Philosophiae Doctor - PhD / Cancer is a complex disease that involves a sequence of gene-environment interactions in a progressive process that cannot occur without dysfunction in multiple systems. From a mathematical point of view, the sequence of gene-environment interactions often leads to mathematical models which are hard to solve analytically. Therefore, this thesis focuses on the design and implementation of reliable numerical methods for nonlinear, first order delay differential equations, second order non-linear time-dependent parabolic partial (integro) differential problems and optimal control problems arising in cancer modeling. The development of cancer modeling is necessitated by the lack of reliable numerical methods, to solve the models arising in the dynamics of this dreadful disease. Our focus is on chemotherapy, biological stoichometry, double infections, micro-environment, vascular and angiogenic signalling dynamics. Therefore, because the existing standard numerical methods fail to capture the solution due to the behaviors of the underlying dynamics. Analysis of the qualitative features of the models with mathematical tools gives clear qualitative descriptions of the dynamics of models which gives a deeper insight of the problems. Hence, enabling us to derive robust numerical methods to solve such models. / 2021-04-30

In this thesis, the dynamics of an isolated cavitation bubble submerged in a steady flow is studied numerically. A Lagrangian-Eulerian approach is considered, in which properties of the fluid are computed first by means of Eulerian methods (in this study the commercial CFD software Ansys Fluent 19 was used) and the trajectory of the bubble is then computed in a Lagrangian fashion, i.e. the bubble is considered as a small particle moving relative to the fluid, due to the effect of several forces depending on fluid's pressure field, fluid's velocity field and bubble's radius. Bubble's radius dynamics, modeled by Rayleigh-Plesset equation, has a big influence on its kinetics, so a special attention is given to it. Two study cases are considered. The first one, motivated by acoustic cavitation is concerned with the response of the bubble's radius in a static flow under the influence of an oscillatory pressure field, the second one studies the trajectory of the bubble submerged in a fluid passing by a Venturi tube and a sharp-edged orifice plate.

Master’s thesis deals with development of software application to calculate contact pressure in eleastohydrodynamic lubricated contact in order to use previously obtained data of the lubricating film thickness. The introduction contains a short overview of methods used for the study of film thickness and contact pressure. Custom work includes a contact pressure solution derived from a film thickness, a description of the developed software and analysis of algorithms used for its compilation. The last part deals with the verification of algorithm, application of software for calculating the contact pressure on the rough surface and analysis of the results. The conclusion includes a summary and possible application of software in practice.

The thesis deals with constructing and solving a discrete mechanical model from the point of view of dynamics. As an example of a real-world problem which can be approximated in this way, the dynamical behaviour of a bow and arrow during loose have been chosen. In order to solve this particular type of model, a specialized piece of software has been developed. This method and software have been deployed to simulate shooting a straight bow of specific parametres. Two variants for the shape of the bowstaff have been evaluated.

The paper describes a storage scheme for sparse symmetric or nonsymmetric matrices which has been developed and used for many years at the Technical University of Chemnitz. An overview of existing library subroutines using such matrices is included.

In this paper, we investigate the question of the spectrally equivalence of the so- called Pseudo-Laplacian to the usual discrete Laplacian in order to use hierarchical preconditioners for this more complicate matrix. The spectral equivalence is shown to be equivalent to a Brezzi-type inequality, which is fulfilled for the finite element spaces considered here.

This paper describes the analysis tool DSPNexpress which has been developed at the Technische Universität Berlin since 1991. The development of DSPNexpress has been motivated by the lack of a powerful software package for the numerical solution of deterministic and stochastic Petri nets (DSPNs) and the complexity requirements imposed by evaluating memory consistency models for multicomputer systems. The development of DSPNexpress has gained by the author's experience with the version 1.4 of the software package GreatSPN. However, opposed to GreatSPN, the software architecture of DSPNexpress is particularly tailored to the numerical evaluation of DSPNs. Furthermore, DSPNexpress contains a graphical interface running under the X11 window system. To the best of the author's knowledge, DSPNexpress is the first software package which contains an efficient numerical algorithm for computing steady-state solutions of DSPNs.

