Abstract
In this article, we consider a Bayesian inverse problem associated to elliptic partial differential equations in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic sequential Monte Carlo (SMC) method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method is established in a function space setting. We also develop an enhancement of the SMC methods for inverse problems which were introduced in Kantas et al. (SIAM/ASA J Uncertain Quantif 2:464–489, 2014); the enhancement is designed to deal with the additional complexity of this elliptic inverse problem. The efficacy of the methodology and its desirable theoretical properties, are demonstrated for numerical examples in both two and three dimensions.
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References
Agapiou, S., Roberts, G.O., Völlmer, S.: Unbiased Monte Carlo: posterior estimation for intractable/infinite-dimensional models (2014). arXiv:1411.7713
Beskos, A., Roberts, G., Stuart, A.M.: Optimal scalings for local Metropolis-Hastings chains on non-product targets in high-dimensions. Ann. Appl. Probab. 19, 863–898 (2009)
Beskos, A., Crisan, D., Jasra, A.: On the stability of sequential Monte Carlo methods in high dimensions. Ann. Appl. Probab. 24, 1396–1445 (2014a)
Beskos, A., Crisan, D., Jasra, A., Whiteley, N.P.: Error bounds and normalizing constants for sequential Monte Carlo samplers in high-dimensions. Adv. Appl. Probab. 46, 279–306 (2014b)
Beskos, A., Crisan, D., Jasra, A., Kamatani, K., Zhou, Y.: A stable particle filter in high-dimensions (2014c). arXiv:1412.3501
Beskos, A., Jasra, A., Kantas, N., Thiery, A.: On the convergence of adaptive sequential Monte Carlo methods (2014d). arxiv:1306.6462
Beskos, A., Jasra, A., Law, K.J.H., Tempone, R., Zhou, Y.: Multilevel sequential Monte Carlo samplers (2015). arXiv preprint
Cotter, S., Roberts, G., Stuart, A.M., White, D.: MCMC methods for functions: modifying old algorithms to make them faster. Stat. Sci. 28, 424–446 (2013)
Dashti, M., Stuart, A.M.: The Bayesian approach to inverse problems. In: Handbook of Uncertainty Quantification. Springer (2015). arxiv:1302.6989
Del Moral, P.: Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York (2004)
Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo samplers. J. R. Stat. Soc. B 68, 411–436 (2006)
Del Moral, P.: Mean Field Simulation for Monte Carlo Integration. Chapman & Hall, London (2013)
Hairer, M., Stuart, A.M., Vollmer, S.J.: Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions. Ann. Appl. Probab. 24, 2455–2490 (2014)
Hoang, V.H., Schwab, C., Stuart, A.M.: Complexity analysis of accelerated MCMC methods for Bayesian inversion. Inverse Prob. 29, 085010 (2013)
Iglesias, M., Law, K.J.H., Stuart, A.M.: Evaluation of Gaussian approximations for data assimilation for reservoir models. Comput. Geosci. 17, 851–885 (2013)
Jasra, A., Stephens, D.A., Doucet, A., Tsagaris, T.: Inference for Lévy driven stochastic volatility models using adaptive sequential Monte Carlo. Scand. J. Stat. 38, 1–22 (2011)
Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Applied Mathematical Sciences, vol. 160. Springer, New York (2005)
Kantas, N., Beskos, A., Jasra, A.: Sequential Monte Carlo for inverse problems: a case study for the Navier Stokes equation. SIAM/ASA J. Uncertain. Quantif. 2, 464–489 (2014)
Kirk, B.S., Peterson, J.W., Stogner, R.H., Carey, G.F.: A C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22, 237–254 (2006)
McLaughlin, D., Townley, L.R.: A reassessment of the groundwater inverse problem. Water Resour. Res. 32, 1131–1161 (1996)
Neal, R.M.: Annealed importance sampling. Stat. Comput. 11, 125–139 (2001)
Rebeschini, P., Van Handel, R.: Can local particle filters beat the curse of dimensionality? Ann. Appl. Probab. (to appear)
Schwab, C., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291–467 (2011)
Schwab, C., Stuart, A.M.: Sparse determinisitc approximation of Bayesian inverse problems. Inverse Prob. 28, 045003 (2012)
Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia (2005)
Völlmer, S.: Posterior Consistency for Bayesian inverse problems through stability and regression results. Inverse Prob. 29, 125011 (2013)
Whiteley, N.: Sequential Monte Carlo samplers: Error bounds and insensitivity to initial conditions. Stoch. Anal. Appl. 30, 774–798 (2013)
Zhou, Y., Johansen, A.M., Aston, J.A.D.: Towards automatic model comparison: an adaptive sequential Monte Carlo approach. J. Comput. Graph. Stat. (to appear)
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Beskos, A., Jasra, A., Muzaffer, E.A. et al. Sequential Monte Carlo methods for Bayesian elliptic inverse problems. Stat Comput 25, 727–737 (2015). https://doi.org/10.1007/s11222-015-9556-7
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DOI: https://doi.org/10.1007/s11222-015-9556-7