doi: 10.3934/ipi.2022033
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Multilevel Markov Chain Monte Carlo for Bayesian inverse problem for Navier-Stokes equation

1. 

Nvidia AI Technology Center, 8 Temasek Blvd, Singapore 038988, Singapore

2. 

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore

* Corresponding author: Viet Ha Hoang

Received  October 2021 Revised  April 2022 Early access July 2022

Bayesian inverse problems for inferring the unknown forcing and initial condition of Navier-Stokes equation play important roles in many practical areas. The computation cost of sampling the posterior probability measure can be exceedingly high. We develop the Finite Element Multilevel Markov Chain Monte Carlo (FE-MLMCMC) sampling method for approximating expectation with respect to the posterior probability measure of quantities of interest for a model problem of Navier-Stokes equation in the two dimensional periodic torus. We first consider the case where the forcing and the initial condition are bounded for all the realizations and depend linearly on a countable set of random variables which are uniformly distributed in a compact interval. We establish the essentially optimal convergence rate of the method and verify it numerically. The method follows from that developed in V. H. Hoang, Ch. Schwab and A. M. Stuart, Inverse problems, vol. 29, 2013 for inferring the coefficients of linear elliptic forward equations under the uniform prior probability measure. In the case of the Gaussian prior probability measure, numerical results, using the MLMCMC method developed for the Gaussian prior in V. H. Hoang, J. H. Quek and Ch. Schwab, Inverse problems, vol. 36, 2020, indicate the essentially optimal convergence rates. However, a rigorous theory for the MLMCMC sampling procedure is not available, due to the non-integrability with respect to the Gaussian prior of the theoretical finite element errors of the forward solvers that are available in the literature.

Citation: Juntao Yang, Viet Ha Hoang. Multilevel Markov Chain Monte Carlo for Bayesian inverse problem for Navier-Stokes equation. Inverse Problems and Imaging, doi: 10.3934/ipi.2022033
References:
[1]

A. BeskosA. JasraK. LawR. Tempone and Y. Zhou, Multilevel sequential Monte Carlo samplers, Stochastic Process. Appl., 127 (2017), 1417-1440.  doi: 10.1016/j.spa.2016.08.004.

[2]

D. BlömkerK. LawA. M. Stuart and K. C. Zygalakis, Accuracy and stability of the continuous-time 3DVAR filter for the Navier-Stokes equation, Nonlinearity, 26 (2013), 2193-2219.  doi: 10.1088/0951-7715/26/8/2193.

[3]

C. E. A. BrettK. F. LamK. J. H. LawD. S. McCormickM. R. Scott and A. M. Stuart, Accuracy and stability of filters for dissipative PDEs, Phys. D, 245 (2013), 34-45.  doi: 10.1016/j.physd.2012.11.005.

[4]

H.-J. Bungartz and M. Griebel, Sparse grids, Acta Numer., 13 (2004), 147-269.  doi: 10.1017/S0962492904000182.

[5]

S. H. CheungT. A. OliverE. E. PrudencioS. Prudhomme and R. D. Moser, Bayesian uncertainty analysis with applications to turbulence modeling, Reliability Engineering & System Safety, 96 (2011), 1137-1149. 

[6]

S. L. Cotter, M. Dashti, J. C. Robinson and A. M. Stuart, Bayesian inverse problems for functions and applications to fluid mechanics, Inverse Problems, 25 (2009), 115008, 43 pp. doi: 10.1088/0266-5611/25/11/115008.

[7]

J. DickM. Feischl and C. Schwab, Improved efficiency of a multi-index FEM for computational uncertainty quantification, SIAM J. Numer. Anal., 57 (2019), 1744-1769.  doi: 10.1137/18M1193700.

[8]

T. J. DodwellC. KetelsenR. Scheichl and A. L. Teckentrup, A hierarchical multilevel Markov chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 1075-1108.  doi: 10.1137/130915005.

