Skip to main content
SpringerLink
Log in

Adjoint Methods for Guiding Adaptive Mesh Refinement in Tsunami Modeling

  • Published:
Pure and Applied Geophysics Aims and scope Submit manuscript

Abstract

One difficulty in developing numerical methods for tsunami modeling is the fact that solutions contain time-varying regions where much higher resolution is required than elsewhere in the domain, particularly when tracking a tsunami propagating across the ocean. The open source GeoClaw software deals with this issue by using block-structured adaptive mesh refinement to selectively refine around propagating waves. For problems where only a target area of the total solution is of interest (e.g., one coastal community), a method that allows identifying and refining the grid only in regions that influence this target area would significantly reduce the computational cost of finding a solution. In this work, we show that solving the time-dependent adjoint equation and using a suitable inner product with the forward solution allows more precise refinement of the relevant waves. We present the adjoint methodology first in one space dimension for illustration and in a broad context since it could also be used in other adaptive software, and potentially for other tsunami applications beyond adaptive refinement. We then show how this adjoint method has been integrated into the adaptive mesh refinement strategy of the open source GeoClaw software and present tsunami modeling results showing that the accuracy of the solution is maintained and the computational time required is significantly reduced through the integration of the adjoint method into adaptive mesh refinement.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  • Akcelik, V., Biros, G., & Ghattas, O. (2002). Parallel multiscale Gauss-Newton-Krylov methods for inverse wave propagation. In: Proceedings of the 2002 ACM/IEEE Conference on Supercomputing, SC ’02 (pp. 1–15).

  • Amante, C., & Eakins, B. W. (2009). ETOPO1 1 Arc-Minute Global Relief Model: Procedures, Data Sources and Analysis. NOAA Technical Memorandum NESDIS NGDC-24, National Geophysical Data Center. USA: NOAA. doi:10.7289/V5C8276M.

  • Asner, L., Tavener, S., & Kay, D. (2012). Adjoint-based a posteriori error estimation for coupled time-dependent systems. SIAM Journal on Scientific Computing, 34, A2394–A2419.

    Article  Google Scholar 

  • Becker, R., & Rannacher, R. (2001). An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, 10, 1–102.

    Article  Google Scholar 

  • Berger, M., & Rigoutsos, I. (1991). An algorithm for point clustering and grid generation. IEEE Transactions on Systems, Man, and Cybernetics, 21(5), 1278–1286.

    Article  Google Scholar 

  • Berger, M. J., George, D. L., LeVeque, R. J., & Mandli, K. T. (2011). The geoclaw software for depth-averaged flows with adaptive refinement. Advances in Water Resources, 24, 1195–1206.

    Article  Google Scholar 

  • Blaise, S., St-Cyr, A., Mavriplis, D., & Lockwood, B. (2013). Discontinuous Galerkin unsteady discrete adjoint method for real-time efficient tsunami simulations. J Comput Phys, 232, 416–430.

  • Borrero, J. C., LeVeque R. J., Greer, D., O’Neill, S., & Davis, B. N. (2015). Observations and modelling of tsunami currents at the Port of Tauranga, New Zealand. In: Australasian Coasts & Ports Conference, Engineers Australia and IPENZ (pp. 90–95).

  • Buffoni, G., & Cupini, E. (2001). The adjoint advection-diffusion equation in stationary and time dependent problems: a reciprocity relation. Rivista di Matematica della Universita di Parma, 4, 9–19.

    Google Scholar 

  • Bunge, H. P., Hagelberg, C. R., & Travis, B. J. (2003). Mantle circulation models with variational data assimilation: inferring past mantle flow and structure from plate motion histories and seismic tomography. Geophysical Journal International, 152, 280–301.

    Article  Google Scholar 

  • Clawpack Development Team. (2015). Clawpack software. http://www.clawpack.org (version 5.3).

  • Davis, B. N. (2015). Adjoint code repository. https://github.com/BrisaDavis/adjoint.

  • Flynt, B. T., & Mavriplis, D. J. (2012). Discrete adjoint based adaptive error control in unsteady flow problems. AIAA Paper 2012-0078.

  • GeoClaw Development Team. (2016). GeoClaw software. http://www.clawpack.org/geoclaw.

  • George, D. L. (2008). Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation. Journal of Computational Physics, 227, 3089–3113.

  • Giles, M. B., & Pierce, N. A. (2000). An introduction to the adjoint approach to design. Flow, Turbulence and Combustion, 65(3–4), 393–415.

  • Giles, M. B., & Suli, E. (2002). Adjoint methods for pdes: a posteriori error analysis and postprocessing by duality. Acta Numerica, 11, 145–236.

    Article  Google Scholar 

  • González, F. I., LeVeque, R. J., Adams, L. M., Goldfinger, C., Priest, G. R., & Wang, K. (2014). Probabilistic Tsunami Hazard Assessment (PTHA) for Crescent City, CA. https://digital.lib.washington.edu/researchworks/handle/1773/25916.

