Fast algorithms for bayesian uncertainty quantification in large-scale linear inverse problems based on low-rank partial hessian approximations

SIAM Journal on Scientific Computing

Volumn 33, Issue 1, 2011, Pages 407-432

  Flath, H P ^a   Wilcox, L C ^a   AkcElik V ^b,c   Hill, J ^d,e   Van Bloemen Waanders, B ^d   Ghattas, O ^f  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b SLAC NATIONAL ACCELERATOR LABORATORY   (United States)

^c EXXONMOBIL UPSTREAM RESEARCH COMPANY   (United States)

^d SANDIA NATIONAL LABORATORIES   (United States)

^e OAK RIDGE NATIONAL LABORATORY   (United States)

^f UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Bayesian inference; Contaminant transport; Convection diffusion; Fast algorithms; Hessian; Lanczos; Large scale statistical inverse problem; Low rank approximation; Uncertainty quantification

Indexed keywords

APPROXIMATION ALGORITHMS; BAYESIAN NETWORKS; COVARIANCE MATRIX; DIFFERENTIAL EQUATIONS; HEAT CONVECTION; INFERENCE ENGINES; PARAMETER ESTIMATION; PROBABILITY DENSITY FUNCTION; PROBLEM SOLVING; UNCERTAINTY ANALYSIS;

BAYESIAN INFERENCE; CONTAMINANTS TRANSPORT; CONVECTION DIFFUSION; FAST ALGORITHMS; HESSIAN; LANCZOS; LARGE-SCALE STATISTICAL INVERSE PROBLEM; LARGE-SCALES; LOW RANK APPROXIMATIONS; STATISTICAL INVERSE PROBLEMS; UNCERTAINTY QUANTIFICATIONS;

INVERSE PROBLEMS;

EID: 79952307064     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/090780717     Document Type: Article

References (32)

1. V. Akçelik, G. Biros, A. Draganescu, O. Ghattas, J. Hill, and B. van Bloeman Waanders, Dynamic data-driven inversion for terascale simulations: Real-time identification of airborne contaminants, in Proceedings of the ACM/IEEE SC2005 Conference on High Performance Networking and Computing, Seattle, WA, 2005.
2. V. Akçelik, G. Biros, O. Ghattas, J. Hill, D. Keyes, and B. van Bloeman Waanders, Parallel PDE constrained optimization, in Parallel Processing for Scientific Computing, M. A. Heroux, P. Raghaven, and H. D. Simon, eds., SIAM, Philadelphia, 2006.
3. V. Akçelik, G. Biros, O. Ghattas, K. R. Long, and B. van Bloemen Waanders, A variational finite element method for source inversion for convective-diffusive transport, Finite Elem. Anal. Des., 39 (2003), pp. 683-705.
4. G. Backus and F. Gilbert, The resolving power of gross earth data, Geophys. J. R. Astron. Soc., 16 (1968), pp. 169-205.
5. S. Balay, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc Users Manual, Technical report ANL-95/11 -Revision 3.1, Argonne National Laboratory, Argonne, IL, 2010.
6. S. Balay, K. Buschelman, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc Web page, http://www.mcs.anl.gov/petsc (2009).
7. S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools for Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds., Birkhäuser Boston, Boston, MA, 1997, pp. 163-202.
8. H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Systems Control Found. Appl. 1, Birkhäuser-Verlag, Boston, Basel, Berlin, 1989.
9. J. G. Berryman, Analysis of approximate inverses in tomography I. Resolution analysis of common inverses, Optim. Eng., 1 (2001), pp. 87-115.
10. J. G. Berryman, Analysis of approximate inverses in tomography II. Iterative inverses, Optim. Eng., 1 (2001), pp. 437-473.
11. L. T. Biegler, O. Ghattas, M. Heinkenschloss, and B. van Bloemen Waanders, eds., Large-Scale PDE-Constrained Optimization, Lect. Notes Comput. Sci. Eng. 30, Springer-Verlag, Heidelberg, 2003.
12. L. Boschi, Measures of resolution in global body wave tomography, Geophys. Res. Lett., 30 (2003), SDE 2-1-2-4.
13. T. Bui-Thanh, O. Ghattas, D. Higdon, Z. Lu, and L. Wilcox, Hessian-Informed Gaussian Process Response Surface Methods for Probability Density Approximation, manuscript, 2011.
14. C. Burstedde, O. Ghattas, J. Martin, and L. C. Wilcox, Uncertainty Quantification in Inverse Problems with Stochastic Newton MCMC, manuscript, 2011.
15. S. Fomel, J. Berryman, R. Clapp, and M. F. Prucha, Iterative resolution estimation in least squares Kirchhoff migration, Geophys. Prospect., 50 (2002), pp. 577-588. (Pubitemid 36000235)
16. W. P. Gouveia and J. A. Scales, Resolution of seismic waveform inversion: Bayes versus Occam, Inverse Problems, 13 (1997), pp. 323-349. (Pubitemid 127549969)
17. W. P. Gouveia and J. A. Scales, Bayesian seismic waveform inversion: Parameter estimation and uncertainty analysis, J. Geophysical Res., 103 (1998), pp. 2759-2779.
18. V. Hernandez, J. E. Roman, A. Tomas, and V. Vidal, Lanczos Methods in SLEPc, Technical report STR-5, Universidad Politécnica de Valencia, Valencia, Spain, 2006; available online from http://www.grycap.upv.es/slepc.
19. V. Hernandez, J. E. Roman, and V. Vidal, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software, 31 (2005), pp. 351-362.
20. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints, Springer-Verlag, New York, 2009.
21. K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, SIAM, Philadelphia, 2008.
22. J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Appl. Math. Sci., 160, Springer-Verlag, New York, 2005.
23. O. Marques, T. Drummond, and D. Vasco, A computational strategy for the solution of large linear inverse problems in geophysics, in Proceedings of the 17th IEEE International Symposium on Parallel and Distributed Processing, 2003, p. 16.2.
24. S. Minkoff, A computationally feasible approximate resolution matrix for seismic inverse problems, Geophys. J. Int., 126 (1996), pp. 345-359.
25. G. Nolet, R. Montelli, and J. Virieux, Explicit, approximate expressions for the resolution and a-posteriori covariance of massive tomographic systems, Geophys. J. Int., 138 (1999), pp. 36-44. (Pubitemid 29462065)
26. A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, 2005.
27. D. W. Vasco, Jr., J. E. Peterson, and E. L. Majer, Resolving seismic anisotropy: Sparse matrix methods for geophysical inverse problems, Geophysics, 63 (1998), pp. 970-983.
28. D. W. Vasco, R. J. Pulliam, and L. R. Johnson, Formal inversion of ISC arrival times for mantle P-velocity structure, Geophys. J. Int., 113 (1993), pp. 586-606. (Pubitemid 23394158)
29. C. R. Vogel, Computational Methods for Inverse Problems, Frontiers Appl. Math. 23, SIAM, Philadelphia, 2002.
30. Z. S. Yao, R. G. Roberts, and A. Tryggvason, Calculating resolution and covariance matrices for seismic tomography with the LSQR method, Geophys. J. Int., 138 (1999), pp. 886-894.
31. H. Zhang and C. H. Thurber, Estimating the model resolution matrix for large seismic tomography problems based on Lanczos bidiagonalization with partial reorthogonalization, Geophys. J. Int., 170 (2007), pp. 337-345. (Pubitemid 47044674)
32. J. Zhang and G. A. McMechan, Estimation of resolution and covariance for large matrix inversions, Geophys. J. Int., 121 (1995), pp. 409-426.