Masters Degrees (Applied Mathematics)

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    Intelligent control for processing solar photovoltaic energy
    (Stellenbosch : Stellenbosch University, 2023-12) Wacira, Joseph Muthui; Bah, Bubacarr; Vargas, Alessandro; Stellenbosch University. Faculty of Science. Dept. of Applied Mathematics.
    ENGLISH ABSTRACT: Maximum Power Point Tracking (MPPT) techniques play a pivotal role in optimizing the performance of photovoltaic systems within renewable energy. Traditional MPPT methods, often reliant on Proportional Integral and Derivative (PID) controllers, face challenges when applied to nonlinear systems with dynamic operating conditions, typical in photovoltaic systems where temperature and irradiance continually fluctuate. The inherent static nature of the PID parameters leads to power losses, thereby reducing their efficiency. Additionally, they rely on trial-and-error approaches to determine the actual Maximum Power Point (MPP). This study introduces two novel MPPT approaches: the Gradient Descent Approach and the Deep Q-Network (DQN) approach. These methods share a common feature: they require knowledge of the maximum power point (MPP). An ANN was employed to predict the MPP under current operating conditions. Once the MPP is known, the Gradient Descent Approach aims to minimize the mean squared error by adjusting the duty cycle, whereas the DQN Approach employs a state-action-reward system that penalizes deviations from the MPP and large actions. To evaluate the effectiveness o f t hese a pproaches, s imulations were conducted under uniform operating conditions using MATLAB/Simulink, with data sourced from the NSRBD website for Brazil. The results were compared with those of the conventional Perturb and Observe algorithm with a PI controller tuned using the Ziegler-Nichols method under Standard Test Conditions. Simulations revealed that the proposed methodologies exhibited significantly higher efficiency than the benchmark algorithm. Furthermore, they demonstrate fast response times and minimal steady-state errors. Although these findings underscore the promise of the proposed approaches, further validation in real-world environments is necessary to confirm their superiority and practical applicability.
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    An ultraspherical spectral element method for solving partial differential equations
    (Stellenbosch : Stellenbosch University, 2023-12) Nel, Emma Alida; Hale, Nicholas; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics Division.
    ENGLISH ABSTRACT: We investigate the ultraspherical spectral element method for solving second-order partial differential equations in two dimensions. Moreover, a novel coordinate transformation is introduced to broaden the scope of the method, making it applicable to rectangular domains with circular holes (square donuts), as well as certain types of curved boundaries. The presented method is an integration of two approaches, namely the ultraspherical spectral method and the hierarchical Poincaré–Steklov (HPS) scheme. The ultraspherical method is a Petrov–Galerkin scheme that presents operators in the form of sparse and almost-banded matrices, enabling both stability and computational efficiency. The HPS method is a recursive domain decomposition strategy that enables fast direct solves. It merges solution operators and Dirichlet-to-Neumann operators between subdomains, enforcing continuity of the solution and its derivative across domain boundaries. The fusion of these two methods, combined with a bilinear mapping, results in an accurate discretisation with an explicit direct solve that can be applied to problems on arbitrary polygonal domains with smooth solutions. A major advantage is the reuse of precomputed solution operators facilitated by the HPS scheme, enhancing the efficiency of elliptic solves within implicit and semi-implicit time-steppers. Additionally, the approach is highly parallelisable, allowing for efficient computation time. An implementation of the method is established as a software system, ultraSEM, which employs the HPS method to solve on rectangular and polygonal domains. We extend this implementation to allow for solving on domains with circular cavities. This extension relies on a nonlinear coordinate mapping and proves to work effectively, achieving near machine level precision accuracy. On some simple test problems, we demonstrate geometric convergence for refinement of the polynomial degree and algebraic convergence for domain refinement. Furthermore, we show that execution times scale comparably to those achieved for a rectangular domain. We demonstrate the application of the method on various time-dependent and fluid dynamics examples, including contaminant transport and reaction-diffusion systems, and underscore the practical applicability of the methodology and the new domain.
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    A catalytic model for SARS-CoV-2 reinfections : performing simulation-based validation and extending the model to include nth infections
    (Stellenbosch : Stellenbosch University, 2023-12) Lombard, Belinda; Van Schalkwyk, Cari; Pulliam, Juliet; Stellenbosch University. Faculty of Science. Dept. of Applied Mathematics.
    ENGLISH SUMMARY: Background: A global pandemic of COVID-19, caused by SARS-CoV-2, was declared in March 2020. Subsequently, studies have revealed a high seroprevalence of SARS-CoV-2 in both South African and global populations, along with instances of multiple reinfections. Among various models, a catalytic model has been developed for detecting population-level increases in risk of reinfection, following primary infection. This thesis aims to assess how potential biases from imperfect data observation processes affect the catalytic model’s ability to detect increases in reinfection risk. Furthermore, the thesis extends the catalytic model to detect increases in the risk of multiple reinfections. Methods: Simulation-based validation involved creating different reinfection scenarios representing real life data, which were then used in the model’s fitting and projection procedure. Observed reinfections were simulated using a time-series of primary infections, representative of South African data. Scenarios included considering both imperfect observation (with constant observation probability or a probability dependent on primary infection count) and mortality. The method’s ability to detect increases in the reinfection risk was measured by determining both the clusters of reinfections and the proportion of points that fell above the projection interval. Following simulation-based validation, the method was extended to detect population-level increases in the risk of 𝑛𝑡ℎ infections. This extended method was applied to observed third infections in South Africa, with an additional model parameter representing increased reinfections during the Omicron wave. Simulation-based validation was conducted on the extended method to assess its ability to detect increases of varying magnitudes in the risk of third infection. Results: During the simulation-based validation of the original catalytic model, model parameters converged in most scenarios. Failure to converge was mostly related to insufficient cases to properly inform the model parameters during the fitting procedure. Scenarios where the model parameters did not converge, or where the simulated data did not accurately fit the model, were excluded from interpretation. Introducing an increase in the reinfection risk resulted in successful detection of an increase (even with small increments), although with delayed timing under lower observed infection numbers. Mortality from first infections, unaccounted for in the model, did not impact the method’s ability to detect increases in the reinfection risk. The method demonstrated high specificity, reliably distinguishing true increases in the reinfection risk from noise. The catalytic model was extended to detect increases in the risk of 𝑛𝑡ℎ infections, and the extended method’s ability to detect increases in the risk of third infections was validated. The additional third infection hazard representing increased reinfection risk observed during the Omicron wave was successfully fitted to the data, and the method effectively detected increases in the risk of third infections. Conclusion: The findings highlight the need for sufficient infection data and the importance of convergence as a prerequisite for result interpretation. The extended model reliably detected increases in the risk of two or more reinfections and demonstrated robustness under different observation processes and increases in reinfection risk scenarios.
