Populationen kanzeröser Zellen können aus verschiedenen Subpopulationen mit distinkten phänotypischen Profilen bestehen. Diese Dissertation verwendet mathematische Modellierung sowie die Analyse von Einzelzell-Genexpressionsdaten zur Beantwortung von Fragen über die Entstehung, das Wachstum und die Behandlung von Tumoren im Kontext einer solchen intratumoralen Heterogenität. / Tumoural cell populations may consist of different subpopulations with distinct phenotypic profiles. This thesis applies mathematical modelling as well as the analysis of single-cell gene expression data to questions related to the emergence, growth and treatment of tumours in the context of such an intratumoural heterogeneity.

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<p>In this work, we develop several special computational structures of Neural Networks based on some existing approaches such as Auto-Encoder and DeepONet. Combined with classic numerical methods in scientific computing, finite difference and SAV method, our model is able to solve the operator learning tasks of partial differential equations accurately in both data-driven and non-data-driven settings. The high dimensional problem requires a large number of samples for training in the normal settings of Neural network training. The proposed</p> <p>model equipped with auto-encoder performs the dimension reduction for the input operator, which discovers the intrinsic hidden features, to reduce the number of samples needed for training. In addition, the non-linear basis of the hidden variables are constructed</p> <p>for both the operator variable and the solution of the equation, leading to a concise representation of the solution. For non data-driven setting, our method derives the solution of the equation with only initial and boundary condition, where the normal network can not manage to do it, with the assistance of SAV method. Besides, it preserves the advantages of DeepONet. It performs the operator learning with various initial conditions or parametric equations. The modified energy is defined to estimate the true energy of the system and it has the monotonic decreasing property. It also serves as an indicator of the suitable time step, allowing the model to adjust the time step. Finally, the optimization is a key procedure of network training. We propose a new optimization method based on SAV. It allows a much</p> <p>larger learning rate compared to SGD and ADAM, which are most popular methods used nowadays. Moreover, It also allows the adaptive learning rate to pursue the faster speed converging to the critical point.</p>

Stochastic effects in cellular systems are usually modeled and simulated with Gillespie's stochastic simulation algorithm (SSA), which follows the same theoretical derivation as the chemical master equation (CME), but the low efficiency of SSA limits its application to large chemical networks. To improve efficiency of stochastic simulations, Haseltine and Rawlings proposed a hybrid of ODE and SSA algorithm, which combines ordinary differential equations (ODEs) for traditional deterministic models and SSA for stochastic models. In this dissertation, accuracy analysis, efficient implementation strategies, and application of of Haseltine and Rawlings's hybrid method (HR) to a budding yeast cell cycle model are discussed. Accuracy of the hybrid method HR is studied based on a linear chain reaction system, motivated from the modeling practice used for the budding yeast cell cycle control mechanism. Mathematical analysis and numerical results both show that the hybrid method HR is accurate if either numbers of molecules of reactants in fast reactions are above certain thresholds, or rate constants of fast reactions are much larger than rate constants of slow reactions. Our analysis also shows that the hybrid method HR allows for a much greater region in system parameter space than those for the slow scale SSA (ssSSA) and the stochastic quasi steady state assumption (SQSSA) method. Implementation of the hybrid method HR requires a stiff ODE solver for numerical integration and an efficient event-handling strategy for slow reaction firings. In this dissertation, an event-handling strategy is developed based on inverse interpolation. Performances of five wildly used stiff ODE solvers are measured in three numerical experiments. Furthermore, inspired by the strategy of the hybrid method HR, a hybrid of ODE and SSA stochastic models for the budding yeast cell cycle is developed, based on a deterministic model in the literature. Simulation results of this hybrid model match very well with biological experimental data, and this model is the first to do so with these recently available experimental data. This study demonstrates that the hybrid method HR has great potential for stochastic modeling and simulation of large biochemical networks. / Ph. D.

