Abstract
With uncertainty quantification, we aim to efficiently propagate the uncertainties in the input parameters of a computer simulation, in order to obtain a probability distribution of its output. In this work, we use multi-fidelity sparse grid interpolation to propagate the uncertainty in the shape of the incoming wave for the Okushiri test-case, which is a wave tank model of a part of the 1993 Hokkaido Nansei-oki tsunami. An important issue with many uncertainty quantification approaches is the ‘curse of dimensionality’: the overall computational cost of the uncertainty propagation increases rapidly when we increase the number of uncertain input parameters. We aim to mitigate the curse of dimensionality by using a multifidelity approach. In the multifidelity approach, we combine results from a small number of accurate and expensive high-fidelity simulations with a large number of less accurate but also less expensive low-fidelity simulations. For the Okushiri test-case, we find an improved scaling when we increase the number of uncertain input parameters. This results in a significant reduction of the overall computational cost. For example, for four uncertain input parameters, accurate uncertainty quantification based on only high-fidelity simulations comes at a normalised cost of 219 high-fidelity simulations; when we use a multifidelity approach, this is reduced to a normalised cost of only 10 high-fidelity simulations.
Similar content being viewed by others
References
Beck, J., & Guillas, S. (2016). Sequential design with mutual information for computer experiments (MICE): emulation of a tsunami model. SIAM/ASA Journal for Uncertainty Quantification, 4, 739–766.
Behrens, J., & Dias, F. (2015). New computational methods in tsunami science. Philosophical Transactions of the Royal Society A, 373, 1–15.
Bungartz, H.-J., & Dirnstorfer, S. (2004). Higher order quadrature on sparse grids. In M. Bubak, G.D. van Albada, J. Dongarra, P.M. Sloot (Eds.), International Conference on Computational Science ICCS 2004: 4th International Conference, Kraköw, Poland, June 6-9, 2004, Proceedings, Part IV (pp. 394–401). Heidelberg: Springer.
Bungartz, H.-J., & Griebel, M. (2004). Sparse grids. Acta Numerica, 13, 147–269.
Cliffe, K. A., Giles, M. B., Scheichl, R., & Teckentrup, A. L. (2011). Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Computing and Visualization in Science, 14, 3–15.
de Baar, J. H. S., & Harding, B. (2015). A gradient-enhanced sparse grid algorithm for uncertainty quantification. International Journal for Uncertainty Quantification, 5(5), 453–468.
de Baar, J. H. S., Dwight, R. P., & Bijl, H. (2013). Speeding up kriging through fast estimation of the hyperparameters in the frequency-domain. Computers & Geosciences, 54, 99–106.
de Baar, J. H. S., Roberts, S. G., Dwight, R. P., & Mallol, B. (2015). Uncertainty quantification for a sailing yacht hull, using multi-fidelity Kriging. Computers & Fluids, 123, 185–201.
Forrester, A. I. J., Sobester, A., & Keane, J. (2007). Multi-fidelity optimization via surrogate modelling. Proceedings of the Royal Society A, 463, 3251–3269.
Garcke, J. (2013). Sparse Grids in a Nutshell. In J. Garcke & M. Griebel (Eds.), Sparse Grids and Applications (Vol. 88, pp. 57–80)., Lecture Notes in Computational Science and Engineering Berlin: Springer.
Griebel, M., Schneider, M., & Zenger, C. (1992). A combination technique for the solution of sparse grid problems. In P. de Groen, R. Beauwens (Ed.), Iterative Methods in Linear Algebra: proceedings of the ImACS International Symposium on Iterative Methods in Linear Algebra, Brussels, Belgium, 2-4 April, 1991 (pp. 263–281).
Kennedy, M. C., & O’Hagan, A. (2000). Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87, 1–13.
Matsuyama, M., & Tanaka, H. (2001). An experimental study of the highest run-up height in 1993 Hokkaido Nansei-oki earthquake tsunami. International Tsunami Symposium.
Metropolis, N., & Ulam, S. (1949). The Monte Carlo method. Journal of the American Statistical Association, 44, 335–341.
