Search for: All records

Inverse models arise in various environmental applications, ranging from atmospheric modeling to geosciences. Inverse models can often incorporate predictor variables, similar to regression, to help estimate natural processes or parameters of interest from observed data. Although a large set of possible predictor variables may be included in these inverse or regression models, a core challenge is to identify a small number of predictor variables that are most informative of the model, given limited observations. This problem is typically referred to as model selection. A variety of criterion-based approaches are commonly used for model selection, but most follow a two-step process: first, select predictors using some statistical criteria, and second, solve the inverse or regression problem with these predictor variables. The first step typically requires comparing all possible combinations of candidate predictors, which quickly becomes computationally prohibitive, especially for large-scale problems. In this work, we develop a one-step approach for linear inverse modeling, where model selection and the inverse model are performed in tandem. We reformulate the problem so that the selection of a small number of relevant predictor variables is achieved via a sparsity-promoting prior. Then, we describe hybrid iterative projection methods based on flexible Krylov subspace methods for efficient optimization. These approaches are well-suited for large-scale problems with many candidate predictor variables. We evaluate our results against traditional, criteria-based approaches. We also demonstrate the applicability and potential benefits of our approach using examples from atmospheric inverse modeling based on NASA's Orbiting Carbon Observatory-2 (OCO-2) satellite. 

Abstract. Inverse models arise in various environmental applications, ranging from atmospheric modeling to geosciences. Inverse models can often incorporate predictor variables, similar to regression, to help estimate natural processes or parameters of interest from observed data. Although a large set of possible predictor variables may be included in these inverse or regression models, a core challenge is to identify a small number of predictor variables that are most informative of the model, given limited observations. This problem is typically referred to as model selection. A variety of criterion-based approaches are commonly used for model selection, but most follow a two-step process: first, select predictors using some statistical criteria, and second, solve the inverse or regression problem with these predictor variables. The first step typically requires comparing all possible combinations of candidate predictors, which quickly becomes computationally prohibitive, especially for large-scale problems. In this work, we develop a one-step approach, where model selection and the inverse model are performed in tandem. We reformulate the problem so that the selection of a small number of relevant predictor variables is achieved via a sparsity-promoting prior. Then, we describe hybrid iterative projection methods based on flexible Krylov subspace methods for efficient optimization.  These approaches are well-suited for large-scale problems with many candidate predictor variables. We evaluate our results against traditional, criteria-based approaches. We also demonstrate the applicability and potential benefits of our approach using examples from atmospheric inverse modeling based on NASA's Orbiting Carbon Observatory 2 (OCO-2) satellite. 

Abstract In this work, we investigate the diffusive optical tomography (DOT) problem in the case that limited boundary measurements are available. Motivated by the direct sampling method (DSM) proposed in Chow et al. (SIAM J Sci Comput 37(4):A1658–A1684, 2015), we develop a deep direct sampling method (DDSM) to recover the inhomogeneous inclusions buried in a homogeneous background. In this method, we design a convolutional neural network to approximate the index functional that mimics the underling mathematical structure. The benefits of the proposed DDSM include fast and easy implementation, capability of incorporating multiple measurements to attain high-quality reconstruction, and advanced robustness against the noise. Numerical experiments show that the reconstruction accuracy is improved without degrading the efficiency, demonstrating its potential for solving the real-world DOT problems. 

This work investigates the electrical impedance tomography problem when only limited boundary measurements are available, which is known to be challenging due to the extreme ill-posedness. Based on the direct sampling method (DSM) introduced in [Y. T. Chow, K. Ito, and J. Zou, Inverse Problems, 30 (2016), 095003], we propose deep direct sampling methods (DDSMs) to locate inhomogeneous inclusions in which two types of deep neural networks (DNNs) are constructed to approximate the index function (functional): fully connected neural networks and convolutional neural networks. The proposed DDSMs are easy to be implemented, capable of incorporating multiple Cauchy data pairs to achieve high-quality reconstruction and highly robust with respect to large noise. Additionally, the implementation of DDSMs adopts offline-online decomposition, which helps to reduce a lot of computational costs and makes DDSMs as efficient as the conventional DSM proposed by Chow, Ito, and Zou. The numerical experiments are presented to demonstrate the efficacy and show the potential benefits of combining DNN with DSM.