Related papers: Taylor-Fourier approximation
This paper discusses model order reduction of LTI systems over limited frequency intervals within the framework of balanced truncation. Two new \emph{frequency-dependent balanced truncation} methods were developed, one is \emph{SF-type…
In this paper we propose an approximation method for high-dimensional $1$-periodic functions based on the multivariate ANOVA decomposition. We provide an analysis on the classical ANOVA decomposition on the torus and prove some important…
Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n \longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions $x \ne 0$ from…
In this paper, we propose a numerical method of computing an integral whose integrand is a slowly decaying oscillatory function. In the proposed method, we consider a complex analytic function in the upper-half complex plane, which is…
Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier transform, waves that leave the computational domain on one…
An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…
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…
We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of…
A poly-log time method to compute the truncated theta function, its derivatives, and integrals is presented. The method is elementary, rigorous, explicit, and suited for computer implementation. We repeatedly apply the Poisson summation…
Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some…
Although Fourier series approximation is ubiquitous in computational physics owing to the Fast Fourier Transform (FFT) algorithm, efficient techniques for the fast evaluation of a three-dimensional truncated Fourier series at a set of…
This paper introduces a novel method for approximating the dynamics of a large autonomous system projected onto a fixed subspace. The core contribution is a novel recursive algorithm to construct an effective time-dependent generator that…
In this article, we present an $O(N \log N)$ rapidly convergent algorithm for the numerical approximation of the convolution integral with radially symmetric weakly singular kernels and compactly supported densities. To achieve the reduced…
Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems in a manner that accounts for either open- or closed-loop observability and controllability aspects of the system. A…
A general solution is found for a large class of time continuous autonomous nonlinear dynamical systems, the so-called quasi-polynomial systems. This solution is expressed in terms of a new type of special functions defined via their Taylor…
Given a time series vector, how can we efficiently compute a specified part of Fourier coefficients? Fast Fourier transform (FFT) is a widely used algorithm that computes the discrete Fourier transform in many machine learning applications.…
This paper introduces a weighted generalized inverse framework for Fourier extensions, designed to suppress spurious oscillations in the extended region while maintaining high approximation accuracy on the original interval. By formulating…
In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method. In the…
In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e., an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and 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…