For both randomized clinical trials and prospective cohort studies, the Cox regression model is a powerful tool for evaluating the effect of a treatment or an explanatory variable on time-to-event outcome. This method assumes proportional hazards over time. Systematic approaches to efficiently evaluate non-proportionality and to model data in the presence of non-proportionality are investigated. Six graphical methods are assessed to verify the proportional hazards assumption based on characteristics of the survival function, cumulative hazard, or the feature of residuals. Their performances are empirically evaluated with simulations by checking their ability to be consistent and sensitive in detecting proportionality or non-proportionality. Two-sample data are generated in three scenarios of proportional hazards and five types of alternatives (that is, non-proportionality). The usefulness of these graphical assessment methods depends on the event rate and type of non-proportionality. Three numerical (statistical testing) methods are compared via simulation studies to investigate the proportional hazards assumption. In evaluating data for proportionality versus non-proportionality, the goal is to test a non-zero slope in a regression of the variable or its residuals on a specific function of time, or a Kolmogorov-type supremum test. Our simulation results show that statistical test performance is affected by the number of events, event rate, and degree of divergence of non-proportionality for a given hazards scenario. Determining which test will be used in practice depends on the specific situation under investigation. Both graphical and numerical approaches have benefits and costs, but they are complementary to each other. Several approaches to model and summarize non-proportionality data are presented, including non-parametric measurements and testing, semi-parametric models, and a parametric approach. Some illustrative examples using simulated data and real data are also presented. In summary, we present a systemic approach using both graphical and numerical methods to identify non-proportionality, and to provide numerous modeling strategies when proportionality is violated in time-to-event data.

<p>High-dimensional regression problems are ubiquitous in science and engineering. Deep learning has been a critical tool for solving a wide range of high-dimensional problems with surprising performance. Even though in theory neural networks have good properties in terms of approximation and optimization, numerically obtaining an accurate neural network solution is a challenging problem due to the highly non-convex objective function and implicit bias of least square optimization. In this dissertation, we mainly discuss two topics involving the high dimensional regression using efficient deep learning algorithms. These two topics include solving PDEs with high dimensional domains and data-driven dynamical modeling. </p> <p><br></p> <p>In the first topic, we aim to develop an efficient solver for PDE problems. Firstly, we focus on neural network structures to increase efficiency. We propose a data-driven activation function called reproducing activation function which can reproduce traditional approximation tools and enable faster convergence of deep neural network training with smaller parameter cost. Secondly, we target the application of neural networks to mitigate the numerical issues that hamper the traditional approach. As an example, we develop a neural network solver for elliptic PDEs on unknown manifolds and verify its effectiveness for the large-scale problem. </p> <p><br></p> <p>In the second topic, we aim to enhance the accuracy of learning the dynamical system from data by incorporating the prior. In the missing dynamics problem, taking advantage of known partial dynamics, we propose a framework that approximates a map that takes the memories of the resolved and identifiable unresolved variables to the missing components in the resolved dynamics. With this framework, we achieve a low error to predict the missing component, enabling the accurate prediction of the resolved variables. In the recovering Hamiltonian dynamics, by the energy conservation property, we learn the conserved Hamiltonian function instead of its associated vector field. To better learn the Hamiltonian from the stiff dynamics, we identify and splits the</p> <p>training data into stiff and nonstiff portions, and adopt different learning strategies based on the classification to reduce the training error. </p>

>Magister Scientiae - MSc / We present the Black-Scholes Merton partial differential equation (BSMPDE) and its analytical solution. We present the Black-Scholes option pricing model and list some limitations of this model. We also present a nonlinear model (the Frey-Patie model) that may improve on one of these limitations. We apply various numerical methods on the BSMPDE and run simulations to compare which method performs best in approximating the value of a European put option based on the maximum errors each method produces when we vary some parameters like the interest rate and the volatility. We re-apply the same finite difference methods on the nonlinear model. / 2025