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Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion

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Computing and Visualization in Science

Abstract

We present a finite element implementation for the steady-state nonlocal Dirichlet problem with homogeneous volume constraints. Here, the nonlocal diffusion operator is defined as integral operator characterized by a certain kernel function. We assume that the domain is an arbitrary d-dimensional hyperrectangle and the kernel is translation and reflection invariant. Under these assumptions, we carefully analyze the structure of the stiffness matrix resulting from a continuous Galerkin method with \(Q_1\) elements and exploit this structure in order to cope with the curse of dimensionality associated to nonlocal problems. For the purpose of illustration we choose a particular kernel, which is related to space-fractional diffusion and present numerical results in 1d, 2d and for the first time also in 3d.

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Acknowledgements

The first author has been supported by the German Research Foundation (DFG) within the Research Training Group 2126: “Algorithmic Optimization”.

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  1. Universitaet Trier, 54286, Trier, Germany

    Christian Vollmann & Volker Schulz

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  1. Christian Vollmann
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  2. Volker Schulz
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Correspondence to Christian Vollmann.

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Communicated by Gabriel Wittum.

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Vollmann, C., Schulz, V. Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion. Comput. Visual Sci. 20, 29–46 (2019). https://doi.org/10.1007/s00791-018-00306-6

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