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 series can be a challenging problem even for relatively well behaved functions. In this paper a method for approximating a sparse collection of Fourier-like coefficients is presented that uses a collocation technique combined with an optimization problem inspired by recent results in compressed sensing research. The discussion includes approximation error rates and numerical examples to illustrate the effectiveness of the method. One example displays the accuracy of the generalized Fourier series approximation for several test functions, while the other is an application of the generalized Fourier series approximation to rotation-invariant pattern recognition in images.
Based on the Fourier extension, we propose an oversampling collocation method for solving the elliptic partial differential equations with variable coefficients over arbitrary irregular domains. This method only uses the function values on the equispaced nodes, which has low computational cost and versatility. While a variety of numerical experiments are presented to demonstrate the effectiveness of this method, it shows that the approximation error fast reaches a plateau with increasing the degrees of freedom, due to the inherent ill-conditioned of frames.
This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for collocation onto high-order piecewise polynomial space that is allowed to be discontinuous. We consider both Lagrange and Hermite interpolation methods on nested collocation points. Our construction includes a wide range of function space, including tho…
This work is a follow-up to our previous contribution ("Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs)", SIAM J. Numer. Anal., 2018), and contains further insights on some aspects of the solution of elliptic PDEs with lognormal diffusion coefficients using sparse grids. Specifically, we first focus on the choice of univariate interpolation rules, advocating the use of Gaussian Leja points as introduced by Narayan and Jakeman ("Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional …
High Dimensional Uncertainty Quantification for an Electrothermal Field Problem using Stochastic Collocation on Sparse Grids and Tensor Train Decompositions
This work suggests an interpolation-based stochastic collocation method for the non-intrusive and adaptive construction of sparse polynomial chaos expansions (PCEs). Unlike pseudo-spectral projection and regression-based stochastic collocation methods, the proposed approach results in PCEs featuring one polynomial term per collocation point. Moreover, the resulting PCEs are interpolating, i.e., they are exact on the interpolation nodes/collocation points. Once available, an interpolating PCE can be used as an inexpensive surrogate model, or be post-processed for the purposes of uncertainty qua…
We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of t…
When dealing with timber structures, the characteristic strength and stiffness of the material are made highly variable and uncertain by the unavoidable, yet hardly predictable, presence of knots and other defects. In this work we apply the sparse grids stochastic collocation method to perform uncertainty quantification for structural engineering in the scenario described above. Sparse grids have been developed by the mathematical community in the last decades and their theoretical background has been rigorously and extensively studied. The document proposes a brief practice-oriented introduct…
In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) …
We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier in the paper ``Multi-dimensional sublinear sparse Fourier algorithm" (2016), an efficient sparse Fourier algorithm with $Θ(ds \log s)$ average-case runtime and $Θ(ds)$ sampling complexity under certain assumptions was developed for signals that are $s$-sparse and bandlimited in the $d$-dimensional Fourier domain, i.e. there are at most $s$ energetic frequencies and they are in $ \left[-N/2, N/2\right)^d\cap \mathbb{Z}^d$. However, in practice the measurements of signals often contain noise, …
In this paper a deterministic sparse Fourier transform algorithm is presented which breaks the quadratic-in-sparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. These functions include, e.g., the oft-considered set of block frequency sparse functions of the form $$f(x) = \sum^{n}_{j=1} \sum^{B-1}_{k=0} c_{ω_j + k} e^{i(ω_j + k)x},~~\{ ω_1, \dots, ω_n \} \subset \left(-\left\lceil \frac{N}{2}\right\rceil, \left\lfloor \frac{N}{2}\right\rfloor\right]\cap\mathbb{Z}$$ as a simple subclass. Theoretical error bounds in combination with numerica…
In this paper we propose a general strategy for rapidly computing sparse Legendre expansions. The resulting methods yield a new class of fast algorithms capable of approximating a given function $f:[-1,1] \rightarrow \mathbb{R}$ with a near-optimal linear combination of $s$ Legendre polynomials of degree $\leq N$ in just $(s \log N)^{\mathcal{O}(1)}$-time. When $s \ll N$ these algorithms exhibit sublinear runtime complexities in $N$, as opposed to traditional $Ω(N \log N)$-time methods for computing all of the first $N$ Legendre coefficients of $f$. Theoretical as well as numerical results dem…
In this paper, we discuss the development of a sublinear sparse Fourier algorithm for high-dimensional data. In ``Adaptive Sublinear Time Fourier Algorithm" by D. Lawlor, Y. Wang and A. Christlieb (2013), an efficient algorithm with $Θ(k\log k)$ average-case runtime and $Θ(k)$ average-case sampling complexity for the one-dimensional sparse FFT was developed for signals of bandwidth $N$, where $k$ is the number of significant modes such that $k\ll N$. In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals, extending some of the ideas in the paper mentioned …
In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form $f(t) = \sum_{j=1}^{K} γ_{j} \, \cos(2πa_{j} t + b_{j})$, where the frequency parameters $a_{j} \in {\mathbb R}$ (or $a_{j} \in {\mathrm i} {\mathbb R}$) are pairwise different. Our method is based on the recently proposed stable iterative rational approximation algorithm in \cite{NST18}. For signal reconstruction we use a set of classical Fourier coefficients of $f$ with regard to a fixed interval $(0, P)$ with $P>0$. Even …
Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields, and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the Fourier Neural Solver (FNS), to address them. FNS is based on deep learning and Fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis…
In this paper we propose a new fast Fourier transform to recover a real nonnegative signal ${\bf x}$ from its discrete Fourier transform. If the signal ${\mathbf x}$ appears to have a short support, i.e., vanishes outside a support interval of length $m < N$, then the algorithm has an arithmetical complexity of only ${\cal O}(m \log m \log (N/m)) $ and requires ${\cal O}(m \log (N/m))$ Fourier samples for this computation. In contrast to other approaches there is no a priori knowledge needed about sparsity or support bounds for the vector ${\bf x}$. The algorithm automatically recognizes and e…
We develop the uniform sparse Fast Fourier Transform (usFFT), an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The algorithm is an adaption of the sparse Fast Fourier Transform (sFFT), a dimension-incremental algorithm, which tries to detect the most important frequencies in a given search domain and therefore adaptively generates a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function. The usFFT does this w.r.t. the stochastic domain of the PDE simultaneously f…
In this paper modified variants of the sparse Fourier transform algorithms from [14] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a near-optimal k-term Fourier series for any given input function, $f : [0, 2 pi] -> C, in O(k^2 \cdot D^4)$ time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexiti…
In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, ``[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless march to larger and larger [bandwidths] continues''. This paper attempts to further the virtue of the Fast Fourier Transform (FFT) as not only bandwidth is pushed to its limits, but also the dimension of the problem. Instead of using the traditional FFT however, we make a key substitution: a high-dimensional, sparse Fourier transform (SFT) paired with rando…
In this paper a sublinear time algorithm is presented for the reconstruction of functions that can be represented by just few out of a potentially large candidate set of Fourier basis functions in high spatial dimensions, a so-called high-dimensional sparse fast Fourier transform. In contrast to many other such algorithms, our method works for arbitrary candidate sets and does not make additional structural assumptions on the candidate set. Our transform significantly improves upon the other approaches available for such a general framework in terms of the scaling of the sample complexity. Our…
We present a general class of compressed sensing matrices which are then demonstrated to have associated sublinear-time sparse approximation algorithms. We then develop methods for constructing specialized matrices from this class which are sparse when multiplied with a discrete Fourier transform matrix. Ultimately, these considerations improve previous sampling requirements for deterministic sparse Fourier transform methods.
We consider fast, provably accurate algorithms for approximating functions on the $d$-dimensional torus, $f: \mathbb{ T }^d \rightarrow \mathbb{C}$, that are sparse (or compressible) in the Fourier basis. In particular, suppose that the Fourier coefficients of $f$, $\{c_{\bf k} (f) \}_{{\bf k} \in \mathbb{Z}^d}$, are concentrated in a finite set $I \subset \mathbb{Z}^d$ so that $$\min_{Ω\subset I s.t. |Ω| =s } \left\| f - \sum_{{\bf k} \in Ω} c_{\bf k} (f) e^{ -2 πi {\bf k} \cdot \circ} \right\|_2 < ε\|f \|_2$$ holds for $s \ll |I|$ and $ε\in (0,1)$. We aim to identify a near-minimizing subset…
In this paper, we theoretically propose a new hashing scheme to establish the sparse Fourier transform in high-dimensional space. The estimation of the algorithm complexity shows that this sparse Fourier transform can overcome the curse of dimensionality. To the best of our knowledge, this is the first polynomial-time algorithm to recover the high-dimensional continuous frequencies.
Defense Innovation Institute, Chinese Academy of Military Science（中国人民解放军军事科学院国防科技创新研究院）
Perception of the full state is an essential technology to support the monitoring, analysis, and design of physical systems, one of whose challenges is to recover global field from sparse observations. Well-known for brilliant approximation ability, deep neural networks have been attractive to data-driven flow and heat field reconstruction studies. However, limited by network structure, existing researches mostly learn the reconstruction mapping in finite-dimensional space and has poor transferability to variable resolution of outputs. In this paper, we extend the new paradigm of neural operat…
We consider the numerical solution of the discrete multi-marginal optimal transport (MOT) by means of the Sinkhorn algorithm. In general, the Sinkhorn algorithm suffers from the curse of dimensionality with respect to the number of marginals. If the MOT cost function decouples according to a tree or circle, its complexity is linear in the number of marginal measures. In this case, we speed up the convolution with the radial kernel required in the Sinkhorn algorithm by non-uniform fast Fourier methods. Each step of the proposed accelerated Sinkhorn algorithm with a tree-structured cost function…
Phase retrieval consists in the recovery of an unknown signal from phaseless measurements of its usually complex-valued Fourier transform. Without further assumptions, this problem is notorious to be severe ill posed such that the recovery of the true signal is nearly impossible. In certain applications like crystallography, speckle imaging in astronomy, or blind channel estimation in communications, the unknown signal has a specific, sparse structure. In this paper, we exploit these sparse structure to recover the unknown signal uniquely up to inevitable ambiguities as global phase shifts, tr…
