Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption

Junxiong Jia; Jigen Peng; Jinghuai Gao

doi:10.3934/ipi.2020061

Article Contents

2021, Volume 15, Issue 2: 201-228. Doi: 10.3934/ipi.2020061

This issue Previous Article Leaf Peeling method for the wave equation on metric tree graphs Next Article Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems

Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption

1.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China
2.
School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China
3.
School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, 710049, China

^* Corresponding author: Jigen Peng
^* Corresponding author: Jigen Peng

Received: February 2020

Revised: August 2020

Early access: October 2020

Published: April 2021

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Abstract

We investigate an empirical Bayesian nonparametric approach to a family of linear inverse problems with Gaussian prior and Gaussian noise. We consider a class of Gaussian prior probability measures with covariance operator indexed by a hyperparameter that quantifies regularity. By introducing two auxiliary problems, we construct an empirical Bayes method and prove that this method can automatically select the hyperparameter. In addition, we show that this adaptive Bayes procedure provides optimal contraction rates up to a slowly varying term and an arbitrarily small constant, without knowledge about the regularity index. Our method needs not the prior covariance, noise covariance and forward operator have a common basis in their singular value decomposition, enlarging the application range compared with the existing results. A simple simulation example is given that illustrates the effectiveness of the proposed method.

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Mathematics Subject Classification: Primary: 35R30, 35R25; Secondary: 45Q05.

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Figure 1. Left panels the empirical Bayes posterior mean (red) and the true curve (blue, dashed). Right panels corresponding normalized likelihood for $ \hat{\tilde{\alpha}} $ (regularity index for the artificial diagonal problem). We have $ n = 10^3, 10^5, 10^8, $ and $ 10^{12} $, from top to bottom

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