Ng, L. W. T. & Eldred, M. S. (2012). Multifidelity uncertainty quantification using non-intrusive polynomial chaos and stochastic collocation. In 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 23–26 April, Honolulu, Hawaii.
Ng, L. W. T., & Willcox, K. E. (2014). Multifidelity approaches for optimization under uncertainty. International Journal for Numerical Methods in Engineering, 10, 746–772.
Ng, L. W. T. & Willcox, K. E. (2015). Monte Carlo information-reuse approach to aircraft conceptual design optimization under uncertainty. Journal of Aircraft (AIAA Early Edition), 53(2), 1–12.
Nielsen, O., Roberts, S., Gray, D., McPherson, A., & Hitchman, A. (2005). Hydrodynamic modelling of coastal inundation. MODSIM International Congress on Modelling and Simulation, Modelling and Simulation Society of Australia & New Zealand (pp. 518–523).
Oden, T., Moser, R., & Ghattas, O. (2010a). Computer predictions with quantified uncertainty (Part 1). SIAM News, 43(9).
Oden, T., Moser, R., & Ghattas O. (2010b). Computer predictions with quantified uncertainty (Part 2). SIAM News, 43(10).
Pellegrini, R., Leotardi, C., Iemma, U., Campana, E. F., & Diez, M. (2016). A multi-fidelity adaptive sampling method for metamodel-based uncertainty quantification of computer simulations. In: ECCOMAS, 5–10 June, Crete Island, Greece.
Roberts, S., Nielsen, O., Gray, D., Sexton, J., & Davies, G. (2015). ANUGA User Manual. Geoscience Australia.
Sacks, J., Welch, W. J., Mitchell, T. J., & Wynn, H. P. (1989). Design and analysis of computer experiments. Statistical Science, 4(4), 409–423.
Salmanidou, D., Guillas, S., Georgiopoulou, A., & Dias, F. (2017). Statistical emulation of landslide-induced tsunamis at the rockall bank, NE Atlantic. Proceedings of Royal Society A, 473(2200), 20170026.
Santiago Padrón, A., Alonso, J. J., & Palacios, F. (2014). Multi-fidelity uncertainty quantification: application to a vertical axis wind turbine under an extreme gust. In 15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 16–20 June, Atlanta, GA.
Sarri, A., Guillas, S., & Dias, F. (2012). Statistical emulation of a tsunami model for sensitivity analysis and uncertainty quantification. Natural Hazards and Earth System Sciences, 12, 2003–2018.
Sraj, I., Mandli, K. T., Knio, O. M., Dawson, C. N., & Hoteit, I. (2014). Uncertainty quantification and inference of Mannings friction coefficients using DART buoy data during the Tohoku tsunami. Ocean Model, 83, 82–97.
Stefanakis, T. S., Contal, E., Vayatis, N., Dias, F., & Synolakis, C. E. (2014). Can small islands protect nearby coasts from tsunamis? An active experimental design approach. Proceedings of the Royal Society A, 470, 1–20.
Toal, D. J. (2015). Some considerations regarding the use of multi-fidelity Kriging in the construction of surrogate models. Structural and Multidisciplinary Optimization, 51(6).
Viana, F. A. C., Simpson, T. W., Balabanov, V., & Toropov, V. (2014). Metamodeling in multidisciplinary design optimization: how far have we really come? AIAA Journal, 52(4), 670–690.
Zenger, C. (1991). Sparse grids. In W. Hackbush (Ed.), Parallel algorithms for partial differential equations, Notes on Numerical Fluid Mechanics, 31. Braunschweig: Vieweg.
Acknowledgements
This research was undertaken with the assistance of resources from the National Computational Infrastructure (NCI), which is supported by the Australian Government.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
de Baar, J.H.S., Roberts, S.G. Multifidelity Sparse-Grid-Based Uncertainty Quantification for the Hokkaido Nansei-oki Tsunami. Pure Appl. Geophys. 174, 3107–3121 (2017). https://doi.org/10.1007/s00024-017-1606-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00024-017-1606-y