Browsing by Subject "inverse problems"
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(2023)In this thesis, we develop a Bayesian approach to the inverse problem of inferring the shape of an asteroid from time-series measurements of its brightness. We define a probabilistic model over possibly non-convex asteroid shapes, choosing parameters carefully to avoid potential identifiability issues. Applying this probabilistic model to synthetic observations and sampling from the posterior via Markov Chain Monte Carlo, we show that the model is able to recover the asteroid shape well in the limit of many well-separated observations, and is able to capture posterior uncertainty in the case of limited observations. We greatly accelerate the computation of the forward problem (predicting the measured light curve given the asteroid’s shape parameters) by using a bounding volume hierarchy and by exploiting data parallelism on a graphics processing unit.
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(2022)The quantification of carbon dioxide emissions pose a significant and multi-faceted problem for the atmospheric sciences as a part of the research regarding global warming and greenhouse gases. Emissions originating from point sources, referred to as plumes, can be simulated using mathematical and physical models, such as a convection-diffusion plume model and a Gaussian plume model. The convection-diffusion model is based on the convection-diffusion partial differential equation describing mass transfer in diffusion and convection fields. The Gaussian model is a special case or a solution for the general convection-diffusion equation when assumptions of homogeneous wind field, relatively small diffusion and time independence are made. Both of these models are used for simulating the plumes in order to find out the emission rate for the plume source. An equation for solving the emission rate can be formulated as an inverse problem written as y=F(x)+ε where y is the observed data, F is the plume model, ε is the noise term and x is an unknown vector of parameters, including the emission rate, which needs to be solved. For an ill-posed inverse problem, where F is not well behaved, the solution does not exist, but a minimum norm solution can be found. That is, the solution is a vector x which minimizes a chosen norm function, referred to as a loss function. This thesis focuses on the convection-diffusion and Gaussian plume models, and studies both the difference and the sensibility of these models. Additionally, this thesis investigates three different approaches for optimizing loss functions: the optimal estimation for linear model, Levenberg–Marquardt algorithm for non-linear model and adaptive Metropolis algorithm. A goodness of different fits can be quantified by comparing values of the root mean square errors; the better fit the smaller value the root mean square error has. A plume inversion program has been implemented in Python programming language using the version 3.9.11 to test the implemented models and different algorithms. Assessing the parameters' effect on the estimated emission rate is done by performing sensitivity tests for simulated data. The plume inversion program is also applied for the satellite data and the validity of the results is considered. Finally, other more advanced plume models and improvements for the implementation will be discussed.
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(2019)The motivation for the methods developed in this thesis rises from solving the severely ill-posed inverse problem of limited angle computed tomography. Breast tomosynthesis provides an example where the inner structure of the breast should be reconstructed from a very limited measurement angle. Some parts of the boundaries of the structure can be recovered from the X-ray measurements and others can not. These are referred to as visible and invisible boundaries. For parallel beam measurement geometry directions of visible and invisible boundaries can be deduced from the measurement angles. This motivates the usage of the concept of wavefront set. Roughly speaking, a wavefront set contains boundary points and their directions. The definition of wavefront set is based on Fourier analysis, but its characterization with the decay properties of functions called shearlets is used in this thesis. Shearlets are functions based on changing resolution, orientation, and position of certain generating functions. The theoretical part of this thesis focuses on studying this connection between shearlets and wavefront sets. This thesis applies neural networks to the limited angle CT problem since neural networks have become state-of-the-art in many computer vision tasks and achieved impressive performance in inverse problems related to imaging. Neural networks are compositions of multiple simple functions, typically alternating linear functions and some element-wise non-linearities. They are trained to learn values for a huge amount of parameters to approximate the desired relation between input and output spaces. Neural networks are very flexible function approximators, but high dimensional optimization of parameters from data makes them hard to interpret. Convolutional neural networks (CNN) are the ones that succeed in tasks with image-like inputs. U-Net is a CNN architecture with very good properties, like learning useful parameters form considerably small data sets. This thesis provides two U-Net based CNN methods for solving limited angle CT problems. The main focus is on method projecting model-based reconstructions such that the projections have the desired wavefront sets. The guiding principle of this projector network is that it should not change reconstruction already projected to the given wavefront set. Another network estimates the invisible part of the wavefront set from the visible one. Few different data sets are simulated to train and evaluate these methods and performance on real data is also tested. A combination of the wavefront set estimator and the projector networks were used to postprocess model-based reconstructions. The fact this postprocessing has two steps increases the interpretability and the control over the processes performed by neural networks. This postprocessing increased the quality of reconstructions significantly and quality was even better when the true wavefront set was given for the projector as a prior.
Now showing items 1-3 of 3