[9]

Y. EfendievB. JinM. Presho and X. Tan, Multilevel Markov Chain Monte Carlo method for high-contrast single-phase flow problems, Commun. Comput. Phys., 17 (2015), 259-286.  doi: 10.4208/cicp.021013.260614a.

[10]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.

[11]

M. B. Giles, Multilevel Monte Carlo methods, Acta Numer., 24 (2015), 259-328.  doi: 10.1017/S096249291500001X.

[12]

M. B. Giles, An introduction to multilevel Monte Carlo methods, Proceedings of the International Congress of Mathematicians, Rio de Janeiro, 2018, 3571–3590.

[13]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[14]

M. HairerA. M. Stuart and S. J. Vollmer, Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions, Ann. Appl. Probab., 24 (2014), 2455-2490.  doi: 10.1214/13-AAP982.

[15]

A.-L. Haji-AliF. Nobile and R. Tempone, Multi-index Monte Carlo: When sparsity meets sampling, Numer. Math., 132 (2016), 767-806.  doi: 10.1007/s00211-015-0734-5.

[16]

Y. He, The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comp., 77 (2008), 2097-2124.  doi: 10.1090/S0025-5718-08-02127-3.

[17]

J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem Part Ⅳ: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.

[18]

V. H. Hoang, Bayesian inverse problems in measure spaces with application to Burgers and Hamilton-Jacobi equations with white noise forcing, Inverse Problems, 28 (2012), 025009, 29 pp. doi: 10.1088/0266-5611/28/2/025009.

[19]

V. H. HoangK. J. H. Law and A. M. Stuart, Determine white noise forcing from Eulerian observations in the Navier-Stokes equation, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 233-261.  doi: 10.1007/s40072-014-0028-4.

[20]

V. H. Hoang, J. H. Quek and C. Schwab, Analysis of a multilevel Markov Chain Monte Carlo finite element method for Bayesian inversion of log-normal diffusions, Inverse Problems, 36 (2020), 035021, 46 pp. doi: 10.1088/1361-6420/ab2a1e.

[21]

V. H. HoangJ. H. Quek and C. Schwab, Multilevel Markov chain Monte Carlo for Bayesian inversion of parabolic partial differential equations under Gaussian prior, SIAM/ASA J. Uncertain. Quantif., 9 (2021), 384-419.  doi: 10.1137/20M1354714.

[22]

V. H. Hoang and C. Schwab, High-dimensional finite elements for elliptic problems with multiple scales, Multiscale Model. Simul., 3 (2004/05), 168-194.  doi: 10.1137/030601077.

[23]

V. H. Hoang, C. Schwab and A. M. Stuart, Complexity analysis of accelerated MCMC methods for Bayesian inversion, Inverse Problems, 29 (2013), 085010, 37 pp. doi: 10.1088/0266-5611/29/8/085010.

[24]

H. HoelK. J. H. Law and R. Tempone, Multilevel ensemble Kalman filtering, SIAM J. Numer. Anal., 54 (2016), 1813-1839.  doi: 10.1137/15M100955X.

[25]

A. JasraK. KamataniK. J. H. Law and Y. Zhou, A multi-index Markov chain Monte Carlo method, Int. J. Uncertain. Quantif., 8 (2018), 61-73.  doi: 10.1615/Int.J.UncertaintyQuantification.2018021551.

[26]

V. John, Finite Element Methods for Incompressible Flow Problems, Springer, Cham, 2016. doi: 10.1007/978-3-319-45750-5.

[27] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511802270.
[28]

K. J. H. Law and A. M. Stuart, Evaluating data assimilation algorithms, Monthly Weather Review, 140 (2012), 3757-3782. 

[29]

M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, Handb. Numer. Anal., VI, North-Holland, Amsterdam, 1998,503–688.

[30]

M. Nodet, Variational assimilation of Lagrangian data in oceanography, Inverse Problems, 22, 245–263.

[31]

R. Peyret, Spectral Methods for Incompressible Viscous Flow, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-6557-1.