  • Grothe, P., Taylor, L., Eakins, B., Carignan, K., Caldwell, R., Lim, E., & Friday, D. (2011). Digital Elevation Models of Crescent City, California: Procedures, Data Sources and Analysis. NOAA Technical Memorandum NESDIS NGDC-51. Boulder, CO: US Dept. of Commerce. http://www.ngdc.noaa.gov/dem/squareCellGrid/download/693.

  • Hall, M. C. G. (1986). Application of adjoint sensitivity theory to an atmospheric general circulation model. Journal of the Atmospheric Sciences, 43, 2644–2652.

    Article  Google Scholar 

  • Jameson, A. (1988). Aerodynamic design via control theory. Journal of Scientific Computing, 3(3), 233–260.

    Article  Google Scholar 

  • Kast, S. M., & Fidkowski, K. J. (2013). Output-based mesh adaptation for high order navier-stokes simulations on deformable domains. Journal of Computational Physics, 252, 468–494.

  • Kennedy, G. J., & Martins, J. R. R. A. (2013). An adjoint-based derivative evaluation method for time-dependent aeroelastic optimization of flexible aircraft. In: Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (p. 1530). Boston, MA.

  • LeVeque, R. J. (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge: Cambridge University Press.

  • LeVeque, R. J., George, D. L., & Berger, M. J. (2011). Tsunami modeling with adaptively refined finite volume methods. Acta Numerica, 20, 211–289.

  • Luo, Y., & Fidkowski, K. J. (2011). Output-based space-time mesh adaptation for unsteady aerodynamics. AIAA Paper 2011-491.

  • Mani, K., & Mavriplis, D. J. (2007). Discrete adjoint based time-step adaptation and error reduction in unsteady flow problems. AIAA Paper 2007-3944.

  • Marburger, J. (2012). Adjoint-based optimal control of time-dependent free boundary problems. arXiv:1212.3789.

  • Mishra, A., Mani, K., Mavriplis, D., & Sitaraman, J. (2013). Time-dependent adjoint-based optimization for coupled aeroelastic problems. In: 31st AIAA Applied Aerodynamic Conference (p. 2906). AIAA: San Diego, CA.

  • Othmer, C. (2014). Adjoint methods for car aerodynamics. Journal of Mathematics in Industry, 4(1), 6.

    Article  Google Scholar 

  • Park, M. A. (2004). Adjoint-based, three-dimensional error prediction and grid adaptation. AIAA Journal, 42, 1854–1862.

    Article  Google Scholar 

  • Pierce, N. A., & Giles, M. B. (2000). Adjoint recovery of superconvergent functionals from pde approximations. SIAM Review, 42, 247–264.

    Article  Google Scholar 

  • Pires, C., & Miranda, P. M. A. (2001) Tsunami waveform inversion by adjoint methods. Journal of Geophysical Research: Oceans, 106(C9):19773–19796.

    Article  Google Scholar 

  • Pires, C., & Miranda, P. M. A. (2003) Sensitivity of the adjoint method in the inversion of tsunami source parameters. Natural Hazards and Earth System Science, 3(5):341–351.

    Article  Google Scholar 

  • Sanders, B. F., & Katopodes, N. D. (2000). Adjoint sensitivity analysis for shallow-water wave control. Journal of Engineering Mechanics, 126(9), 909919.

    Article  Google Scholar 

  • Tang, L., Chamberlin, C., Tolkova, E., Spillane, M., Titov, V. V., Bernard, E. N., & Mofjeld, H. O. (2006). Assessment of porential tsunami impact for Pearl Harbor. Hawaii: NOAA Technical Memorandum OAR PMEL-131.

  • Tromp, J., Tape, C., & Liu, Q. (2005). Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophysical Journal International, 160, 195–216.

    Article  Google Scholar 

  • Venditti, D. A., & Darmofal, D. L. (2000). Adjoint error estimation and grid adaptation for functional outputs: Application to quasi-one-dimensional flow. Journal of Computational Physics, 164, 204–227.

    Article  Google Scholar 

  • Venditti, D. A., & Darmofal, D. L. (2002). Grid adaptation for functional outputs: application to two-dimensional inviscid flows. Journal of Computational Physics, 176, 40–69.

  • Venditti, D. A., & Darmofal, D. L. (2003). Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows. Journal of Computational Physics, 187, 22–46.

  • Wang, Q., Moin, P., & Iaccarino, G. (2009). Minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation. SIAM Journal on Scientific Computing, 31(4), 2549–2567.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, Seattle, WA, USA

    B. N. Davis & R. J. LeVeque

Authors
  1. B. N. Davis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. J. LeVeque
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. N. Davis.

Additional information

Supported in part by an NSF Graduate Research Fellowship DGE-1256082 and NSF grants DMS-1216732 and EAR-133141.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davis, B.N., LeVeque, R.J. Adjoint Methods for Guiding Adaptive Mesh Refinement in Tsunami Modeling. Pure Appl. Geophys. 173, 4055–4074 (2016). https://doi.org/10.1007/s00024-016-1412-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00024-016-1412-y

Keywords

Search

Navigation