doi: 10.3934/era.2021044
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Accelerating the Bayesian inference of inverse problems by using data-driven compressive sensing method based on proper orthogonal decomposition

1. 

Department of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

2. 

Department of Mathematical Sciences, Hanbat National University, Daejeon, Republic of Korea

* Corresponding author: jming@hust.edu.cn

Received  November 2020 Revised  March 2021 Early access June 2021

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. Electronic Research Archive, doi: 10.3934/era.2021044
References:
[1]

I. BabuškaR. Tempone and G. E. Zouraris, Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation, Comput. Methods Appl. Mech. Engrg., 194 (2005), 1251-1294.  doi: 10.1016/j.cma.2004.02.026.  Google Scholar

[2]

K. S. Bormotin, Inverse problems of optimal control in creep theory, J. Appl. Ind. Math., 6 (2012), 421-430.  doi: 10.1134/S1990478912040035.  Google Scholar

[3]

S. Brooks, Markov Chain Monte Carlo method and its application, Journal of the Royal Statistical Society, 47 (1998), 69-100.  doi: 10.1111/1467-9884.00117.  Google Scholar

[4]

O. Bruno and J. Chaubell, One-dimensional inverse scattering problem for optical coherence tomography, Inverse Problems, 21 (2005), 499-524.  doi: 10.1088/0266-5611/21/2/006.  Google Scholar

[5]

E. J. Candès, The restricted isometry property and its implications for compressed sensing, C. R. Math. Acad. Sci. Paris, 346 (2008), 589-592.  doi: 10.1016/j.crma.2008.03.014.  Google Scholar

[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.  Google Scholar

[7]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[8]

A. Doostan and H. Owhadi, A non-adapted sparse approximation of PDEs with stochastic inputs, J. Comput. Phys., 230 (2011), 3015-3034.  doi: 10.1016/j.jcp.2011.01.002.  Google Scholar

[9]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problem, Kluwer Academic Publishers, 1996.  Google Scholar

[10]

E. M. Fernández-Berdaguer and G. B. Savioli, An inverse problem arising from the displacement of oil by water in porous media, Appl. Numer. Math., 59 (2009), 2452-2466.  doi: 10.1016/j.apnum.2009.04.009.  Google Scholar

[11]

M. Fornasier and H. Rauhut, Compressive sensing, in: O. Scherzer (Ed.), Handbook of mathematical methods in imaging, Springer New York, (2015), 205-256.  Google Scholar

[12]

M. Gunzburger and J. Ming, Optimal control of stochastic flow over a backward-facing step using reduced-order modeling, SIAM J. Sci. Comput., 33 (2011), 2641-2663.  doi: 10.1137/100817279.  Google Scholar

[13]

M. D. GunzburgerC. G. Webster and G. Zhang, Stochastic finite element methods for partial differential equations with random input data, Acta Numer., 23 (2014), 521-650.  doi: 10.1017/S0962492914000075.  Google Scholar

[14]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer-Verlag, New York, 2005.  Google Scholar

[15]

J. Kaipio and E. Somersalo, Statistical inverse problems: Discretization, model reduction and inverse crimes, J. Comput. Appl. Math., 198 (2007), 493-504.  doi: 10.1016/j.cam.2005.09.027.  Google Scholar

[16]

I. Knowles and M. A. LaRussa, Conditional well-posedness for an elliptic inverse problem, SIAM J. Appl. Math., 71 (2011), 952-971.  doi: 10.1137/09077566X.  Google Scholar

[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.  Google Scholar

[18]

H. LiangQ. Sun and Q. Du, Data-driven compressive sensing and applications in uncertainty quantification, J. Comput. Phys., 374 (2018), 787-802.  doi: 10.1016/j.jcp.2018.07.056.  Google Scholar

[19]

C. LiebermanK. Willcox and O. Ghattas, Parameter and state model reduction for large- scale statistical inverse problems, SIAM J. Sci. Comput., 32 (2010), 2523-2542.  doi: 10.1137/090775622.  Google Scholar

[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.  Google Scholar

[21]

Y. M. MarzoukH. N. Najm and L. A. Rahn, Stochastic spectral methods for efficient Bayesian solution of inverse problems, J. Comput. Phys., 224 (2007), 560-586.  doi: 10.1016/j.jcp.2006.10.010.  Google Scholar

[22]

Y. M. Marzouk and H. N. Najm, Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems, J. Comput. Phys., 228 (2009), 1862-1902.  doi: 10.1016/j.jcp.2008.11.024.  Google Scholar