[32]

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

[33]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing, Amsterdam-New York-Oxford, 1977.

show all references

References:
[1]

A. BeskosA. JasraK. LawR. Tempone and Y. Zhou, Multilevel sequential Monte Carlo samplers, Stochastic Process. Appl., 127 (2017), 1417-1440.  doi: 10.1016/j.spa.2016.08.004.

[2]

D. BlömkerK. LawA. M. Stuart and K. C. Zygalakis, Accuracy and stability of the continuous-time 3DVAR filter for the Navier-Stokes equation, Nonlinearity, 26 (2013), 2193-2219.  doi: 10.1088/0951-7715/26/8/2193.

[3]

C. E. A. BrettK. F. LamK. J. H. LawD. S. McCormickM. R. Scott and A. M. Stuart, Accuracy and stability of filters for dissipative PDEs, Phys. D, 245 (2013), 34-45.  doi: 10.1016/j.physd.2012.11.005.

[4]

H.-J. Bungartz and M. Griebel, Sparse grids, Acta Numer., 13 (2004), 147-269.  doi: 10.1017/S0962492904000182.

[5]

S. H. CheungT. A. OliverE. E. PrudencioS. Prudhomme and R. D. Moser, Bayesian uncertainty analysis with applications to turbulence modeling, Reliability Engineering & System Safety, 96 (2011), 1137-1149. 

[6]

S. L. Cotter, M. Dashti, J. C. Robinson and A. M. Stuart, Bayesian inverse problems for functions and applications to fluid mechanics, Inverse Problems, 25 (2009), 115008, 43 pp. doi: 10.1088/0266-5611/25/11/115008.

[7]

J. DickM. Feischl and C. Schwab, Improved efficiency of a multi-index FEM for computational uncertainty quantification, SIAM J. Numer. Anal., 57 (2019), 1744-1769.  doi: 10.1137/18M1193700.

[8]

T. J. DodwellC. KetelsenR. Scheichl and A. L. Teckentrup, A hierarchical multilevel Markov chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 1075-1108.  doi: 10.1137/130915005.

[9]

Y. EfendievB. JinM. Presho and X. Tan, Multilevel Markov Chain Monte Carlo method for high-contrast single-phase flow problems, Commun. Comput. Phys., 17 (2015), 259-286.  doi: 10.4208/cicp.021013.260614a.

[10]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.

[11]

M. B. Giles, Multilevel Monte Carlo methods, Acta Numer., 24 (2015), 259-328.  doi: 10.1017/S096249291500001X.

[12]

M. B. Giles, An introduction to multilevel Monte Carlo methods, Proceedings of the International Congress of Mathematicians, Rio de Janeiro, 2018, 3571–3590.

[13]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[14]

M. HairerA. M. Stuart and S. J. Vollmer, Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions, Ann. Appl. Probab., 24 (2014), 2455-2490.  doi: 10.1214/13-AAP982.

[15]

A.-L. Haji-AliF. Nobile and R. Tempone, Multi-index Monte Carlo: When sparsity meets sampling, Numer. Math., 132 (2016), 767-806.  doi: 10.1007/s00211-015-0734-5.

[16]

Y. He, The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comp., 77 (2008), 2097-2124.  doi: 10.1090/S0025-5718-08-02127-3.

[17]

J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem Part Ⅳ: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.

[18]

V. H. Hoang, Bayesian inverse problems in measure spaces with application to Burgers and Hamilton-Jacobi equations with white noise forcing, Inverse Problems, 28 (2012), 025009, 29 pp. doi: 10.1088/0266-5611/28/2/025009.

[19]

V. H. HoangK. J. H. Law and A. M. Stuart, Determine white noise forcing from Eulerian observations in the Navier-Stokes equation, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 233-261.  doi: 10.1007/s40072-014-0028-4.

[20]

V. H. Hoang, J. H. Quek and C. Schwab, Analysis of a multilevel Markov Chain Monte Carlo finite element method for Bayesian inversion of log-normal diffusions, Inverse Problems, 36 (2020), 035021, 46 pp. doi: 10.1088/1361-6420/ab2a1e.

