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Accelerating the Bayesian inference of inverse problems by using data-driven compressive sensing method based on proper orthogonal decomposition

  • Received: 01 November 2020 Revised: 01 March 2021 Published: 24 June 2021
  • Primary: 65D15, 65N21, 62F15; Secondary: 60H15

  • In Bayesian inverse problems, using the Markov Chain Monte Carlo method to sample from the posterior space of unknown parameters is a formidable challenge due to the requirement of evaluating the forward model a large number of times. For the purpose of accelerating the inference of the Bayesian inverse problems, in this work, we present a proper orthogonal decomposition (POD) based data-driven compressive sensing (DCS) method and construct a low dimensional approximation to the stochastic surrogate model on the prior support. Specifically, we first use POD to generate a reduced order model. Then we construct a compressed polynomial approximation by using a stochastic collocation method based on the generalized polynomial chaos expansion and solving an $ l_1 $-minimization problem. Rigorous error analysis and coefficient estimation was provided. Numerical experiments on stochastic elliptic inverse problem were performed to verify the effectiveness of our POD-DCS method.

    Citation: Meixin Xiong, Liuhong Chen, Ju Ming, Jaemin Shin. Accelerating the Bayesian inference of inverse problems by using data-driven compressive sensing method based on proper orthogonal decomposition[J]. Electronic Research Archive, 2021, 29(5): 3383-3403. doi: 10.3934/era.2021044

    Related Papers:

  • In Bayesian inverse problems, using the Markov Chain Monte Carlo method to sample from the posterior space of unknown parameters is a formidable challenge due to the requirement of evaluating the forward model a large number of times. For the purpose of accelerating the inference of the Bayesian inverse problems, in this work, we present a proper orthogonal decomposition (POD) based data-driven compressive sensing (DCS) method and construct a low dimensional approximation to the stochastic surrogate model on the prior support. Specifically, we first use POD to generate a reduced order model. Then we construct a compressed polynomial approximation by using a stochastic collocation method based on the generalized polynomial chaos expansion and solving an $ l_1 $-minimization problem. Rigorous error analysis and coefficient estimation was provided. Numerical experiments on stochastic elliptic inverse problem were performed to verify the effectiveness of our POD-DCS method.



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    [1] Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation. Comput. Methods Appl. Mech. Engrg. (2005) 194: 1251-1294.
    [2] Inverse problems of optimal control in creep theory. J. Appl. Ind. Math. (2012) 6: 421-430.
    [3] Markov Chain Monte Carlo method and its application. Journal of the Royal Statistical Society (1998) 47: 69-100.
    [4] One-dimensional inverse scattering problem for optical coherence tomography. Inverse Problems (2005) 21: 499-524.
    [5] The restricted isometry property and its implications for compressed sensing. C. R. Math. Acad. Sci. Paris (2008) 346: 589-592.
    [6] S. S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM Rev., 43 (2001), 129-159. doi: 10.1137/S003614450037906X
    [7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5
    [8] A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. (2011) 230: 3015-3034.
    [9] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problem, Kluwer Academic Publishers, 1996.
    [10] An inverse problem arising from the displacement of oil by water in porous media. Appl. Numer. Math. (2009) 59: 2452-2466.
    [11] M. Fornasier and H. Rauhut, Compressive sensing, in: O. Scherzer (Ed.), Handbook of mathematical methods in imaging, Springer New York, (2015), 205-256.
    [12] Optimal control of stochastic flow over a backward-facing step using reduced-order modeling. SIAM J. Sci. Comput. (2011) 33: 2641-2663.
    [13] Stochastic finite element methods for partial differential equations with random input data. Acta Numer. (2014) 23: 521-650.
    [14] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer-Verlag, New York, 2005.
    [15] Statistical inverse problems: Discretization, model reduction and inverse crimes. J. Comput. Appl. Math. (2007) 198: 493-504.
    [16] Conditional well-posedness for an elliptic inverse problem. SIAM J. Appl. Math. (2011) 71: 952-971.
    [17] J. Li and Y. M. Marzouk, Adaptive construction of surrogates for the Bayesian solution of inverse problems, SIAM J. Sci. Comput., 36 (2014), A1163-A1186. doi: 10.1137/130938189
    [18] Data-driven compressive sensing and applications in uncertainty quantification. J. Comput. Phys. (2018) 374: 787-802.
    [19] Parameter and state model reduction for large- scale statistical inverse problems. SIAM J. Sci. Comput. (2010) 32: 2523-2542.
    [20] J. Martin, L. C. Wilcox, C. Burstedde and O. Ghattas, A stochastic newton MCMC method for large-scale statistical inverse problems with application to seismic inversion, SIAM J. Sci. Comput., 34 (2012), A1460-A1487. doi: 10.1137/110845598
    [21] Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. (2007) 224: 560-586.
    [22] Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comput. Phys. (2009) 228: 1862-1902.
    [23] N. Petra, J. Martin, G. Stadler and O. Ghattas, A computational framework for infinite-dimensional Bayesian inverse problems, Part II: Stochastic Newton MCMC with application to ice sheet flow inverse problems, SIAM J. Sci. Comput., 36 (2014), A1525-A1555. doi: 10.1137/130934805
    [24] Sparse Legendre expansions via $l_1$-minimization. J. Approx. Theory (2012) 164: 517-533.
    [25] An inverse problem for the steady state diffusion equation. SIAM J. Appl. Math. (1981) 41: 210-221.
    [26] Inverse problems: A Bayesian perspective. Acta Numer. (2010) 19: 451-559.
    [27] (1977) Solutions of Ill-Posed Problems,.Halsted Press.
    [28] Model reduction and pollution source identification from remote sensing data. Inverse Probl. Imaging (2009) 3: 711-730.
    [29] (2010) Numerical Methods for Stochastic Computations: A Spectral Method Approach,.Princeton University Press.
    [30] L. Yan and L. Guo, Stochastic collocation algorithms using $l_1$-minimization for Bayesian solution of inverse problems, SIAM J. Sci. Comput., 37 (2015), A1410-A1435. doi: 10.1137/140965144
    [31] Review of parameter identification procedures in groundwater hydrology: The inverse problem. Water Resources Research (1986) 22: 95-108.
    [32] A wavelet adaptive-homotopy method for inverse problem in the fluid-saturated porous media. Appl. Math. Comput. (2009) 208: 189-196.
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