approximation using frames, i.e. complete systems that are generally redundant but provide infinite representations with coefficients of bounded norm. Frames offer enormous flexibility compared to bases. We showed that, in spite of extreme ill-conditioning, a regularized projection onto a finite truncated frame can provide accuracy up to order √ ǫ, where ǫ is an arbitrarily small threshold. Here, we generalize the setting in two ways. First, we assume information or samples from f from a wide class of linear operators acting on f , rather than inner products with the frame elements. Second, we allow oversampling, leading to least-squares approximations. The first property enables the analysis of fully discrete approximations based, for instance, on function values only. We show that the second property, oversampling, crucially leads to much improved accuracy on the order of ǫ rather than √ ǫ. Overall, we demonstrate that numerical function approximation using truncated frames leads to highly accurate approximations in spite of having to solve ill-conditioned systems of equations. Once the approximations start to converge, i.e. once sufficiently many degrees of freedom are used, any function f can be approximated to within order ǫ with coefficients of small norm.
We obtain exponentially accurate Fourier series for non-periodic functions on the interval [−1, 1] by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal polynomials that resemble Chebyshev polynomials of the first and second kinds. We analyze a previously proposed numerical method, which is unstable in theory but stable in practice. We propose a new numerical method that is stable both in theory and in practice.Keywords : Fourier series, orthogonal polynomials, frames, numerical integration AMS(MOS) Classification : Primary : 42A10, Secondary : 42C15, 65D32 On the Fourier extension of non-periodic functionsDaan Huybrechs * † AbstractWe obtain exponentially accurate Fourier series for non-periodic functions on the interval [−1, 1] by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal polynomials that resemble Chebyshev polynomials of the first and second kinds. We analyze a previously proposed numerical method, which is unstable in theory but stable in practice. We propose a new numerical method that is stable both in theory and in practice.
An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation.In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte, Trefethen & Kuijlaars states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner. arXiv:1206.4111v3 [math.NA] 12 May 2013 2. The condition number κ(F N ) of the exact continuous FE mapping is exponentially large in N . The condition number of the exact discrete FE mapping satisfies κ(F N ) = 1 for all N (see §3.4).
We consider two-dimensional scattering problems, formulated as an integral equation defined on the boundary of the scattering obstacle. The oscillatory nature of high-frequency scattering problems necessitates a large number of unknowns in classical boundary element methods. In addition, the corresponding discretization matrix of the integral equation is dense. We formulate a boundary element method with basis functions that incorporate the asymptotic behavior of the solution at high frequencies. The method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns, but retains accuracy for lower frequencies. New in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals. As a result, we obtain a sparse discretization matrix for the oscillatory problem. Moreover, numerical experiments indicate that the accuracy of the solution actually increases with increasing frequency. The sparse discretization applies to problems where the phase of the solution can be predicted a priori, for example in the case of smooth and convex scatterers.
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