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On a Multilevel Preconditioner and its Condition Numbers for the Discretized Laplacian on Full and Sparse Grids in Higher Dimensions

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Singular Phenomena and Scaling in Mathematical Models

Abstract

We first discretize the d-dimensional Laplacian in (0, 1)d for varying d on a full uniform grid and build a new preconditioner that is based on a multilevel generating system. We show that the resulting condition number is bounded by a constant that is independent of both, the level of discretization J and the dimension d. Then, we consider so-called sparse grid spaces, which offer nearly the same accuracy with far less degrees of freedom for function classes that involve bounded mixed derivatives. We introduce an analogous multilevel preconditioner and show that it possesses condition numbers which are at least as good as these of the full grid case. In fact, for sparse grids we even observe falling condition numbers with rising dimension in our numerical experiments. Furthermore, we discuss the cost of the algorithmic implementations. It is linear in the degrees of freedom of the respective multilevel generating system. For completeness, we also consider the case of a sparse grid discretization using prewavelets and compare its properties to those obtained with the generating system approach.

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Notes

  1. 1.

    Non-local basis functions (p-version) are likely to result in a Toeplitz-type matrix, which can be inverted in log-linear time.

  2. 2.

    Note that it is even possible to execute this matrix-vector product in \(\mathcal{O}(d \cdot N_{d,J})\) operations by the successive multiplications of M d, J and of \(\mathbf{A}_{d,J}\mathbf{M}_{d,J}^{-1} =\sum _{ p=1}^{d}(\bigotimes _{q=1}^{p-1}\mathbf{I}_{J}) \otimes \mathbf{A}_{1,J}\mathbf{M}_{1,J}^{-1} \otimes (\bigotimes _{q=p+1}^{d}\mathbf{I}_{J})\).

  3. 3.

    In this paper, we restrict ourselves to homogeneous boundary conditions and do not introduce functions at the boundary. However, by \(u = u_{{\varOmega }^{d}} + u_{\varGamma }\) with \(u_{{\varOmega }^{d}}\vert _{\varGamma } = 0\) and \(-\varDelta u_{{\varOmega }^{d}} = f +\varDelta u_{\varGamma }\), we cover any case with Dirichlet boundary functions u Γ  = g on Γ.

  4. 4.

    This holds for a range of parameters 0 ≤ s < t ≤ r with r being the order of the spline of the space construction. In our case of linear splines r = 2 holds.

  5. 5.

    Note that an additional logarithmic term appears in the error estimate for s = 0, cf. [15].

  6. 6.

    Here, the level J of the full grid is to be equal to the level J of the regular sparse grid or equal to the finest level involved in \(\mathcal{I}\) for the general sparse grid.

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  1. Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstr. 6, D-53115, Bonn, Germany

    Michael Griebel & Alexander Hullmann

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  1. Michael Griebel
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    Michael Griebel

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Griebel, M., Hullmann, A. (2014). On a Multilevel Preconditioner and its Condition Numbers for the Discretized Laplacian on Full and Sparse Grids in Higher Dimensions. In: Griebel, M. (eds) Singular Phenomena and Scaling in Mathematical Models. Springer, Cham. https://doi.org/10.1007/978-3-319-00786-1_12

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