Fast cubature of high dimensional biharmonic potential based on approximate approximations

Lanzara, Flavia; Maz′ya, Vladimir; Schmidt, G.

doi:10.1007/s11565-019-00328-z

Cited by 4 publications

(3 citation statements)

References 23 publications

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“…By combining cubature formulas for volume potentials based on approximate approximations with the strategy of separated representations (cf., e.g., [4], [12]), it is possible derive a method for approximating volume potentials which is accurate and fast also in the multidimensional case and provides approximation formulas of high order. This procedure was applied successfully for the fast integration of the harmonic [15], biharmonic [19], diffraction [18], elastic and hydrodynamic [20] potentials. In [16], [17] this approach was extended to parabolic problems.…”

Section: Introductionmentioning

confidence: 99%

Fast computation of the multidimensional fractional Laplacian

Lanzara

Maz′ya

Schmidt

2021

Applicable Analysis

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The paper discusses new cubature formulas for the Riesz potential and the fractional Laplacian (−∆) α/2 , 0 < α < 2, in the framework of the method approximate approximations. This approach, combined with separated representations, makes the method successful also in high dimensions. We prove error estimates and report on numerical results illustrating that our formulas are accurate and provide the predicted convergence rate 2, 4, 6, 8 up to dimension 10 4 .

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Section: Introductionmentioning

confidence: 99%

Fast computation of the multidimensional fractional Laplacian

Lanzara

Maz′ya

Schmidt

2021

Applicable Analysis

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“…Fast cubature formulas of advection-diffusion potentials in the frame of approximate approximations have been obtained in [10]. In [11] the procedure has been extended to parabolic problems of second order and in [12] to the biharmonic operator.…”

Section: Introductionmentioning

confidence: 99%

Fast computation of elastic and hydrodynamic potentials using approximate approximations

Lanzara

Maz′ya

Schmidt³

2020

Anal.Math.Phys.

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We propose fast cubature formulas for the elastic and hydrodynamic potentials based on the approximate approximation of the densities with Gaussian and related functions. For densities with separated representation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures. We obtain high order approximations up to a small saturation error, which is negligible in computations. Results of numerical experiments which show approximation order $$\mathscr {O}(h^{2M})$$ O ( h 2 M ) , $$M=1,2,3,4$$ M = 1 , 2 , 3 , 4 , are provided.

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“…In this paper we modify these formulas and show that also the diffraction potential can be treated with our approach. Note also that this approach was applied to parabolic problems in [9] and to higher order operators in [11] and [13].…”

Section: Introductionmentioning

confidence: 99%

Accurate computation of the high dimensional diffraction potential over hyper-rectangles

Lanzara,

Maz'ya,

Schmidt

2018

Preprint

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We propose a fast method for high order approximation of potentials of the Helmholtz type operator ∆ + κ 2 over hyper-rectangles in R n . By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of onedimensional integrals with separable integrands. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Numerical tests show that these formulas are accurate and provide approximations of order 6 up to dimension 100 and κ 2 = 100.

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Fast cubature of high dimensional biharmonic potential based on approximate approximations

Cited by 4 publications

References 23 publications

Fast computation of the multidimensional fractional Laplacian

Fast computation of the multidimensional fractional Laplacian

Fast computation of elastic and hydrodynamic potentials using approximate approximations

Accurate computation of the high dimensional diffraction potential over hyper-rectangles

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