Matching Items (6)
Filtering by
- All Subjects: Applied Mathematics
Description
Solving partial differential equations on surfaces has many applications including modeling chemical diffusion, pattern formation, geophysics and texture mapping. This dissertation presents two techniques for solving time dependent partial differential equations on various surfaces using the partition of unity method. A novel spectral cubed sphere method that utilizes the windowed Fourier technique is presented and used for both approximating functions on spherical domains and solving partial differential equations. The spectral cubed sphere method is applied to solve the transport equation as well as the diffusion equation on the unit sphere. The second approach is a partition of unity method with local radial basis function approximations. This technique is also used to explore the effect of the node distribution as it is well known that node choice plays an important role in the accuracy and stability of an approximation. A greedy algorithm is implemented to generate good interpolation nodes using the column pivoting QR factorization. The partition of unity radial basis function method is applied to solve the diffusion equation on the sphere as well as a system of reaction-diffusion equations on multiple surfaces including the surface of a red blood cell, a torus, and the Stanford bunny. Accuracy and stability of both methods are investigated.
ContributorsIslas, Genesis Juneiva (Author) / Platte, Rodrigo (Thesis advisor) / Cochran, Douglas (Committee member) / Espanol, Malena (Committee member) / Kao, Ming-Hung (Committee member) / Renaut, Rosemary (Committee member) / Arizona State University (Publisher)
Created2021
Description
High-dimensional systems are difficult to model and predict. The underlying mechanisms of such systems are too complex to be fully understood with limited theoretical knowledge and/or physical measurements. Nevertheless, redcued-order models have been widely used to study high-dimensional systems, because they are practical and efficient to develop and implement. Although model errors (biases) are inevitable for reduced-order models, these models can still be proven useful to develop real-world applications. Evaluation and validation for idealized models are indispensable to serve the mission of developing useful applications. Data assimilation and uncertainty quantification can provide a way to assess the performance of a reduced-order model. Real data and a dynamical model are combined together in a data assimilation framework to generate corrected model forecasts of a system. Uncertainties in model forecasts and observations are also quantified in a data assimilation cycle to provide optimal updates that are representative of the real dynamics. In this research, data assimilation is applied to assess the performance of two reduced-order models. The first model is developed for predicting prostate cancer treatment response under intermittent androgen suppression therapy. A sequential data assimilation scheme, the ensemble Kalman filter (EnKF), is used to quantify uncertainties in model predictions using clinical data of individual patients provided by Vancouver Prostate Center. The second model is developed to study what causes the changes of the state of stratospheric polar vortex. Two data assimilation schemes: EnKF and ES-MDA (ensemble smoother with multiple data assimilation), are used to validate the qualitative properties of the model using ECMWF (European Center for Medium-Range Weather Forecasts) reanalysis data. In both studies, the reduced-order model is able to reproduce the data patterns and provide insights to understand the underlying mechanism. However, significant model errors are also diagnosed for both models from the results of data assimilation schemes, which suggests specific improvements of the reduced-order models.
ContributorsWu, Zhimin (Author) / Kostelich, Eric (Thesis advisor) / Moustaoui, Mohamed (Thesis advisor) / Jones, Chris (Committee member) / Espanol, Malena (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2021
Description
The variable projection method has been developed as a powerful tool for solvingseparable nonlinear least squares problems. It has proven effective in cases where
the underlying model consists of a linear combination of nonlinear functions, such as
exponential functions. In this thesis, a modified version of the variable projection
method to address a challenging semi-blind deconvolution problem involving mixed
Gaussian kernels is employed. The aim is to recover the original signal accurately
while estimating the mixed Gaussian kernel utilized during the convolution process.
The numerical results obtained through the implementation of the proposed algo-
rithm are presented. These results highlight the method’s ability to approximate the
true signal successfully. However, accurately estimating the mixed Gaussian kernel
remains a challenging task. The implementation details, specifically focusing on con-
structing a simplified Jacobian for the Gauss-Newton method, are explored. This
contribution enhances the understanding and practicality of the approach.
