| Popis: |
Computational inverse problems frequently give rise to linear or nonlinear least squares problems, and if the problems are large, iterative solvers are typically the method of choice, with the Krylov subspace methods a frequently used family of iterative schemes. In the case of nonlinear least squares problems, local linearization is an additional step that must be completed before applying an iterative solver. In this thesis, we outline an iterative scheme to solve the inverse problem of electrical impedance tomography (EIT), an ill-posed and nonlinear inverse problem in which we look to approximate the admittivity distribution inside a body given a discrete set of current and voltage measurements taken at the boundary. 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, rather than using one modality on its own.We solve the EIT inverse problem in the Bayesian framework, using a statistically preconditioned Krylov iterative scheme and applying prior information from the x-ray image to guide the solution. We present an inner-outer iterative algorithm for computing the maximum a posteriori estimate of the Bayesian solution. The covariance of the prior distribution acts as the right preconditioner and the covariance of the combined modeling and measurement error acts as the left preconditioner to the linearized problem, which is solved in the inner iteration using a Krylov subspace method with a stopping criterion that is coupled with the progress made in the outer iteration. The discretization of the forward model enhances the computational efficiency of the method and allows for a fast computation of the Jacobian. We add a classification algorithm to interpret the results of the EIT inverse problem by way of a quantitative image, mapping out the tissue classification inside of the breast. We present a summary on the background of the EIT problem and discuss at length our computational methods to solve both the forward and inverse problems. Computed examples are discussed in detail to demonstrate the effectiveness of the inner-outer algorithm and the simulated high sensitivity and specificity of the method. |
