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A transport-based multifidelity preconditioner for Markov chain Monte Carlo

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Abstract

Markov chain Monte Carlo (MCMC) sampling of posterior distributions arising in Bayesian inverse problems is challenging when evaluations of the forward model are computationally expensive. Replacing the forward model with a low-cost, low-fidelity model often significantly reduces computational cost; however, employing a low-fidelity model alone means that the stationary distribution of the MCMC chain is the posterior distribution corresponding to the low-fidelity model, rather than the original posterior distribution corresponding to the high-fidelity model. We propose a multifidelity approach that combines, rather than replaces, the high-fidelity model with a low-fidelity model. First, the low-fidelity model is used to construct a transport map that deterministically couples a reference Gaussian distribution with an approximation of the low-fidelity posterior. Then, the high-fidelity posterior distribution is explored using a non-Gaussian proposal distribution derived from the transport map. This multifidelity “preconditioned” MCMC approach seeks efficient sampling via a proposal that is explicitly tailored to the posterior at hand and that is constructed efficiently with the low-fidelity model. By relying on the low-fidelity model only to construct the proposal distribution, our approach guarantees that the stationary distribution of the MCMC chain is the high-fidelity posterior. In our numerical examples, our multifidelity approach achieves significant speedups compared with single-fidelity MCMC sampling methods.

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References

  1. Allaire, D., Willcox, K.: A mathematical and computational framework for multifidelity design and analysis with computer models. Int. J. Uncertain. Quantif. 4(1), 1–20 (2014)

    MathSciNet  Google Scholar 

  2. Balakrishnan, S., Roy, A., Ierapetritou, M.G., Flach, G.P., Georgopoulos, P.G.: Uncertainty reduction and characterization for complex environmental fate and transport models: an empirical Bayesian framework incorporating the stochastic response surface method. Water Resour. Res. 39(12), 1–13 (2003)

    Google Scholar 

  3. Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Beskos, A., Jasra, A., Law, K., Tempone, R., Zhou, Y.: Multilevel sequential Monte Carlo samplers. Stochastic Processes and their Applications 127 (5), 1417–1440 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Bigoni, D., Spantini, A., Morrison, R., Baptista, R.M.: Documentation of TransportMaps software package. Tech. rep., Uncertainty Quantification Group, Massachusetts Institute of Technology. http://transportmaps.mit.edu (2018)

  6. Bonnotte, N.: From Knothe’s rearrangement to Brenier’s optimal transport map. SIAM J. Math. Anal. 45(1), 64–87 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Boyaval, S.: A fast Monte–Carlo method with a reduced basis of control variates applied to uncertainty propagation and Bayesian estimation. Comput. Methods Appl. Mech. Eng. 241–244, 190–205 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Boyaval, S., Lelièvre, T.: A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm. Commun. Math. Sci. 8(3), 735–762 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Brooks, S., Gelman, A., Jones, G.L., Meng, X.L.: Handbook of Markov Chain Monte Carlo. Chapman and Hall/CRC, London (2011)

    MATH  Google Scholar 

  10. Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numerica 13, 147–269 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Carlier, G., Galichon, A., Santambrogio, F.: From Knothe’s transport to Brenier’s map and a continuation method for optimal transport. SIAM J. Math. Anal. 41(6), 2554–2576 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Chen, P., Quarteroni, A.: Accurate and efficient evaluation of failure probability for partial different equations with random input data. Comput. Methods Appl. Mech. Eng. 267, 233–260 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Chen, P., Quarteroni, A., Rozza, G.: Reduced basis methods for uncertainty quantification. SIAM/ASA J. Uncertain. Quantif. 5(1), 813–869 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Chen, P., Schwab, C.: Sparse-grid, reduced-basis Bayesian inversion. Comput. Methods Appl. Mech. Eng. 297, 84–115 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Christen, J.A., Fox, C.: Markov chain Monte Carlo using an approximation. J. Comput. Graph. Stat. 14(4), 795–810 (2005)

    MathSciNet  Google Scholar 

  16. Conrad, P., Davis, A., Marzouk, Y., Pillai, N., Smith, A.: Parallel local approximation MCMC for expensive models. SIAM/ASA J. Uncertain. Quantif. 6 (1), 339–373 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Conrad, P.R., Marzouk, Y., Pillai, N.S., Smith, A.: Accelerating asymptotically exact MCMC for computationally intensive models via local approximations. J. Am. Stat. Assoc. 111(516), 1591–1607 (2016)

    MathSciNet  Google Scholar 

  18. Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20(3), 273–297 (1995)

