We introduce a new numerical method for the computation of the inverse nonlinear Fourier transform and compare its computational complexity and accuracy to those of other methods available in the literature. For a given accuracy, the proposed method requires the lowest number of operations
Fourier Extension Based on Weighted Generalized Inverse
School of Mathematics and Statistics, Shandong University of Technology(山东理工大学数学与统计学院);Key Laboratory of Deep Petroleum Intelligent Exploration and Development, Institute of Geology and Geophysics, Chinese Academy of Sciences(中国科学院地质与地球物理研究所深地油气勘探开发重点实验室);Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University(莫斯科国立大学物理系数学系)
This paper introduces a factorization for the inverse of discrete Fourier integral operators that can be applied in quasi-linear time. The factorization starts by approximating the operator with the butterfly factorization. Next, a hierarchical matrix representation is constructed for the hermitian matrix arising from composing the Fourier integral operator with its adjoint. This representation is inverted efficiently with a new algorithm based on the hierarchical interpolative factorization. By combining these two factorizations, an approximate inverse factorization for the Fourier integral o…
Solving the multi-frequency electromagnetic inverse source problem by the Fourier method
School of Mathematics, Jilin University(吉林大学数学学院);Department of Mathematics, Harbin Institute of Technology(哈尔滨工业大学数学系);Department of Mathematics, Southern University of Science and Technology(南方科技大学数学系)
Resolving the Gibbs Phenomenon in Fractional Fourier Series via Inverse Polynomial Reconstruction
School of Mathematics, China University of Mining and Technology(中国矿业大学数学学院);Department of Mathematics, Riphah International University Faisalabad(菲萨尔巴德立夫海国际大学数学系);School of Mathematics and Statistics, Guangxi Normal University(广西师范大学数学与统计学院)
This paper is concerned with the inverse random source problem for a stochastic time fractional diffusion equation, where the source is assumed to be driven by a Gaussian random field. The direct problem is shown to be well-posed by examining the well-posedness and regularity of the solution for the equivalent stochastic two-point boundary value problem in the frequency domain. For the inverse problem, the Fourier modulus of the diffusion coefficient of the random source is proved to be uniquely determined by the variance of the Fourier transform of the boundary data. As a phase retrieval for …
This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman Weight Function in it. In the previous works the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave fi…
We investigate an interpolation/extrapolation method that, given scattered observations of the Fourier transform, approximates its inverse. The interpolation algorithm takes advantage of modelling the available data via a shape-driven interpolation based on Variably Scaled Kernels (VSKs), whose implementation is here tailored for inverse problems. The so-constructed interpolants are used as inputs for a standard iterative inversion scheme. After providing theoretical results concerning the spectrum of the VSK collocation matrix, we test the method on astrophysical imaging benchmarks.