[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.  Google Scholar

[24]

H. Rauhut and R. Ward, Sparse Legendre expansions via $l_1$-minimization, J. Approx. Theory, 164 (2012), 517-533.  doi: 10.1016/j.jat.2012.01.008.  Google Scholar

[25]

G. R. Richter, An inverse problem for the steady state diffusion equation, SIAM J. Appl. Math., 41 (1981), 210-221.  doi: 10.1137/0141016.  Google Scholar

[26]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

[27] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems,, Halsted Press, 1977.   Google Scholar
[28]

A. Voutilainen and J. P. Kaipio, Model reduction and pollution source identification from remote sensing data, Inverse Probl. Imaging, 3 (2009), 711-730.  doi: 10.3934/ipi.2009.3.711.  Google Scholar

[29] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach,, Princeton University Press, 2010.   Google Scholar
[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.  Google Scholar

[31]

W. W. Yeh, Review of parameter identification procedures in groundwater hydrology: The inverse problem, Water Resources Research, 22 (1986), 95-108.  doi: 10.1029/WR022i002p00095.  Google Scholar

[32]

H. Ying and B. Han, A wavelet adaptive-homotopy method for inverse problem in the fluid-saturated porous media, Appl. Math. Comput., 208 (2009), 189-196.  doi: 10.1016/j.amc.2008.11.033.  Google Scholar

show all references

References:
[1]

I. BabuškaR. Tempone and G. E. Zouraris, Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation, Comput. Methods Appl. Mech. Engrg., 194 (2005), 1251-1294.  doi: 10.1016/j.cma.2004.02.026.  Google Scholar

[2]

K. S. Bormotin, Inverse problems of optimal control in creep theory, J. Appl. Ind. Math., 6 (2012), 421-430.  doi: 10.1134/S1990478912040035.  Google Scholar

[3]

S. Brooks, Markov Chain Monte Carlo method and its application, Journal of the Royal Statistical Society, 47 (1998), 69-100.  doi: 10.1111/1467-9884.00117.  Google Scholar

[4]

O. Bruno and J. Chaubell, One-dimensional inverse scattering problem for optical coherence tomography, Inverse Problems, 21 (2005), 499-524.  doi: 10.1088/0266-5611/21/2/006.  Google Scholar

[5]

E. J. Candès, The restricted isometry property and its implications for compressed sensing, C. R. Math. Acad. Sci. Paris, 346 (2008), 589-592.  doi: 10.1016/j.crma.2008.03.014.  Google Scholar

[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.  Google Scholar

[7]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[8]

A. Doostan and H. Owhadi, A non-adapted sparse approximation of PDEs with stochastic inputs, J. Comput. Phys., 230 (2011), 3015-3034.  doi: 10.1016/j.jcp.2011.01.002.  Google Scholar

[9]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problem, Kluwer Academic Publishers, 1996.  Google Scholar

[10]

E. M. Fernández-Berdaguer and G. B. Savioli, An inverse problem arising from the displacement of oil by water in porous media, Appl. Numer. Math., 59 (2009), 2452-2466.  doi: 10.1016/j.apnum.2009.04.009.  Google Scholar

[11]

M. Fornasier and H. Rauhut, Compressive sensing, in: O. Scherzer (Ed.), Handbook of mathematical methods in imaging, Springer New York, (2015), 205-256.  Google Scholar

[12]

M. Gunzburger and J. Ming, Optimal control of stochastic flow over a backward-facing step using reduced-order modeling, SIAM J. Sci. Comput., 33 (2011), 2641-2663.  doi: 10.1137/100817279.  Google Scholar

[13]

M. D. GunzburgerC. G. Webster and G. Zhang, Stochastic finite element methods for partial differential equations with random input data, Acta Numer., 23 (2014), 521-650.  doi: 10.1017/S0962492914000075.  Google Scholar

[14]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer-Verlag, New York, 2005.  Google Scholar

[15]

J. Kaipio and E. Somersalo, Statistical inverse problems: Discretization, model reduction and inverse crimes, J. Comput. Appl. Math., 198 (2007), 493-504.  doi: 10.1016/j.cam.2005.09.027.  Google Scholar

[16]

I. Knowles and M. A. LaRussa, Conditional well-posedness for an elliptic inverse problem, SIAM J. Appl. Math., 71 (2011), 952-971.  doi: 10.1137/09077566X.  Google Scholar

[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.  Google Scholar

[18]

