Fast discrete convolution in ℝ2$\mathbb {R}^{2}$ with radial kernels using non-uniform fast Fourier transform with nonequispaced frequencies

Averseng, Martin

doi:10.1007/s11075-019-00670-5

Cited by 3 publications

(4 citation statements)

References 21 publications

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“…. For problems such that N > 5 × 10 3 , we compress the matrices using the Efficient Bessel Decomposition [7], in association with the "fiNUFFT" code from the Flat Iron institute [10,11].…”

Section: Numerical Resultsmentioning

confidence: 99%

New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results

Alouges

Averseng

2021

Numer. Math.

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Helmholtz wave scattering by open screens in 2D can be formulated as first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introduce two new preconditioners in the form of square-roots of on-curve differential operators both for the Dirichlet and Neumann boundary conditions on the screen. They generalize the so-called “analytical” preconditioners available for Lipschitz scatterers. We introduce a functional setting adapted to the singularity of the problem and enabling the analysis of those preconditioners. The efficiency of the method is demonstrated on several numerical examples.

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Section: Numerical Resultsmentioning

confidence: 99%

New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results

Alouges

Averseng

2021

Numer. Math.

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“…For problems such that |Γ| λ ≤ 200, where λ stands for the wavelength, the problems are solved in full BEM without compression method. For the cases with |Γ| λ > 200, (N > 10 4 where N is the number of unknowns) the memory of the computer is insufficient to store the full problem, and in this case the Efficient Bessel Decomposition (EBD) [8] is used to compress and accelerate the matrix-vector products.…”

Section: Preconditioning Resultsmentioning

confidence: 99%

New preconditioners for Laplace and Helmholtz integral equations on open curves: Analytical framework and Numerical results

Alouges¹,

Averseng²

2019

Preprint

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The numerical resolution of wave scattering problems by open curves leads to ill-conditioned linear systems which are difficult to precondition due to the geometrical singularities at the edges. We introduce two new preconditioners to tackle this problem respectively for Dirichlet or Neumann boundary data, that take the form of square roots of local operators. We describe an adapted analytical setting to analyze them and demonstrate the efficiency of this method on several numerical examples. A complete new pseudo-differential calculus suited to the study of such operators is postponed to the second part of this work.

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“…The numerical solution of the system (4.9) requires a priori to inverse a dense matrix, which can be potentially challenging for large values of N with direct methods. In order to solve (4.9) in reasonable computational time, we rely on the Efficient Bessel Decomposition (EBD) algorithm of Averseng [6]. This algorithm allows to evaluate…”

Section: (E[a]mentioning

confidence: 99%

“…in a mean-square sense at the rate O(N − 1 2 ). Let us note that linear systems analogous to (1.8) occur in many applications (such as in the classical Nystrom method for solving (1.9)) and can be solved efficiently with the Fast Multipole Method (FMM) from Greengard and Rohklin [22] or some alternatives such as the Efficient Bessel Decomposition [6]. For instance, the FMM was used in [36,20] to speed up the computation of matrix-vector products by iterative linear solvers, or by [27] for computing the wave scattered by a collection of large number of acoustic obstacles.…”

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confidence: 99%

Analysis of a Monte-Carlo Nystrom Method

Feppon¹,

Ammari²

2022

SIAM J. Numer. Anal.

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This paper considers a Monte-Carlo Nystrom method for solving integral equations of the second kind, whereby the values (z(y i )) 1≤i≤N of the solution z at a set of N random and independent points (y i ) 1≤i≤N are approximated by the solution (z N,i ) 1≤i≤N of a discrete, Ndimensional linear system obtained by replacing the integral with the empirical average over the samples (y i ) 1≤i≤N . Under the unique assumption that the integral equation admits a unique solution z(y), we prove the invertibility of the linear system for sufficiently large N with probability one, and the convergence of the solution (z N,i ) 1≤i≤N towards the point values (z(y i )) 1≤i≤N in a meansquare sense at a rate O(N − 1 2 ). For particular choices of kernels, the discrete linear system arises as the Foldy-Lax approximation for the scattered field generated by a system of N sources emitting waves at the points (y i ) 1≤i≤N . In this context, our result can equivalently be considered as a proof of the well-posedness of the Foldy-Lax approximation for systems of N point scatterers, and of its convergence as N → +∞ in a mean-square sense to the solution of a Lippmann-Schwinger equation characterizing the effective medium. The convergence of Monte-Carlo solutions at the rate O(N −1/2 ) is numerically illustrated on 1D examples and for solving a 2D Lippmann-Schwinger equation.

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Fast discrete convolution in ℝ2$\mathbb {R}^{2}$ with radial kernels using non-uniform fast Fourier transform with nonequispaced frequencies

Cited by 3 publications

References 21 publications

New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results

New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results

New preconditioners for Laplace and Helmholtz integral equations on open curves: Analytical framework and Numerical results

Analysis of a Monte-Carlo Nystrom Method

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