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

Meixin Xiong; Liuhong Chen; Ju Ming; Jaemin Shin; Meixin Xiong; Liuhong Chen; Ju Ming; Jaemin Shin

doi:10.3934/era.2021044

Electronic Research Archive

2021, Volume 29, Issue 5: 3383-3403. 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

Meixin Xiong ^{1
,},
Liuhong Chen ^{1
,},
Ju Ming ^{1
,
,},
Jaemin Shin ^{2
,}

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

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.
- Bayesian inference,
- inverse problems,
- data-driven method,
- compressive sensing,
- proper orthogonal decomposition
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

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Abstract

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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