Related papers: A Fourier continuation framework for high-order ap…
Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighbourhoods of the endpoints. Fourier extensions circumvent…
The FC-Gram trigonometric polynomial approximation of a non-periodic function that interpolates the function on equispaced grids was introduced in 2010 by Bruno and Lyon [J. Comput. Phys, 229(6):2009-2033, 2010]. Since then, the…
Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases…
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…
The FC-Gram algorithm approximates non-periodic functions to high order by constructing a periodic extension with controlled boundary behavior and applying trigonometric interpolation. In this paper we introduce a generalized FC-Gram…
This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci.…
Fourier series of smooth, non-periodic functions on $[-1,1]$ are known to exhibit the Gibbs phenomenon, and exhibit overall slow convergence. One way of overcoming these problems is by using a Fourier series on a larger domain, say $[-T,T]$…
We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show…
We prove that any stable method for resolving the Gibbs phenomenon - that is, recovering high-order accuracy from the first $m$ Fourier coefficients of an analytic and nonperiodic function - can converge at best root-exponentially fast in…
High-dimensional Partial Differential Equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically…
Computationally efficient numerical methods for high-order approximations of convolution integrals involving weakly singular kernels find many practical applications including those in the development of fast quadrature methods for…
In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions from the approximation point of view. For a simple step…
The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is…
This paper presents a "two-dimensional Fourier Continuation" method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly…
We consider functions $f$ of two real variables, given as trigonometric functions over a finite set $F$ of frequencies. This set is assumed to be closed under rotations in the frequency plane of angle $\frac{2k\pi}{M}$ for some integer $M$.…
This paper presents a novel boundary-optimized fast Fourier extension algorithm for efficient approximation of non-periodic functions. The proposed methodology constructs periodic extensions through strategic utilization of boundary…
Generalized Fourier series with orthogonal polynomial bases have useful applications in several fields, including differential equations, pattern recognition, and image and signal processing. However, computing the generalized Fourier…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…
The convolution method for the numerical solution of forward-backward stochastic differential equations (FBSDEs), introduced in [21], uses a uniform space grid. In this paper we utilize a tree-like spatial discretization that approximates…
We present a numerical method for rigorous over-approximation of a reachable set of differential inclusions. The method gives high-order error bounds for single step approximations and a uniform bound on the error over the finite time…