H. LiangQ. Sun and Q. Du, Data-driven compressive sensing and applications in uncertainty quantification, J. Comput. Phys., 374 (2018), 787-802.  doi: 10.1016/j.jcp.2018.07.056.  Google Scholar

[19]

C. LiebermanK. Willcox and O. Ghattas, Parameter and state model reduction for large- scale statistical inverse problems, SIAM J. Sci. Comput., 32 (2010), 2523-2542.  doi: 10.1137/090775622.  Google Scholar

[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.  Google Scholar

[21]

Y. M. MarzoukH. N. Najm and L. A. Rahn, Stochastic spectral methods for efficient Bayesian solution of inverse problems, J. Comput. Phys., 224 (2007), 560-586.  doi: 10.1016/j.jcp.2006.10.010.  Google Scholar

[22]

Y. M. Marzouk and H. N. Najm, Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems, J. Comput. Phys., 228 (2009), 1862-1902.  doi: 10.1016/j.jcp.2008.11.024.  Google Scholar

[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.  Google Scholar

[24]

H. Rauhut and R. Ward, Sparse Legendre expansions via $l_1$-minimization, J. Approx. Theory, 164 (2012), 517-533.  doi: 10.1016/j.jat.2012.01.008.  Google Scholar

[25]

G. R. Richter, An inverse problem for the steady state diffusion equation, SIAM J. Appl. Math., 41 (1981), 210-221.  doi: 10.1137/0141016.  Google Scholar

[26]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

[27] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems,, Halsted Press, 1977.   Google Scholar
[28]

A. Voutilainen and J. P. Kaipio, Model reduction and pollution source identification from remote sensing data, Inverse Probl. Imaging, 3 (2009), 711-730.  doi: 10.3934/ipi.2009.3.711.  Google Scholar

[29] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach,, Princeton University Press, 2010.   Google Scholar
[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.  Google Scholar

[31]

W. W. Yeh, Review of parameter identification procedures in groundwater hydrology: The inverse problem, Water Resources Research, 22 (1986), 95-108.  doi: 10.1029/WR022i002p00095.  Google Scholar

[32]

H. Ying and B. Han, A wavelet adaptive-homotopy method for inverse problem in the fluid-saturated porous media, Appl. Math. Comput., 208 (2009), 189-196.  doi: 10.1016/j.amc.2008.11.033.  Google Scholar