    MATH  Google Scholar 

  19. Cotter, S., Dashti, M., Stuart, A.: Approximation of Bayesian inverse problems for PDEs. SIAM J. Numer. Anal. 48(1), 322–345 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Cui, T., Marzouk, Y., Willcox, K.: Data-driven model reduction for the Bayesian solution of inverse problems. Int. J. Numer. Methods Eng. 102(5), 966–990 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Dodwell, T.J., Ketelsen, C., Scheichl, R., Teckentrup, A.L.: A hierarchical multilevel Markov chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow. SIAM/ASA J. Uncertain. Quantif. 3 (1), 1075–1108 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Drohmann, M., Carlberg, K.: The ROMES method for statistical modeling of reduced-order-model error. SIAM/ASA J. Uncertain. Quantif. 3(1), 116–145 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Efendiev, Y., Hou, T., Luo, W.: Preconditioning Markov chain Monte Carlo simulations using coarse-scale models. SIAM J. Sci. Comput. 28(2), 776–803 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Eldred, M.S., Ng, L.W.T., Barone, M.F., Domino, S.P.: Multifidelity uncertainty quantification using spectral stochastic discrepancy models. In: Ghanem, R., Higdon, D., Owhadi, H. (eds.) Handbook of Uncertainty Quantification, pp 1–45. Springer, Cham (2016)

    Google Scholar 

  25. Forrester, A.I.J., Keane, A.J.: Recent advances in surrogate-based optimization. Prog. Aerosp. Sci. 45(1–3), 50–79 (2009)

    Google Scholar 

  26. Frangos, M., Marzouk, Y., Willcox, K., van Bloemen Waanders, B.: Surrogate and Reduced-Order Modeling: a Comparison of Approaches for Large-Scale Statistical Inverse Problems, pp 123–149. Wiley, New York (2010)

    Google Scholar 

  27. Gilks, W.R., Richardson, S., Spiegelhalter, D.J.: Markov Chain Monte Carlo in Practice. Chapman & Hall, London (1996)

    MATH  Google Scholar 

  28. Gugercin, S., Antoulas, A.: A survey of model reduction by balanced truncation and some new results. Int. J. Control. 77(8), 748–766 (2004)

    MathSciNet  MATH  Google Scholar 

  29. Haario, H., Laine, M., Mira, A., Saksman, E.: DRAM: efficient adaptive MCMC. Stat. Comput. 16(4), 339–354 (2006)

    MathSciNet  Google Scholar 

  30. Haario, H., Saksman, E., Tamminen, J.: An adaptive Metropolis algorithm. Bernoulli 7(2), 223–242 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springer, Berlin (2005)

    MATH  Google Scholar 

  32. Latz, J., Papaioannou, I., Ullmann, E.: Multilevel sequential2 Monte Carlo for Bayesian inverse problems. J. Comput. Phys. 368, 154–178 (2018)

    MathSciNet  MATH  Google Scholar 

  33. Li, J., Marzouk, Y.: Adaptive construction of surrogates for the Bayesian solution of inverse problems. SIAM J. Sci. Comput. 36(3), A1163–A1186 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Lieberman, C., Willcox, K., Ghattas, O.: Parameter and state model reduction for large-scale statistical inverse problems. SIAM J. Sci. Comput. 32(5), 2523–2542 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, Berlin (2008)

    MATH  Google Scholar 

  36. Manzoni, A., Pagani, S., Lassila, T.: Accurate solution of Bayesian inverse uncertainty quantification problems combining reduced basis methods and reduction error models. SIAM/ASA J. Uncertain. Quantif. 4(1), 380–412 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Marzouk, Y., Moselhy, T., Parno, M., Spantini, A.: Sampling via measure transport: an introduction. In: Ghanem, R., Higdon, D., Owhadi, H. (eds.) Handbook of Uncertainty Quantification, pp 1–41. Springer, Cham (2016)

    Google Scholar 

  38. Marzouk, Y., Najm, H.N.: Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comput. Phys. 228(6), 1862–1902 (2009)

    MathSciNet  MATH  Google Scholar 

  39. Marzouk, Y., Najm, H.N., Rahn, L.A.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. 224(2), 560–586 (2007)

    MathSciNet  MATH  Google Scholar 

  40. Marzouk, Y., Xiu, D.: A stochastic collocation approach to Bayesian inference in inverse problems. Commun. Comput. Phys. 6, 826–847 (2009)

    MathSciNet  MATH  Google Scholar 

  41. Moselhy, T.A.E., Marzouk, Y.: Bayesian inference with optimal maps. J. Comput. Phys. 231(23), 7815–7850 (2012)

    MathSciNet  MATH  Google Scholar 

  42. Ng, L., Willcox, K.: Multifidelity approaches for optimization under uncertainty. Int. J. Numer. Methods Eng. 100(10), 746–772 (2014)

    MathSciNet  MATH  Google Scholar 

  43. Ng, L., Willcox, K.: Monte-carlo information-reuse approach to aircraft conceptual design optimization under uncertainty. J. Aircr., 1–12 (2015)

  44. Parno, M.: Transport maps for accelerated Bayesian computation. Ph.D. thesis Massachusetts Institute of Technology (2015)

  45. Parno, M., Marzouk, Y.: Transport map accelerated Markov chain Monte Carlo. SIAM/ASA J. Uncertain. Quantif. 6(2), 645–682 (2018)

    MathSciNet  MATH  Google Scholar 

  46. Peherstorfer, B.: Multifidelity Monte Carlo estimation with adaptive low-fidelity models. SIAM/ASA J. Uncertain. Quantif. (accepted) (2019)