[21]

V. H. HoangJ. H. Quek and C. Schwab, Multilevel Markov chain Monte Carlo for Bayesian inversion of parabolic partial differential equations under Gaussian prior, SIAM/ASA J. Uncertain. Quantif., 9 (2021), 384-419.  doi: 10.1137/20M1354714.

[22]

V. H. Hoang and C. Schwab, High-dimensional finite elements for elliptic problems with multiple scales, Multiscale Model. Simul., 3 (2004/05), 168-194.  doi: 10.1137/030601077.

[23]

V. H. Hoang, C. Schwab and A. M. Stuart, Complexity analysis of accelerated MCMC methods for Bayesian inversion, Inverse Problems, 29 (2013), 085010, 37 pp. doi: 10.1088/0266-5611/29/8/085010.

[24]

H. HoelK. J. H. Law and R. Tempone, Multilevel ensemble Kalman filtering, SIAM J. Numer. Anal., 54 (2016), 1813-1839.  doi: 10.1137/15M100955X.

[25]

A. JasraK. KamataniK. J. H. Law and Y. Zhou, A multi-index Markov chain Monte Carlo method, Int. J. Uncertain. Quantif., 8 (2018), 61-73.  doi: 10.1615/Int.J.UncertaintyQuantification.2018021551.

[26]

V. John, Finite Element Methods for Incompressible Flow Problems, Springer, Cham, 2016. doi: 10.1007/978-3-319-45750-5.

[27] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511802270.
[28]

K. J. H. Law and A. M. Stuart, Evaluating data assimilation algorithms, Monthly Weather Review, 140 (2012), 3757-3782. 

[29]

M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, Handb. Numer. Anal., VI, North-Holland, Amsterdam, 1998,503–688.

[30]

M. Nodet, Variational assimilation of Lagrangian data in oceanography, Inverse Problems, 22, 245–263.

[31]

R. Peyret, Spectral Methods for Incompressible Viscous Flow, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-6557-1.

[32]

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

[33]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing, Amsterdam-New York-Oxford, 1977.