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    Mathematical modelling of tuberculosis in South Africa : investigating the impact of interventions on population-level incidence and mortality
    (Stellenbosch : Stellenbosch University, 2023-03) Brown, Lauren; Van Schalkwyk, Cari; Marx, Florian; Stellenbosch University. Faculty of Science. Dept. of Applied Mathematics.
    ENGLISH SUMMARY: Background. Tuberculosis (TB) remains a major public health threat in South Africa. Substantial additional efforts are therefore needed to prevent, find, and successfully treat the disease. An increasing number of mathematical modelling studies have investigated the population-level impact of TB prevention and care interventions; however, this evidence has not yet been assessed in the South African context. Of particular concern for TB care in South Africa is the high proportion of initial loss to follow-up (ILTFU), defined as loss to follow-up of individuals who were diagnosed with TB but who did not (yet) initiate TB treatment. The aim of this thesis was to review existing literature on TB mathematical modelling research to determine the most effective intervention strategies to reduce TB burden in South Africa, to identify potential gaps in TB modelling research, and further, to conduct a mathematical modelling study to estimate the impact of reducing ILTFU in South Africa. Methods. A systematic review of studies that used transmission-dynamic models of TB in South Africa was conducted. PubMed, Scopus, and Web of Science databases were searched. Target populations, types of interventions, and estimates of impact on outcomes related to the End TB strategy targets were summarized. For country-level studies, average annual percentage declines (AAPDs) in TB incidence and mortality were estimated to compare the impact of interventions. Additionally, an existing TB transmission-dynamic model was adapted to estimate the impact of reducing ILTFU in South Africa. Data from the LINKEDIn study, a large quasi-experimental study that was conducted in three South African provinces, were used to inform model scenarios and intervention parameter estimates. The impact of scaling-up the LINKEDIn intervention to country level was specified as the number of incident cases and deaths averted over a 13-year period (2023-2035). Results. Twenty-nine studies were identified in the systematic review, of which seven modelled TB preventive interventions, 12 considered interventions along the TB care cascade, and 10 modelled combinations of both. One study considered reductions in TB-related catastrophic costs. The highest impact of a single intervention was estimated in studies of TB vaccination, preventive treatment among people living with HIV, and scale up of antiretroviral treatment. For preventive interventions, AAPDs for incidence varied between 0.06% and 7.07%, and for care-cascade interventions between 0.05% and 3.27%. In the modelling study, reducing ILTFU by 50% in the population was projected to avert 49,812 (95% uncertainty interval [UI]: 21,258-84,644) incident TB cases and 21,479 (UI: 9,500-32,661) deaths between 2023 and 2035. Sensitivity analyses showed that population-level impact would depend on rapid implementation and maximum effect of the intervention. Conclusion. This thesis describes a body of mathematical modelling research with focus on TB prevention and care in South Africa. Higher estimates of impact reported in studies of preventive interventions were found, highlighting the need to invest in TB prevention in South Africa. The population-level impact of reducing ILTFU was projected to be modest. Combinations rather than single interventions, such as the LINKEDIn intervention, are likely needed to reach the End TB Strategy targets in South Africa.
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    Bayesian forecasting of stock returns using simultaneous graphical dynamic linear models
    (Stellenbosch : Stellenbosch University, 2022-12) Kyakutwika, Nelson; Bartlett, Bruce; Becker, Ronnie; Stellenbosch University. Faculty of Science. Dept. of Applied Mathematics.
    ENGLISH ABSTRACT: Cross-series dependencies are crucial in obtaining accurate forecasts when forecast- ing a multivariate time series. Simultaneous Graphical Dynamic Linear Models (SGDLMs) are Bayesian models that elegantly capture cross-series dependencies. This study aims to forecast returns of a 40-dimensional time series of stock data using SGDLMs. The SGDLM approach involves constructing a customised dy- namic linear model (DLM) for each univariate time series. Every day, the DLMs are recoupled using importance sampling and decoupled using mean-field varia- tional Bayes. We summarise the standard theory on DLMs to set the foundation for studying SGDLMs. We discuss the structure of SGDLMs in detail and give de- tailed explanations of the proofs of the formulae involved. Our analyses are run on a CPU-based computer; an illustration of the intensity of the computations is given. We give an insight into the efficacy of the recoupling/decoupling techniques. Our results suggest that SGDLMs forecast the stock data accurately and respond to market gyrations nicely.