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Computation
Special Issues
Inverse Problems with Partial Data
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Inverse Problems with Partial Data

A special issue of Computation (ISSN 2079-3197). This special issue belongs to the section "Computational Engineering".

Deadline for manuscript submissions: closed (20 November 2021) | Viewed by 22568

Special Issue Editors

Dr. Qin Li

E-Mail Website
Guest Editor
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53705, USA
Interests: inverse problems (theory); sampling problems; semiclassical limits for quantum systems
Dr. Li Wang

E-Mail Website
Guest Editor
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Interests: numerical analysis; scientific computing; applied math
Dr. Leonardo Andrés Zepeda Núñez

E-Mail Website
Guest Editor
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53705, USA
Interests: deep learning; inverse problems; high-frequency wave propagation; numerical linear algebra

Special Issue Information

Dear Colleagues,

Inverse problems are ubiquitous in science and engineering. In nearly all engineering applications, ranging from optical tomography to seismic inversion, measurements are taken to infer parameters in certain partial differential equation models that are used to describe the dynamical systems in the forward setting. While the full measurements are ideal for the reconstruction of parameters, in real applications, only partial data, mostly polluted, are available, degrading the accuracy of the reconstruction. It is of great significance, both mathematically and practically, to theoretically understand the impact of partial polluted data and numerically recover the unknown.

In this Special Issue, we collect several contributions addressing the state-of-art research on this topic, encompassing both theoretical and numerical aspects. For the numerical aspects, the Special Issue addresses emerging tools from data science, optimization, Bayesian sampling, and machine learning. For the theoretical aspects, it discusses multiple topics, such as stability deterioration due to the partial data, CGO solutions, and qualitative methods. The applications of these methods range from biomedical imaging, geophysics to atmospheric science. The issue provides various angles to examine systems with unknown parameters when only partial information can be measured.

Dr. Qin Li
Dr. Li Wang
Dr. Leonardo Andrés Zepeda Núñez
Guest Editors

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Published Papers (7 papers)

