Multilevel Markov Chain Monte Carlo for Bayesian inverse problem for Navier-Stokes equation

Juntao Yang; Viet Ha Hoang

doi:10.3934/ipi.2022033

Article Contents

2023, Volume 17, Issue 1: 106-135. Doi: 10.3934/ipi.2022033

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Multilevel Markov Chain Monte Carlo for Bayesian inverse problem for Navier-Stokes equation

Juntao Yang^1,2, and
Viet Ha Hoang^2, ,

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
^* Corresponding author: Viet Ha Hoang

Received: October 2021

Revised: April 2022

Early access: July 2022

Published: February 2023

Abstract / Introduction Full Text(HTML) Figure(8) / Table(6) Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: Primary: 65C05, 62F15, 65J22; Secondary: 76D05.

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Figure 1. MLMCMC error for 2D Navier-Stokes equation with uniform prior, a = 2

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Figure 2. MLMCMC error for 2D Navier-Stokes equation with uniform prior, a = 3

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Figure 3. CPU time for MLMCMC, a = 2

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Figure 4. CPU time for MLMCMC, a = 3

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Figure 5. MLMCMC error for 2D Navier-Stokes equation with Gaussian prior, a = 2

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Figure 6. MLMCMC error for 2D Navier-Stokes equation with Gaussian prior, a = 3

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Figure 7. MLMCMC error for 2D Navier-Stokes equation with Gaussian prior by pCN sampler, a = 2

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Figure 8. MLMCMC error for 2D Navier-Stokes equation with Gaussian prior by pCN sampler, a = 3

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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) $

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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

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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

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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

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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

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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

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