Sparse Grids and Applications - Stuttgart 2014 pp 191-220 | Cite as

# An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient

## 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.

## Keywords

Adaptive Algorithm Collocation Point Quadrature Rule Sparse Grid Work Contribution## Notes

### 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.

## References

- 1.I. Babuška, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev.
**52**(2), 317–355 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 2.J. Bäck, F. Nobile, L. Tamellini, R. Tempone, Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison, in
*Spectral and High Order Methods for Partial Differential Equations*, ed. by J. Hesthaven, E. Ronquist. Volume 76 of Lecture Notes in Computational Science and Engineering (Springer, Berlin/Heidelberg, 2011), pp. 43–62. Selected papers from the ICOSAHOM’09 conference, 22–26 June 2009, TrondheimGoogle Scholar - 3.V. Barthelmann, E. Novak, K. Ritter, High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math.
**12**(4), 273–288 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 4.J. Beck, F. Nobile, L. Tamellini, R. Tempone, On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods. Math. Models Methods Appl. Sci.
**22**(09), 1250023 (2012)Google Scholar - 5.J. Beck, F. Nobile, L. Tamellini, R. Tempone, A Quasi-optimal sparse grids procedure for groundwater flows, in
*Spectral and High Order Methods for Partial Differential Equations – ICOSAHOM’12*, ed. by M. Azaïez, H. El Fekih, J. S. Hesthaven. Volume 95 of Lecture Notes in Computational Science and Engineering (Springer International Publishing, Switzerland, 2014), pp. 1–16. Selected papers from the ICOSAHOM’12 conferenceGoogle Scholar - 6.H. Bungartz, M. Griebel, Sparse grids. Acta Numer.
**13**, 147–269 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 7.J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal.
**50**(1), 216–246 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 8.A. Chkifa, A. Cohen, C. Schwab, High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs.
*Found. Comput. Math.***14**, 601–633 (2014)Google Scholar - 9.K. Cliffe, M. Giles, R. Scheichl, A. Teckentrup, Multilevel monte carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci.
**14**(1), 3–15 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 10.B.A. Davey, H.A. Priestley,
*Introduction to Lattices and Order*, 2nd edn. (Cambridge University Press, New York, 2002)CrossRefzbMATHGoogle Scholar - 11.P. Diggle, P.J. Ribeiro,
*Model-Based Geostatistics*(Springer, New York, 2007)zbMATHGoogle Scholar - 12.M.S. Eldred, J. Burkardt, Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. American Institute of Aeronautics and Astronautics Paper 2009–0976 (2009)Google Scholar
- 13.H.C. Elman, C.W. Miller, E.T. Phipps, R.S. Tuminaro, Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int. J. Uncertain. Quantif.
**1**(1), 19–33 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 14.O. Ernst, B. Sprungk, Stochastic collocation for elliptic PDEs with random data: the lognormal case, in Sparse Grids and Applications – Munich 2012, ed. by J. Garcke, D. Pflüger. Volume 97 of Lecture Notes in Computational Science and Engineering (Springer International Publishing, Switzerland, 2014), pp. 29–53Google Scholar
- 15.J. Foo, X. Wan, G. Karniadakis, The multi-element probabilistic collocation method (ME-PCM): error analysis and applications. J. Comput. Phys.
**227**(22), 9572–9595 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 16.A. Genz, B.D. Keister, Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. J. Comput. Appl. Math.
**71**(2), 299–309 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 17.T. Gerstner, M. Griebel, Dimension-adaptive tensor-product quadrature. Computing
**71**(1), 65–87 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 18.C.J. Gittelson, Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci.
**20**(2), 237–263 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 19.H. Harbrecht, M. Peters, M. Siebenmorgen, Multilevel accelerated quadrature for PDEs with log-normal distributed random coefficient. Preprint 2013–18 (Universität Basel, 2013)Google Scholar
- 20.J.D. Jakeman, R. Archibald, D. Xiu, Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids. J. Comput. Phys.
**230**(10), 3977–3997 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 21.A. Klimke, Uncertainty modeling using fuzzy arithmetic and sparse grids, PhD thesis, Universität Stuttgart, Shaker Verlag, Aachen, 2006Google Scholar
- 22.S. Martello, P. Toth,
*Knapsack Problems: Algorithms and Computer Implementations*. Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley, New York, 1990)Google Scholar - 23.A. Narayan, J.D. Jakeman, Adaptive leja sparse grid constructions for stochastic collocation and high-dimensional approximation. SIAM J. Sci. Comput.
**36**(6), A2952–A2983 (2014)CrossRefzbMATHGoogle Scholar - 24.F. Nobile, L. Tamellini, R. Tempone, Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs. Numerische Mathematik, doi:10.1007/s00211-015-0773-yGoogle Scholar
- 25.F. Nobile, L. Tamellini, R. Tempone, Comparison of Clenshaw–Curtis and Leja quasi-optimal sparse grids for the approximation of random PDEs, in
*Spectral and High Order Methods for Partial Differential Equations – ICOSAHOM’14*, ed. by R.M. Kirby, M. Berzins, J.S. Hesthaven. Volume 106 of Lecture Notes in Computational Science and Engineering (Springer International Publishing, Switzerland, 2015)Google Scholar - 26.F. Nobile, R. Tempone, C. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal.
**46**(5), 2411–2442 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 27.F. Nobile, F. Tesei, A Multi Level Monte Carlo method with control variate for elliptic PDEs with log-normal coefficients. Stoch PDE: Anal Comp (2015) 3:398–444MathSciNetCrossRefzbMATHGoogle Scholar
- 28.C. Schillings, C. Schwab, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems. Inverse Probl.
**29**(6), 065011 (2013)Google Scholar - 29.S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR
**4**, 240–243 (1963)zbMATHGoogle Scholar - 30.L. Tamellini, F. Nobile,
*Sparse Grids Matlab kit*v.15-8. http://csqi.epfl.ch, 2011–2015 - 31.G. Wasilkowski, H. Wozniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complex.
**11**(1), 1–56 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 32.G. Zhang, D. Lu, M. Ye, M. Gunzburger, C. Webster, An adaptive sparse-grid high-order stochastic collocation method for bayesian inference in groundwater reactive transport modeling. Water Resour. Res.
**49**(10), 6871–6892 (2013)CrossRefGoogle Scholar