Figure 1.  MLMCMC error for 2D Navier-Stokes equation with uniform prior, a = 2
Figure 2.  MLMCMC error for 2D Navier-Stokes equation with uniform prior, a = 3
Figure 3.  CPU time for MLMCMC, a = 2
Figure 4.  CPU time for MLMCMC, a = 3
Figure 5.  MLMCMC error for 2D Navier-Stokes equation with Gaussian prior, a = 2
Figure 6.  MLMCMC error for 2D Navier-Stokes equation with Gaussian prior, a = 3
Figure 7.  MLMCMC error for 2D Navier-Stokes equation with Gaussian prior by pCN sampler, a = 2
Figure 8.  MLMCMC error for 2D Navier-Stokes equation with Gaussian prior by pCN sampler, a = 3
Table 1.  Total MLMCMC error with different sample size choices for uniform prior
$ a $ $ M_{l l^{\prime}}, l, l^{\prime}>1 $ $ M_{l 0}=M_{0 l} $ $ M_{00} $ Total error
0 $ 2^{2\left(L-\left(l+l'\right)\right)} $ $ 2^{2(L-l)} / L^{2} $ $ 2^{2 L} / L^{4} $ $ O\left(L^{2} 2^{-L}\right) $
2 $ \left(l+l^{\prime}\right)^{2} 2^{2\left(L-\left(l+l^{\prime}\right)\right)} $ $ 2^{2(L-l)} $ $ 2^{2 L} / L^{2} $ $ O\left(L \log L 2^{-L}\right) $
3 $ \left(l+l^{\prime}\right)^{3} 2^{2\left(L-\left(l+l^{\prime}\right)\right)} $ $ l 2^{2(L-l)} $ $ 2^{2 L} / L $ $ O\left(L^{1 / 2} 2^{-L}\right) $
4 $ \left(l+l^{\prime}\right)^{4} 2^{2\left(L-\left(l+l^{\prime}\right)\right)} $ $ l^{2} 2^{2(L-l)} $ $ 2^{2 L} /\left(\log L^{2}\right) $ $ O\left(\log L 2^{-L}\right) $
$ a $ $ M_{l l^{\prime}}, l, l^{\prime}>1 $ $ M_{l 0}=M_{0 l} $ $ M_{00} $ Total error
0 $ 2^{2\left(L-\left(l+l'\right)\right)} $ $ 2^{2(L-l)} / L^{2} $ $ 2^{2 L} / L^{4} $ $ O\left(L^{2} 2^{-L}\right) $
2 $ \left(l+l^{\prime}\right)^{2} 2^{2\left(L-\left(l+l^{\prime}\right)\right)} $ $ 2^{2(L-l)} $ $ 2^{2 L} / L^{2} $ $ O\left(L \log L 2^{-L}\right) $
3 $ \left(l+l^{\prime}\right)^{3} 2^{2\left(L-\left(l+l^{\prime}\right)\right)} $ $ l 2^{2(L-l)} $ $ 2^{2 L} / L $ $ O\left(L^{1 / 2} 2^{-L}\right) $
4 $ \left(l+l^{\prime}\right)^{4} 2^{2\left(L-\left(l+l^{\prime}\right)\right)} $ $ l^{2} 2^{2(L-l)} $ $ 2^{2 L} /\left(\log L^{2}\right) $ $ O\left(\log L 2^{-L}\right) $
Table 2.  MLMCMC error for 2D Navier-Stokes equation with uniform prior, a = 2
L average error for $ a=2 $
2 0.0756
3 0.0640
4 0.0505
5 0.0316
6 0.0199
7 0.0126
L average error for $ a=2 $
2 0.0756
3 0.0640
4 0.0505
5 0.0316
6 0.0199
7 0.0126
Table 3.  MLMCMC error for 2D Navier-Stokes equation with uniform prior, a = 3
L average error for $ a=3 $
1 0.0866
2 0.0690
3 0.0425
4 0.0252
5 0.0171
6 0.0093
L average error for $ a=3 $
1 0.0866
2 0.0690
3 0.0425
4 0.0252
5 0.0171
6 0.0093
Table 4.  MLMCMC error for 2D Navier-Stokes equation with Gaussian prior by MLMCMC method developed in [20]
L average error for $ a=2 $ average error for $ a=3 $
1 0.1318 0.1318
2 0.1233 0.0930
3 0.0872 0.0650
4 0.0718 0.0481
5 0.0578 0.0254
6 0.0345 0.0172
L average error for $ a=2 $ average error for $ a=3 $
1 0.1318 0.1318
2 0.1233 0.0930
3 0.0872 0.0650
4 0.0718 0.0481
5 0.0578 0.0254
6 0.0345 0.0172
Table 5.  MLMCMC error for 2D Navier-Stokes equation with Gaussian prior by pCN sampler
L average error for $ a=2 $ average error for $ a=3 $
1 9.5562 9.5562
2 12.1887 9.7116
3 12.9223 7.6241
4 10.7129 5.2595
5 7.6000 3.0512
6 4.8135 1.5752
L average error for $ a=2 $ average error for $ a=3 $
1 9.5562 9.5562
2 12.1887 9.7116
3 12.9223 7.6241
4 10.7129 5.2595
5 7.6000 3.0512
6 4.8135 1.5752
Table 6.  MLMCMC error for 2D Navier-Stokes equation with Gaussian prior by the MLMCMC method developed for uniform prior
L average error for $ a=2 $ average error for $ a=3 $
1 2.8709e61 6.7939e180
2 3.5405e86 1.0810e218
3 1.3148e82 6.6531e220
4 3.562e155 8.1358e269
5 4.5362e89 9.9992e267
L average error for $ a=2 $ average error for $ a=3 $
1 2.8709e61 6.7939e180
2 3.5405e86 1.0810e218
3 1.3148e82 6.6531e220
4 3.562e155 8.1358e269
5 4.5362e89 9.9992e267
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Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008

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