Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform

Jiang, null Shidong; Greengard, Leslie

doi:10.4208/cicp.oa-2021-0031

Cited by 5 publications

(3 citation statements)

References 28 publications

(67 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Jiang and Greengard [24] proposed to approximate the Gaussian on [0, ∞) using the inverse Laplace transform…”

Section: Related Resultsmentioning

confidence: 99%

“…The known other methods for completely monotone functions employed the representation by the Laplace transform, see [8], which can then be discretized by a quadrature rule. In [24], a quadrature rule for the inverse Laplace transform formula (1.5) has been applied. We conclude that the convergence analysis for the approximation with exponential sums is always closely related to quadrature formulas that converge exponentially for special analytic functions.…”

Section: Discussionmentioning

confidence: 99%

“…Gaussian functions are widely used in statistics, signal processing, molecular modeling, computational chemistry, as well as in approximation theory, see e.g. [24,22,3,16]. However, since the Gaussian is not always simple to handle on finite intervals, an exact approximation of the Gaussian is helpful in different contexts.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

Derevianko

Plonka²,

Razavi³

2022

Numer Algor

View full text Add to dashboard Cite

In this paper we introduce two algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz + Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case.

show abstract

“…Jiang and Greengard [24] proposed to approximate the Gaussian on [0, ∞) using the inverse Laplace transform…”

Section: Related Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

Derevianko

Plonka²,

Razavi³

2022

Numer Algor

View full text Add to dashboard Cite

show abstract

A Kernel-Independent Sum-of-Exponentials Method

Gao

Liang

2022

J Sci Comput

View full text Add to dashboard Cite

Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion

Kaye,

Huang,

Strand

et al. 2024

Phys. Rev. X

View full text Add to dashboard Cite

We present a deterministic algorithm for the efficient evaluation of imaginary-time diagrams based on the recently introduced discrete Lehmann representation (DLR) of imaginary-time Green’s functions. In addition to the efficient discretization of diagrammatic integrals afforded by its approximation properties, the DLR basis is separable in imaginary-time, allowing us to decompose diagrams into linear combinations of nested sequences of one-dimensional products and convolutions. Focusing on the strong-coupling bold-line expansion of generalized Anderson impurity models, we show that our strategy reduces the computational complexity of evaluating an Mth-order diagram at inverse temperature β and spectral width ωmax from O((βωmax)2M−1) for a direct quadrature to O(M(log(βωmax))M+1), with controllable high-order accuracy. We benchmark our algorithm using third-order expansions for multiband impurity problems with off-diagonal hybridization and spin-orbit coupling, presenting comparisons with exact diagonalization and quantum Monte Carlo approaches. In particular, we perform a self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of Ca2RuO4, demonstrating the promise of the method for modeling realistic strongly correlated multiband materials. For both strong and weak coupling expansions of low and intermediate order, in which diagrams can be enumerated, our method provides an efficient, straightforward, and robust blackbox evaluation procedure. In this sense, it fills a gap between diagrammatic approximations of the lowest order, which are simple and inexpensive but inaccurate, and those based on Monte Carlo sampling of high-order diagrams. Published by the American Physical Society 2024

show abstract

Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform

Cited by 5 publications

References 28 publications

ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

A Kernel-Independent Sum-of-Exponentials Method

Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion

Contact Info

Product

Resources

About