This work is a numerical study of the 2-D heat equation with Dirichlet boundary conditions over a polygonal domain. The motivation for this study is a chemical vapor deposition (CVD) reactor in which a substrate is heated while being exposed to a gas containing precursor molecules. The interaction between the gas and the substrate results in the deposition of a compound thin film on the substrate. Two different numerical approximations are implemented to produce numerical solutions describing the conduction of thermal energy in the reactor. The first method used is a Crank-Nicholson finite difference technique which tranforms the 2-D heat equation into an algebraic system of equations. For the second method, a semi-discrete method is used which transforms the partial differential equation into a system of ordinary differential equations. The goal of this work is to investigate the influence of boundary conditions, domain geometry, and initial condition on thermal conduction throughout the reactor. Once insight is gained with respect to the aforementioned conditions, optimal design and control can be investigated. This work represents a first step in our long term goal of developing optimal design and control of such CVD systems. This work has been funded through Partnerships in Research Excellence and Transition (PRET) grant number F49620-96-1-0329. / Master of Science

This research paper focuses on analog computers and creating a modular low-cost analog computer system in a single board computer form factor. The single-board analog computer will have the capacity to solve second-order differential equations. The capabilities and possibilities of the single board Analog computer will be explored as well as analog computing in general. The paper follows design science research methodology (DSRM) with the goal of creating and evaluating a working artifact. The artifacts' functionality is evaluated based on a demonstration of its ability to solve Mathieu’s differential equation as well as simulate a spring-mass dampening system. This paper proves that it is possible to create a low-cost analog computer in a modern form factor. The artifact is also placed in a larger contextual setting based on the empirical material provided where its value of it in a digital society is presented. For the world to continue its progression in computational power, but still, limit the already high energy usage, a drastic change is needed. This paper suggests adapting to analog/hybrid technology. To further the progression of analog/hybrid technology it must be made accessible to a wider group of people compared to today. The artifact in this paper offers a solution to this.

Les fuites non détectées dans les réseaux d'eau potable sont responsables en moyenne de la perte de 30% de l'eau transportée. Il s'avère donc primordial de pouvoir contrôler ces fuites. Pour atteindre cet objectif, la modélisation de l'écoulement de l'eau dans les conduites en tenant compte des fuites a été formulée de différente manière. La première formulation est un système d'équations différentielles ordinaires représentant des fuites constantes, réparties uniformément le long des conduites. Le système peut s'avérer être numériquement raide lorsque des organes hydrauliques sont rajoutés. Deux méthodes implicites ont été proposées pour sa résolution : la méthode de Rosenbrock et la méthode de Gear. Les résultats obtenus montrent que le débit varie linéairement le long des conduites et que les pertes en eau par unité de longueur sont identiques sur chaque conduite. La seconde formulation prend en compte la relation entre les fuites et la pression. Un système de deux équations aux dérivées partielles a été proposé. L'EDP de transport-diffusion-réaction, contenant l'opérateur du p-Laplacien, est résolue par une méthode à pas fractionnaires. Deux méthodes ont été testées. Dans la première, la réaction est couplée avec la diffusion et dans la seconde, elle est couplée avec le transport. Les résultats indiquent que les pertes en eau ne sont pas réparties de façon homogène sur le réseau. Cette formulation décrit de manière plus réaliste les réseaux d'eau potable. Enfin, le problème du contrôle du volume des fuites par action sur la pression a été étudié. Pour cela, un problème d'optimisation est résolu sous la contrainte que la pression doit être minimale pour réduire les fuites et être suffisante pour garantir un bon service aux consommateurs. Les résultats trouvés confirment que la réduction de la pression permet de réduire le volume des fuites de façon significative et que le choix de l'emplacement du ou des points de contrôle est primordial pour optimiser cette réduction. / Leakage represents a large part, in average more than 30%, of the water supplied. Consequently, it is important to control leakage in Water Distribution System (WDS). For this purpose different methods, which take leakage into account, are proposed to model the hydraulics of WDS. The first formulation considers constant leakage in a network and leads to an ordinary differential equation. It turns out to be a hydraulic stiff problem due to valve and pump operations. This equation is solved using two methods: the first one is a generalised Runge-Kutta method and the second one the Gear method. The results show that the flow rate varies linearly along a pipe and that the water loss per unit of length is identical for each pipe. Magnitude of inertia terms has also been studied. The second formulation takes pressure-dependent leakage into account. We propose to introduce partial differential equations in order to predict more accurately hydraulic flows in WDS. Thus, the physical advection-diffusion-reaction model is presented. A nonlinear operator, called p-Laplacian, related to the diffusion is included into the model. Two resolutions of this model based on a splitting method are detailed. The results confirm that losses vary nonlinearly with pressure. Finally, the leakage-control problem is studied. For this purpose, we solve an optimisation problem with the objective to minimize the distributed volume in order to reduce leakage. The condition of sufficient pressure to satisfy consumers is imposed in this optimisation. The results prove that pressure control significantly reduces leakage and that the emplacement of the valve is important to optimise this reduction.