Figure 1.  Radial basis functions used to define the permeability field
Figure 2.  Bayesian inverse problems framework with direct sampling algorithm
Figure 3.  (Left): The first 15 eigenvalues of correlation matrix $ R $ for $ K_s = 50, 100, 200 $ and 300. (Right): The cumulative energy ratios of the first 15 POD basis functions with $ K_s = 100, 200 $ and 300
Figure 4.  The statistics estimation of $ L^2(D) $-norm error between the full finite element solution and the POD-based ROM solution with $ K_s = 100, 200 $ and 300. (Left): Expectation. (Right): Variance
Figure 5.  Sparsity and error estimations of the POD-DCS method with different $ K_c $ and different threshold $ \tau $. (Left): The proportion $ \widetilde{ \mathcal{R}}_{\tau} $ of non-zero coefficients in matrix $ \widetilde{ \mathbf{ c}} $; (Middle): The expectation $ \widetilde{ \mathcal{E}} $ of $ L^2(D) $-norm error; (Right): The variance $ \widetilde{ \mathcal{ V }} $ of $ L^2(D) $-norm error
Figure 6.  Error estimations of the POD-DCS method w.r.t. different collocation point sizes and $ \tau = 0.01 $
Figure 7.  The eigenvalues and coefficients in the POD-DCS method with $ K_s = 100 $, $ m = 9 $ and $ K_{c} = 200 $
Figure 8.  (Left): The true permeability field $ a(\mathbf{x}) $ used for generating the synthesize data $ \mathbf{z} $; (Right): The model outputs associated with true permeability field, where black dots are the measurement sensors
Figure 9.  Bayesian inverse problems framework with POD-DCS approximate solution
Figure 10.  The posterior marginal densities of unknown parameters with $ \sigma_e = 0.05 $ noise in the observations
Figure 11.  The estimated permeability field and error corresponding to the MAP parameters generated by the POD-DCS method. (Left): Estimated permeability field; (Right): Error
Table 1.  The statistics estimation of $ L^2(D) $-norm error between the finite element solution and the POD-based approximate solution with $ K_s = 100, 200, 300 $ and $ m = 3, 6, 9, 12 $
$ m $ 3 6 9 12
$ K_s = 100 $ $ \nu_m $ 0.9906 0.9988 0.9998 0.9999
$ \widehat{ \mathcal{E}} \times 10^{-2} $ 5.2424 0.9679 0.2840 0.1267
$ \widehat{ \mathcal{V}} \times 10^{-3} $ 8.1367 0.8915 0.1475 0.0282
$ K_s = 200 $ $ \nu_m $ 0.9908 0.9988 0.9998 0.9999
$ \widehat{ \mathcal{E}} \times 10^{-2} $ 4.9558 1.0836 0.2422 0.0907
$ \widehat{ \mathcal{V}} \times 10^{-3} $ 6.7935 0.9507 0.1023 0.0126
$ K_s = 300 $ $ \nu_m $ 0.9900 0.9986 0.9997 0.9999
$ \widehat{ \mathcal{E}} \times 10^{-2} $ 4.6537 0.8260 0.2229 0.0622
$ \widehat{ \mathcal{V}} \times 10^{-3} $ 5.5292 0.4418 0.0898 0.0037
$ m $ 3 6 9 12
$ K_s = 100 $ $ \nu_m $ 0.9906 0.9988 0.9998 0.9999
$ \widehat{ \mathcal{E}} \times 10^{-2} $ 5.2424 0.9679 0.2840 0.1267
$ \widehat{ \mathcal{V}} \times 10^{-3} $ 8.1367 0.8915 0.1475 0.0282
$ K_s = 200 $ $ \nu_m $ 0.9908 0.9988 0.9998 0.9999
$ \widehat{ \mathcal{E}} \times 10^{-2} $ 4.9558 1.0836 0.2422 0.0907
$ \widehat{ \mathcal{V}} \times 10^{-3} $ 6.7935 0.9507 0.1023 0.0126
$ K_s = 300 $ $ \nu_m $ 0.9900 0.9986 0.9997 0.9999
$ \widehat{ \mathcal{E}} \times 10^{-2} $ 4.6537 0.8260 0.2229 0.0622
$ \widehat{ \mathcal{V}} \times 10^{-3} $ 5.5292 0.4418 0.0898 0.0037
Table 2.  Error estimates and sparsity for different methods
POD-DCS POD CS
$ \mathcal{E} \times 10^{-3} $ 4.1758 2.8399 7.3896
$ \mathcal{V} \times 10^{-4} $ 2.4931 1.4750 11.1484
$ \mathcal{R}_{0.01} $ 0.2024 - 0.2182
POD-DCS POD CS
$ \mathcal{E} \times 10^{-3} $ 4.1758 2.8399 7.3896
$ \mathcal{V} \times 10^{-4} $ 2.4931 1.4750 11.1484
$ \mathcal{R}_{0.01} $ 0.2024 - 0.2182
Table 3.  Computational times, in seconds, given by different methods
$ \# $ of full FE solution Time for per FE solution $ \# $ of POD solution Time for per POD solution Time for BP Time for per model output
POD-DCS 100 2.1970 200 1.0085 21 0.2002
POD 100 2.1970 - - - 1.0085
CS 100 2.1970 - - 5045 0.2095
$ \# $ of full FE solution Time for per FE solution $ \# $ of POD solution Time for per POD solution Time for BP Time for per model output
POD-DCS 100 2.1970 200 1.0085 21 0.2002
POD 100 2.1970 - - - 1.0085
CS 100 2.1970 - - 5045 0.2095
Table 4.  The 95% confidence intervals of each input parameter obtained by different methods
FEM POD-DCS CS
$ \xi_1 $ [0.2898, 0.3173] [0.2883, 0.3164] [0.2963, 0.3248]
$ \xi_2 $ [0.2858, 0.3134] [0.2844, 0.3126] [0.2913, 0.3175]
$ \xi_3 $ [0.2913, 0.3191] [0.2927, 0.3200] [0.2906, 0.3208]
$ \xi_4 $ [0.2750, 0.3016] [0.2762, 0.3047] [0.2736, 0.3022]
FEM POD-DCS CS
$ \xi_1 $ [0.2898, 0.3173] [0.2883, 0.3164] [0.2963, 0.3248]
$ \xi_2 $ [0.2858, 0.3134] [0.2844, 0.3126] [0.2913, 0.3175]
$ \xi_3 $ [0.2913, 0.3191] [0.2927, 0.3200] [0.2906, 0.3208]
$ \xi_4 $ [0.2750, 0.3016] [0.2762, 0.3047] [0.2736, 0.3022]
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