The sparse cardinal sine decomposition and its application for fast numerical convolution

Alouges, François; Aussal, Matthieu

doi:10.1007/s11075-014-9953-6

Cited by 17 publications

(30 citation statements)

References 12 publications

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“…As shown in Alouges and Aussal [6], it is possible to extend the procedure to the approximation of the convolution integral…”

Section: Principle For a Radial Kernelmentioning

confidence: 99%

“…For a given tolerance ε the associated values of P, λ p , β p , W pl , ξ pl and (R min, R max ) have been obtained in Alouges and Aussal [6] for the Laplace kernel K(r) = 1/(4πr) and the Helmholtz kernel K(r) = exp(ikr)/(4πr).…”

Section: Principle For a Radial Kernelmentioning

confidence: 99%

“…the second integral appearing in eqn (6). This is done taking on ∂Ω a rotation with velocity w = e 1 ˄ x = w i e i .…”

Section: The Stressletmentioning

confidence: 99%

“…We now test a term similar to the last integral occurring in eqn (6). To do so, we this time define on ∂Ω the vector field Q by…”

Section: The Stokesletmentioning

confidence: 99%

“…Recently, a new accelerating technique has been proposed in Alouges and Aussal [6] for the boundary-integral equations encountered in potential and Helmholtz problems. This method appeals to a suitable sparse integration grid in the Fourier space, and a back and forth non-uniform Fast Fourier transform [7,8].…”

Section: Introductionmentioning

confidence: 99%

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Application of the Sparse Cardinal Sine Decomposition to 3D Stokes Flows

Alouges¹,

Aussal²,

Lefebvre-Lepot³

et al. 2017

Int. J. CMEM

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In boundary element method (BEM), one encounters linear system with a dense and non-symmetric square matrix which might be so large that inverting the linear system is too prohibitive in terms of cpu time and/or memory. Each usual powerful treatment (Fast Multipole Method, H-matrices) developed to deal with this issue is optimized to efficiently perform matrix vector products. This work presents a new technique to adequately and quickly handle such products: the Sparse Cardinal Sine Decomposition. This approach, recently pioneered for the Laplace and Helmholtz equations, rests on the decomposition of each encountered kernel as series of radial Cardinal Sine functions. Here, we achieve this decomposition for the Stokes problem and implement it in MyBEM, a new fast solver for multi-physical BEM. The reported computational examples permit us to compare the advocated method against a usual BEM in terms of both accuracy and convergence.

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“…As shown in Alouges and Aussal [6], it is possible to extend the procedure to the approximation of the convolution integral…”

Section: Principle For a Radial Kernelmentioning

confidence: 99%

Section: Principle For a Radial Kernelmentioning

confidence: 99%

“…the second integral appearing in eqn (6). This is done taking on ∂Ω a rotation with velocity w = e 1 ˄ x = w i e i .…”

Section: The Stressletmentioning

confidence: 99%

“…We now test a term similar to the last integral occurring in eqn (6). To do so, we this time define on ∂Ω the vector field Q by…”

Section: The Stokesletmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Application of the Sparse Cardinal Sine Decomposition to 3D Stokes Flows

Alouges¹,

Aussal²,

Lefebvre-Lepot³

et al. 2017

Int. J. CMEM

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Two-level preconditioning for $$h$$-version boundary element approximation of hypersingular operator with GenEO

et al. 2020

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High-Frequency Asymptotic Compression of Dense BEM Matrices for General Geometries Without Ray Tracing

Huybrechs

Opsomer

2018

J Sci Comput

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Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the sparsifying modification of the Green's function with other accelerating schemes, such as the fast multipole method, appears possible in principle and is a future research topic. * daan.huybrechs@cs.kuleuven.be † peter.opsomer@cs.kuleuven.be arXiv:1606.09178v4 [math.NA]

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The sparse cardinal sine decomposition and its application for fast numerical convolution

Cited by 17 publications

References 12 publications

Application of the Sparse Cardinal Sine Decomposition to 3D Stokes Flows

Application of the Sparse Cardinal Sine Decomposition to 3D Stokes Flows

Two-level preconditioning for $$h$$-version boundary element approximation of hypersingular operator with GenEO

High-Frequency Asymptotic Compression of Dense BEM Matrices for General Geometries Without Ray Tracing

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