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Case Western Reserve University DEPARTMENT OF MATHEMATICS |
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DEPARTMENT OF MATHEMATICS SEMINARS and COLLOQUIAAPPLIED MATHEMATICS |
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PhD DefenseDebra MCGIVNEY, Graduate Student Thesis Advisors: Dr. Erkki Somersalo and Dr. Daniela Calvetti ABSTRACT Computational inverse problems frequently give rise to linear or nonlinear least squares problems and an effective way to solve such problems is with Krylov subspace iterative methods. We present a computational scheme to be applied to electrical impedance tomography (EIT), a nonlinear, ill-posed inverse problem in which we look to estimate the electrical admittivity inside a body given current/voltage measurements made on the boundary. The problem is addressed in the Bayesian statistical framework with an inner-outer iterative scheme to compute the maximum a posteriori estimate with the aid of statistically inspired preconditioners. The right preconditioner arises from a structural prior covariance, while the left preconditioner accounts for the noise, which consists of the assumed measurement error and the modeling error due to a coarse discretization of the problem. The admittivity distribution is updated in the outer iteration and the linearized sub-problem is solved in the inner iteration via conjugate gradient for least squares, with an inexact Newton stopping criterion. Computational efficiency is also addressed in the solution of the forward problem with a finite difference discretization scheme resulting in a fast adjoint method to compute the Jacobian matrix. The motivation of this work is the diagnosis of breast cancer, as the admittivity spectra of benign and malignant breast tumors are significantly different at low frequencies. As x-ray mammography is the current standard imaging modality used to screen for breast cancer, it is natural to see if we can combine the high spatial resolution of the mammogram image along with the EIT solution to improve the sensitivity and specificity of the diagnosis.
PhD DefenseJing QIN, Graduate Student Thesis Advisor: Dr. Weihong Guo ABSTRACT Prior information, including geometric priors and spatially variant intensity variations, assists to establish the mathematical models by providing more accurate description of the underlying signal/image. We have been exploring the applications of the extracted prior information in two directions: image denoising and signal/image reconstruction. To balance the removal of excessive noise and preservation of fine features, we developed a segmentation boosted image denoising scheme. Regarding compressive sensing (CS) signal/image reconstruction with the integration of prior information, we developed a high-frequency guided signal reconstruction, two-stage geometric information guided image reconstruction (GeoCS) and one total generalized variation (TGV) based image reconstruction model.
PhD DefenseLaura HOMA, Graduate Student Thesis Advisors: Dr. Erkki Somersalo and Dr. Daniela Calvetti ABSTRACT Magnetoencephalography (MEG) is a non-invasive brain imaging modality which localizes the electrical activity within the brain based on measurements of the induced magnetic field outside the head. MEG can potentially be used to localize the foci of the onset of seizures in order to assist with surgery planning. Since the data are severely corrupted by noise generated by both external sources and biological noise sources within the brain itself. Therefore, it is of paramount interest in MEG to develop methods to distinguish between the signal generated by the sources of interest from that arising from noise sources. We address the source separation problem within the Bayesian framework for both single time slice data and time series data. For single time slice data, we propose a mixture prior which incorporates the different statistical characteristics of the sources of interest and the noise sources. In addition, we propose a novel depth-scanning algorithm to identify and localize deep focal sources, overcoming the tendency of MEG inverse methods to explain all data with cortical sources. When considering source separation for time series data, we specifically address the problem of separating the signal of interest from the noise signal generated by spontaneous brain activity. It is well-known that this brain noise is correlated in both space and time. We take the novel approach of solving the MEG inverse problem using a Krylov subspace iterative method combined with statistically inspired left and right preconditioners. In particular, the left preconditioner is related to the covariance structure of the brain noise, while the right preconditioner is used to convey our prior beliefs about the statistical behavior of the unknown sources of interest.
PhD DefenseTaina Tuulia IMMONEN, Graduate Student Thesis Advisors: Dr. Erkki Somersalo and Dr. Daniela Calvetti ABSTRACT When modeling multicellular systems, we are often interested in mapping collective macroscopic behavior to cell level characteristics, and vice versa. One of the difficulties in bridging cell level and macroscopic scales is how to interpret the meaning of model parameters when moving from one scale to another. We explore this issue in the context of growth competition assays, used to quantify the ex vivo replicative fitness of HIV-1 isolates. We present a detailed, spatially distributed, hybrid stochastic-deterministic cell level model of the dynamics of two competing virus variants, and use it to perform in silico growth competition experiments to better understand how the characteristics of a virus strain influence its ex vivo fitness. We approximate the cell level model with low resolution and high resolution deterministic mean-field models, which do not account for cell level biological details or spatial structure. To establish a link between the models at the different scales, we estimate the parameters of the two macroscopic models from various data sets generated by the cell level model, approaching the parameter estimation problem from the Bayesian perspective. We investigate how well the macroscopic models specified by the estimated parameters are able to explain the data, and elucidate the intricacies of inferring underlying virus characteristics from macroscopic kinetics by exploring how the parameters of the cell level model relate to those of its macroscopic approximations.
Case Western Reserve University
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