Abstract
In this work we build on the classical adaptive sparse grid algorithm (T. Gerstner and M. Griebel, Dimension-adaptive tensor-product quadrature), obtaining an enhanced version capable of using non-nested collocation points, and supporting quadrature and interpolation on unbounded sets. We also consider several profit indicators that are suitable to drive the adaptation process. We then use such algorithm to solve an important test case in Uncertainty Quantification problem, namely the Darcy equation with lognormal permeability random field, and compare the results with those obtained with the quasi-optimal sparse grids based on profit estimates, which we have proposed in our previous works (cf. e.g. Convergence of quasi-optimal sparse grids approximation of Hilbert-valued functions: application to random elliptic PDEs). To treat the case of rough permeability fields, in which a sparse grid approach may not be suitable, we propose to use the adaptive sparse grid quadrature as a control variate in a Monte Carlo simulation. Numerical results show that the adaptive sparse grids have performances similar to those of the quasi-optimal sparse grids and are very effective in the case of smooth permeability fields. Moreover, their use as control variate in a Monte Carlo simulation allows to tackle efficiently also problems with rough coefficients, significantly improving the performances of a standard Monte Carlo scheme.
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Notes
- 1.
A sublinear functional over a vector space X is a function \(\Theta: X \rightarrow \mathbb{R}\) such that
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\(\Theta (\alpha x) =\alpha \Theta (x)\), \(\forall \alpha> 0\) and x ∈ X;
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\(\Theta (x + y) \leq \Theta (x) + \Theta (y)\), \(\forall x,y \in X\).
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- 2.
Here the n-th component of q denotes the polynomial degree with respect to y n .
- 3.
A function \(f: \overline{D} \subset \mathbb{R}^{d} \rightarrow \mathbb{R}\) is said to be Hölder continuous with parameter s ∈ (0, 1], \(f \in C^{0,s}(\overline{D})\), if there exist non-negative real constants C and s such that
$$\displaystyle{\vert f(\mathbf{x}) - f(\mathbf{y})\vert \leq C\vert \mathbf{x} -\mathbf{y}\vert _{2}^{s}\quad \forall \mathbf{x},\mathbf{y} \in \overline{D}.}$$
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Acknowledgements
F. Nobile, F. Tesei and L. Tamellini have received support from the Center for ADvanced MOdeling Science (CADMOS) and partial support by the Swiss National Science Foundation under the Project No. 140574 “Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media”. R. Tempone is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.
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Nobile, F., Tamellini, L., Tesei, F., Tempone, R. (2016). An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient. In: Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Stuttgart 2014. Lecture Notes in Computational Science and Engineering, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-319-28262-6_8
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