ContributorsDworaczyk, Jordan Taylor (Author) / Espanol, Malena (Thesis advisor) / Welfert, Bruno (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2023
Description
A pneumonia-like illness emerged late in 2019 (coined COVID-19), caused by SARSCoV-2, causing a devastating global pandemic on a scale never before seen sincethe 1918/1919 influenza pandemic. This dissertation contributes in providing deeper
qualitative insights into the transmission dynamics and control of the disease in the
United States. A basic mathematical model, which incorporates the key pertinent
epidemiological features of SARS-CoV-2 and fitted using observed COVID-19 data,
was designed and used to assess the population-level impacts of vaccination and face
mask usage in mitigating the burden of the pandemic in the United States. Conditions
for the existence and asymptotic stability of the various equilibria of the model were
derived. The model was shown to undergo a vaccine-induced backward bifurcation
when the associated reproduction number is less than one. Conditions for achieving
vaccine-derived herd immunity were derived for three of the four FDA-approved vaccines (namely Pfizer, Moderna and Johnson & Johnson vaccine), and the vaccination
coverage level needed to achieve it decreases with increasing coverage of moderately and highly-effective face masks. It was also shown that using face masks as a singular
intervention strategy could lead to the elimination of the pandemic if moderate or
highly-effective masks are prioritized and pandemic elimination prospects are greatly
enhanced if the vaccination program is combined with a face mask use strategy that
emphasizes the use of moderate to highly-effective masks with at least moderate coverage. The model was extended in Chapter 3 to allow for the assessment of the
impacts of waning and boosting of vaccine-derived and natural immunity against
the BA.1 Omicron variant of SARS-CoV-2. It was shown that vaccine-derived herd
immunity can be achieved in the United States via a vaccination-boosting strategy
which entails fully vaccinating at least 72% of the susceptible populace. Boosting
of vaccine-derived immunity was shown to be more beneficial than boosting of natural immunity. Overall, this study showed that the prospects of the elimination of
the pandemic in the United States were highly promising using the two intervention
measures.
ContributorsSafdar, Salman (Author) / Gumel, Abba (Thesis advisor) / Kostelich, Eric (Committee member) / Kang, Yun (Committee member) / Fricks, John (Committee member) / Espanol, Malena (Committee member) / Arizona State University (Publisher)
Created2023
Description
During the inversion of discrete linear systems, noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion. This is a process called regularization. The influence of the provided prior information is controlled by the introduction of non-negative regularization parameter(s). Many methods are available for both the selection of appropriate regularization parame- ters and the inversion of the discrete linear system. Generally, for a single problem there is just one regularization parameter. Here, a learning approach is considered to identify a single regularization parameter based on the use of multiple data sets de- scribed by a linear system with a common model matrix. The situation with multiple regularization parameters that weight different spectral components of the solution is considered as well. To obtain these multiple parameters, standard methods are modified for identifying the optimal regularization parameters. Modifications of the unbiased predictive risk estimation, generalized cross validation, and the discrepancy principle are derived for finding spectral windowing regularization parameters. These estimators are extended for finding the regularization parameters when multiple data sets with common system matrices are available. Statistical analysis of these estima- tors is conducted for real and complex transformations of data. It is demonstrated that spectral windowing regularization parameters can be learned from these new esti- mators applied for multiple data and with multiple windows. Numerical experiments evaluating these new methods demonstrate that these modified methods, which do not require the use of true data for learning regularization parameters, are effective and efficient, and perform comparably to a supervised learning method based on es- timating the parameters using true data. The theoretical developments are validated for one and two dimensional image deblurring. It is verified that the obtained estimates of spectral windowing regularization parameters can be used effectively on validation data sets that are separate from the training data, and do not require known data.
ContributorsByrne, Michael John (Author) / Renaut, Rosemary (Thesis advisor) / Cochran, Douglas (Committee member) / Espanol, Malena (Committee member) / Jackiewicz, Zdzislaw (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2023
Description
Dengue is a mosquito-borne arboviral disease that causes significant public health burden in many trophical and sub-tropical parts of the world (where dengue is endemic). This dissertation is based on using mathematical modeling approaches, coupled with rigorous analysis and computation, to study the transmission dynamics and control of dengue disease. In Chapter 2, a new deterministic model was designed and used to assess the impact of local fluctuation of temperature and mosquito vertical (transvasorial) transmission on the population abundance of dengue mosquitoes and disease in a population. The model, which takes the form of a deterministic system of nonlinear differential equations, was parametrized using data from the Chiang Mai province of Thailand. The disease-free equilibrium of the model was shown to be globally-asymptotically stable when a certain epidemiological quantity is less than unity. Vertical transmission was shown to only have marginal impact on the disease dynamics, and its effect is temperature-dependent. Dengue burden in the province is maximized when the mean monthly temperature lie in the range [26-28] C. A new deterministic model was designed in Chapter 3 to assess the impact of the release of Wolbachia-infected mosquitoes on curtailing the mosquito population and dengue disease in a population. The model, which stratifies the mosquito population in terms of sex and Wolbachia-infection status, was rigorously analysed to characterize the bifurcation property of the model as well as the asymptotic stability of the various disease-free equilibria. Simulations, using Wolbachia-based mosquito control from Queensland, Australia, showed that the frequent release of mosquitoes infected with the bacterium can lead to the effective control of the local wild mosquito population, and that such effective control increases with increasing number of Wolbachia-infected mosquitoes released (up to 90% reduction in the wild mosquito population, from their baseline values, can be achieved). It was also shown that the well-known feature of cytoplasmic incompatibility has very little effect on the effectiveness of the Wolbachia-based mosquito control.
ContributorsTaghikhani, Rahim (Author) / Gumel, Abba (Thesis advisor) / Crook, Sharon (Committee member) / Espanol, Malena (Committee member) / Kuang, Yang (Committee member) / Scotch, Matthew (Committee member) / Arizona State University (Publisher)
Created2020