在人口統計領域中，早期習慣將人口變化視為時間的函數，企圖以Deterministic Function來刻劃，例如：1798年Malthus提出的Malthusian Growth Model ；1825年Gompertz提出的Gompertz Model以及1838年Verhulst主張以Logistic Function描述人口成長。而近年來則是傾向於逐項分析各種因素的隨機性模型，例如：1983年Holford加入世代的APC模型；1992年Lee 和Carter提出的Lee-Carter死亡率模型以及2003年Renshaw與Haberman提出改善Lee-Carter死亡率模型的Reduction Factor模型。 人口變化主要分成自然增加與社會增加，而自然增加是為出生扣掉死亡，社會增加則為移入扣掉移出。首先，本文先不考慮遷移的部分，各別以出生與死亡人口的變化為研究對象，視其變化為一隨時間變動的動態系統，以常微分方程來刻劃。由台灣地區人口統計資料顯示，出生率或死亡率都有逐年下降的趨勢，而且隨著時間而變化加劇的傾向，使得以往使用的模型不易捕捉變化，因此我們提出「二次逼近法」，從出生、死亡人數對時間的變化率與曲度利用數值分析的方式來估計出生與死亡數，進而從中找出在此動態系統背後隱藏的規則。而後再同時考慮其他各種變項，以偏微分方程來刻劃，最後即可建立台灣地區人口變化模型。 / In early population statistics, the population changes were regarded as a function of time so that people tended to describe the variations by deterministic functions. For instance, Malthus proposed the Malthusian Growth Model in 1798; Gompertz presented Gompertz Model in 1825; Verhulst advocated using logistic function to describe an increase in population. In recent years, people tend to use the stochastic forecast method to analyse every factor term by term. For instance, the Age-Period-Cohort (APC) Model which was proposed by Holford in 1983; Lee and Carter proposed the Lee-Carter Mortality Model in 2003; and Renshaw and Haberman proposed the Reduction Factor Model in 2003 that improve the Lee-Carter Mortality Model. The population changes equal to nature and social increase, where the nature increase is the difference between birth and death population, and the social increase is the difference between immigrants and emigrants. First, we focus on natural increase rather than social increase. Moreover, we use ordinary differential equation to decribe the variation as a dynamic system over time. From the data obtained from the Ministry of Interior Taiwan, we know that the fertility and mortality has been decreasing, and the change is getting more violent year by year. Under the consideration that previous models are not able to accurately present the changes of birth and death, we proposed "second-order (or parabola) approximation method." From the variation rates and curvatures of birth and death population, we estimated the population size. Furthermore, we want to find the rule in the dynamic system. Later we will consider other factors simultaneously, and describe them by partial differential equation. Finally, the population model is constructed.