Fast computation of the multidimensional fractional Laplacian

Abstract

The paper discusses new cubature formulas for the Riesz potential and the fractional Laplacian $(-\mathrm{\Delta }{)}^{\alpha /2}$, $0<\alpha <2$, 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 ${10}^4$.

Keywords:

• Multidimensional convolution
• Riesz potential
• fractional Laplacian
• separated representation
• approximate approximations

AMS Classifications:

• 65D32
• 41A30
• 41A63

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author has been supported by the RUDN University Strategic Academic Leadership Program.