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Research

18 pages, 1852 KiB  
Article
Inverse Modeling of Hydrologic Parameters in CLM4 via Generalized Polynomial Chaos in the Bayesian Framework
by Georgios Karagiannis, Zhangshuan Hou, Maoyi Huang and Guang Lin
Computation 2022, 10(5), 72; https://doi.org/10.3390/computation10050072 - 5 May 2022
Cited by 1 | Viewed by 1973
Abstract
In this work, generalized polynomial chaos (gPC) expansion for land surface model parameter estimation is evaluated. We perform inverse modeling and compute the posterior distribution of the critical hydrological parameters that are subject to great uncertainty in the Community Land Model (CLM) for [...] Read more.
In this work, generalized polynomial chaos (gPC) expansion for land surface model parameter estimation is evaluated. We perform inverse modeling and compute the posterior distribution of the critical hydrological parameters that are subject to great uncertainty in the Community Land Model (CLM) for a given value of the output LH. The unknown parameters include those that have been identified as the most influential factors on the simulations of surface and subsurface runoff, latent and sensible heat fluxes, and soil moisture in CLM4.0. We set up the inversion problem in the Bayesian framework in two steps: (i) building a surrogate model expressing the input–output mapping, and (ii) performing inverse modeling and computing the posterior distributions of the input parameters using observation data for a given value of the output LH. The development of the surrogate model is carried out with a Bayesian procedure based on the variable selection methods that use gPC expansions. Our approach accounts for bases selection uncertainty and quantifies the importance of the gPC terms, and, hence, all of the input parameters, via the associated posterior probabilities. Full article
(This article belongs to the Special Issue Inverse Problems with Partial Data)
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32 pages, 7533 KiB  
Article
DIAS: A Data-Informed Active Subspace Regularization Framework for Inverse Problems
by Hai Nguyen, Jonathan Wittmer and Tan Bui-Thanh
Computation 2022, 10(3), 38; https://doi.org/10.3390/computation10030038 - 11 Mar 2022
Cited by 2 | Viewed by 2972
Abstract
This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and [...] Read more.
This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and forward model are integrated, should remain untouched. We propose to employ the active subspace method to determine the data-informativeness of a parameter. The resulting framework is thus called a data-informed (DI) active subspace (DIAS) regularization. Four proposed DIAS variants are rigorously analyzed, shown to be robust with the regularization parameter and capable of avoiding polluting solution features informed by the data. They are thus well suited for problems with small or reasonably small noise corruptions in the data. Furthermore, the DIAS approaches can effectively reuse any Tikhonov regularization codes/libraries. Though they are readily applicable for nonlinear inverse problems, we focus on linear problems in this paper in order to gain insights into the framework. Various numerical results for linear inverse problems are presented to verify theoretical findings and to demonstrate advantages of the DIAS framework over the Tikhonov, truncated SVD, and the TSVD-based DI approaches. Full article
(This article belongs to the Special Issue Inverse Problems with Partial Data)
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24 pages, 579 KiB  
Article
Bayesian Instability of Optical Imaging: Ill Conditioning of Inverse Linear and Nonlinear Radiative Transfer Equation in the Fluid Regime
by Qin Li, Kit Newton and Li Wang
Computation 2022, 10(2), 15; https://doi.org/10.3390/computation10020015 - 19 Jan 2022
Viewed by 2524
Abstract
For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, [...] Read more.
For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio-)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well- to ill-posed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback–Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is ill-posed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime. Full article
(This article belongs to the Special Issue Inverse Problems with Partial Data)
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20 pages, 1509 KiB  
Article
Direct Sampling for Recovering Sound Soft Scatterers from Point Source Measurements
by Isaac Harris
Computation 2021, 9(11), 120; https://doi.org/10.3390/computation9110120 - 14 Nov 2021
Cited by 5 | Viewed by 2748
Abstract
In this paper, we consider the inverse problem of recovering a sound soft scatterer from the measured scattered field. The scattered field is assumed to be induced by a point source on a curve/surface that is known. Here, we propose and analyze new [...] Read more.
In this paper, we consider the inverse problem of recovering a sound soft scatterer from the measured scattered field. The scattered field is assumed to be induced by a point source on a curve/surface that is known. Here, we propose and analyze new direct sampling methods for this problem. The first method we consider uses a far-field transformation of the near-field data, which allows us to derive explicit bounds in the resolution analysis for the direct sampling method’s imaging functional. Two direct sampling methods are studied, using the far-field transformation. For these imaging functionals, we use the Funk–Hecke identities to study the resolution analysis. We also study a direct sampling method for the case of the given Cauchy data. Numerical examples are given to show the applicability of the new imaging functionals for recovering a sound soft scatterer with full and partial aperture data. Full article
(This article belongs to the Special Issue Inverse Problems with Partial Data)
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15 pages, 341 KiB  
Article
Multiscale Convergence of the Inverse Problem for Chemotaxis in the Bayesian Setting
by Kathrin Hellmuth, Christian Klingenberg, Qin Li and Min Tang
Computation 2021, 9(11), 119; https://doi.org/10.3390/computation9110119 - 11 Nov 2021
Cited by 3 | Viewed by 2655
Abstract
Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe [...] Read more.
Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe the organism’s movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller–Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms’ population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions. Full article
(This article belongs to the Special Issue Inverse Problems with Partial Data)
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23 pages, 1894 KiB  
Article
Conditional Variational Autoencoder for Learned Image Reconstruction
by Chen Zhang, Riccardo Barbano and Bangti Jin
Computation 2021, 9(11), 114; https://doi.org/10.3390/computation9110114 - 28 Oct 2021
Cited by 9 | Viewed by 5273
Abstract
Learned image reconstruction techniques using deep neural networks have recently gained popularity and have delivered promising empirical results. However, most approaches focus on one single recovery for each observation, and thus neglect information uncertainty. In this work, we develop a novel computational framework [...] Read more.
Learned image reconstruction techniques using deep neural networks have recently gained popularity and have delivered promising empirical results. However, most approaches focus on one single recovery for each observation, and thus neglect information uncertainty. In this work, we develop a novel computational framework that approximates the posterior distribution of the unknown image at each query observation. The proposed framework is very flexible: it handles implicit noise models and priors, it incorporates the data formation process (i.e., the forward operator), and the learned reconstructive properties are transferable between different datasets. Once the network is trained using the conditional variational autoencoder loss, it provides a computationally efficient sampler for the approximate posterior distribution via feed-forward propagation, and the summarizing statistics of the generated samples are used for both point-estimation and uncertainty quantification. We illustrate the proposed framework with extensive numerical experiments on positron emission tomography (with both moderate and low-count levels) showing that the framework generates high-quality samples when compared with state-of-the-art methods. Full article
(This article belongs to the Special Issue Inverse Problems with Partial Data)
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14 pages, 1608 KiB  
Article
Parameter Estimation of Partially Observed Turbulent Systems Using Conditional Gaussian Path-Wise Sampler
by Ziheng Zhang and Nan Chen
Computation 2021, 9(8), 91; https://doi.org/10.3390/computation9080091 - 13 Aug 2021
Viewed by 2920
Abstract
Parameter estimation of complex nonlinear turbulent dynamical systems using only partially observed time series is a challenging topic. The nonlinearity and partial observations often impede using closed analytic formulae to recover the model parameters. In this paper, an exact path-wise sampling method is [...] Read more.
Parameter estimation of complex nonlinear turbulent dynamical systems using only partially observed time series is a challenging topic. The nonlinearity and partial observations often impede using closed analytic formulae to recover the model parameters. In this paper, an exact path-wise sampling method is developed, which is incorporated into a Bayesian Markov chain Monte Carlo (MCMC) algorithm in light of data augmentation to efficiently estimate the parameters in a rich class of nonlinear and non-Gaussian turbulent systems using partial observations. This path-wise sampling method exploits closed analytic formulae to sample the trajectories of the unobserved variables, which avoid the numerical errors in the general sampling approaches and significantly increase the overall parameter estimation efficiency. The unknown parameters and the missing trajectories are estimated in an alternating fashion in an adaptive MCMC iteration algorithm with rapid convergence. It is shown based on the noisy Lorenz 63 model and a stochastically coupled FitzHugh–Nagumo model that the new algorithm is very skillful in estimating the parameters in highly nonlinear turbulent models. The model with the estimated parameters succeeds in recovering the nonlinear and non-Gaussian features of the truth, including capturing the intermittency and extreme events, in both test examples. Full article
(This article belongs to the Special Issue Inverse Problems with Partial Data)
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