  47. Peherstorfer, B., Cui, T., Marzouk, Y., Willcox, K.: Multifidelity importance sampling. Comput. Methods Appl. Mech. Eng. 300, 490–509 (2016)

    MathSciNet  MATH  Google Scholar 

  48. Peherstorfer, B., Kramer, B., Willcox, K.: Multifidelity preconditioning of the cross-entropy method for rare event simulation and failure probability estimation. SIAM/ASA J. Uncertain. Quantif. 6(2), 737–761 (2018)

    MathSciNet  MATH  Google Scholar 

  49. Peherstorfer, B., Willcox, K., Gunzburger, M.: Optimal model management for multifidelity Monte Carlo estimation. SIAM J. Sci. Comput. 38(5), A3163–A3194 (2016)

    MathSciNet  MATH  Google Scholar 

  50. Peherstorfer, B., Willcox, K., Gunzburger, M.: Survey of multifidelity methods in uncertainty propagation, inference, and optimization. SIAM Rev. (2018)

  51. Pflüger, D., Peherstorfer, B., Bungartz, H.: Spatially adaptive sparse grids for high-dimensional data-driven problems. J. Complex. 26(5), 508–522 (2010)

    MathSciNet  MATH  Google Scholar 

  52. Qian, E., Peherstorfer, B., O’Malley, D., Vesselinov, V.V., Willcox, K.: Multifidelity Monte Carlo estimation of variance and sensitivity indices. SIAM/ASA J. Uncertain. Quantif. 6(2), 683–706 (2018)

    MathSciNet  MATH  Google Scholar 

  53. Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer, Berlin (2004)

    MATH  Google Scholar 

  54. Roberts, G.O., Rosenthal, J.S.: Examples of adaptive MCMC. J. Comput. Graph. Stat. 18(2), 349–367 (2009)

    MathSciNet  Google Scholar 

  55. Rosenblatt, M.: Remarks on a multivariate transformation. Ann. Math. Statist. 23(3), 470–472 (1952)

    MathSciNet  MATH  Google Scholar 

  56. Rozza, G., Huynh, D., Patera, A.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Meth. Eng. 15(3), 1–47 (2007)

    Google Scholar 

  57. Santin, G., Wittwar, D., Haasdonk, B.: Greedy regularized kernel interpolation. arXiv:1807.09575 (2018)

  58. Sirovich, L.: Turbulence and the dynamics of coherent structures. Q. Appl. Math. 45, 561–571 (1987)

    MathSciNet  MATH  Google Scholar 

  59. Spall, J.: Introduction to Stochastic Search and Optimization, Estimation, Simulation, and Control. Wiley, New York (2003)

    MATH  Google Scholar 

  60. Spantini, A.: On the low-dimensional structure of Bayesian inference. Ph.D. thesis Massachusetts Institute of Technology (2017)

  61. Spantini, A., Bigoni, D., Marzouk, Y.: Inference via low-dimensional couplings. J. Mach. Learn. Res. 19(66), 1–71 (2018)

    MathSciNet  MATH  Google Scholar 

  62. Stuart, A.M.: Inverse problems: a Bayesian perspective. Acta Numerica 19, 451–559 (2010)

    MathSciNet  MATH  Google Scholar 

  63. Tarantola, A.: Inverse Problem Theory. Elsevier, Amsterdam (1987)

    MATH  Google Scholar 

  64. Tierney, L.: Markov chains for exploring posterior distributions. Ann. Statist. 22(4), 1701–1728 (1994)

    MathSciNet  MATH  Google Scholar 

  65. Vapnik, V.: Statistical Learning Theory. Wiley, New York (1998)

    MATH  Google Scholar 

  66. Vershik, A.M.: Long history of the Monge-Kantorovich transportation problem. Math. Intell. 35(4), 1–9 (2013)

    MathSciNet  MATH  Google Scholar 

  67. Villani, C.: Topics in optimal transportation. American Mathematical Society (2003)

  68. Villani, C.: Optimal Transport: Old and New. Springer, Berlin (2009)

    MATH  Google Scholar 

  69. Wang, J., Zabaras, N.: Using Bayesian statistics in the estimation of heat source in radiation. Int. J. Heat Mass Transf. 48(1), 15–29 (2005)

    MATH  Google Scholar 

  70. Wirtz, D., Haasdonk, B.: A vectorial kernel orthogonal greedy algorithm. Dolomites Research Notes on Approximation 6, 83–100 (2013)

    Google Scholar 

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Benjamin Peherstorfer

  2. Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Youssef Marzouk

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  1. Benjamin Peherstorfer
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  2. Youssef Marzouk
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Correspondence to Benjamin Peherstorfer.

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Communicated by: Anthony Nouy

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The authors acknowledge support of the AFOSR MURI on multi-information sources of multi-physics systems under Award Number FA9550-15-1-0038.

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Peherstorfer, B., Marzouk, Y. A transport-based multifidelity preconditioner for Markov chain Monte Carlo. Adv Comput Math 45, 2321–2348 (2019). https://doi.org/10.1007/s10444-019-09711-y

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