Special Issue "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).

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

Manuscript Submission Information

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Published Papers (4 papers)

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Research

Article
Direct Sampling for Recovering Sound Soft Scatterers from Point Source Measurements
Computation 2021, 9(11), 120; https://0-doi-org.brum.beds.ac.uk/10.3390/computation9110120 - 14 Nov 2021
Viewed by 268
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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Article
Multiscale Convergence of the Inverse Problem for Chemotaxis in the Bayesian Setting
Computation 2021, 9(11), 119; https://0-doi-org.brum.beds.ac.uk/10.3390/computation9110119 - 11 Nov 2021
Viewed by 246
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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Article
Conditional Variational Autoencoder for Learned Image Reconstruction
Computation 2021, 9(11), 114; https://0-doi-org.brum.beds.ac.uk/10.3390/computation9110114 - 28 Oct 2021
Viewed by 283
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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Article
Parameter Estimation of Partially Observed Turbulent Systems Using Conditional Gaussian Path-Wise Sampler
Computation 2021, 9(8), 91; https://0-doi-org.brum.beds.ac.uk/10.3390/computation9080091 - 13 Aug 2021
Viewed by 609
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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