Browsing by Author "Carracedo Rodriguez, Andrea"

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    Approximation of Parametric Dynamical Systems
    Carracedo Rodriguez, Andrea (Virginia Tech, 2020-09-02)
    Dynamical systems are widely used to model physical phenomena and, in many cases, these physical phenomena are parameter dependent. In this thesis we investigate three prominent problems related to the simulation of parametric dynamical systems and develop the analysis and computational framework to solve each of them. In many cases we have access to data resulting from simulations of a parametric dynamical system for which an explicit description may not be available. We introduce the parametric AAA (p-AAA) algorithm that builds a rational approximation of the underlying parametric dynamical system from its input/output measurements, in the form of transfer function evaluations. Our algorithm generalizes the AAA algorithm, a popular method for the rational approximation of nonparametric systems, to the parametric case. We develop p-AAA for both scalar and matrix-valued data and study the impact of parameter scaling. Even though we present p-AAA with parametric dynamical systems in mind, the ideas can be applied to parametric stationary problems as well, and we include such examples. The solution of a dynamical system can often be expressed in terms of an eigenvalue problem (EVP). In many cases, the resulting EVP is nonlinear and depends on a parameter. A common approach to solving (nonparametric) nonlinear EVPs is to approximate them with a rational EVP and then to linearize this approximation. An existing algorithm can then be applied to find the eigenvalues of this linearization. The AAA algorithm has been successfully applied to this scheme for the nonparametric case. We generalize this approach by using our p-AAA algorithm to find a rational approximation of parametric nonlinear EVPs. We define a corresponding linearization that fits the format of the compact rational Krylov (CORK) algorithm for the approximation of eigenvalues. The simulation of dynamical systems may be costly, since the need for accuracy may yield a system of very large dimension. This cost is magnified in the case of parametric dynamical systems, since one may be interested in simulations for many parameter values. Interpolatory model order reduction (MOR) tackles this problem by creating a surrogate model that interpolates the original, is of much smaller dimension, and captures the dynamics of the quantities of interest well. We generalize interpolatory projection MOR methods from parametric linear to parametric bilinear systems. We provide necessary subspace conditions to guarantee interpolation of the subsystems and their first and second derivatives, including the parameter gradients and Hessians. Throughout the dissertation, the analysis is illustrated via various benchmark numerical examples.
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    Mathematical Models of Hepatitis B Virus Dynamics during Antiviral Therapy
    Carracedo Rodriguez, Andrea (Virginia Tech, 2016-04-21)
    Antiviral therapy for patients infected with hepatitis B virus is only partially efficient. The field is in high demand for understanding the connections between the virus, immune responses, short-term and long-term drug efficacy and the overall health of the liver. A mathematical model was introduced in 2009 to help elucidate the host-virus dynamics after the start of therapy. The model allows the study of complicated viral patterns observed in HBV patients. In our research, we will analyze this model to determine the biological markers (e.g. liver proliferation, immune responses, and drug efficacy) that determine the different decay patterns. We will also investigate how such markers affect the length of therapy and the amount of liver damage.