An effective medium approximation and a refined Monte Carlo simulation procedure for solving the stochastic groundwater flow problem are presented. The effective medium approximation permits one to solve the stochastic groundwater flow problem in a single run to generate the expected pressure head field. The proposed effective hydraulic conductivity expression for the effective medium is of the same form as the local Gardner’s equation and is easy to use. The refined Monte Carlo simulation procedure uses analytical means to estimate the sample size by controlling the error incurred in using the sample average in place of its population mean at a chosen confidence level. This estimator consistently performs well. Also, a variance reducing estimator which is different from the simple average for pressure head is developed. This estimator takes advantage of the correlation between the saturated conductivity and the pressure head distribution to reduce the output variance and is unbiased. This reduced variance results in a smaller width of uncertainty about the predicted pressure head. Both the effective medium approximation and the Monte Carlo approaches perform well when applied to several problems. / Ph. D.

De nombreuses sources de variabilité impactent la fabrication des circuits intégrés analogiques et RF et peuvent conduire à une dégradation du rendement. Il est donc nécessaire de mesurer leur influence le plus tôt possible dans le processus de fabrications. Les méthodes de simulation statistiques permettent ainsi d'estimer le rendement paramétrique des circuits durant la phase de conception. Cependant, les méthodes traditionnelles telles que la méthode de Monte Carlo ne sont pas assez précises lorsqu'un faible nombre de circuits est simulé. Par conséquent, il est nécessaire de créer un estimateur précis du rendement paramétrique basé sur un faible nombre de simulations. Dans cette thèse, les méthodes statistiques existantes provenant à la fois de publications en électroniques et non-Électroniques sont d'abord décrites et leurs limites sont mises en avant. Ensuite, trois nouveaux estimateurs de rendement sont proposés: une méthode de type quasi-Monte Carlo avec tri automatique des dimensions, une méthode des variables de contrôle basée sur l'estimation par noyau, et une méthode par tirage d'importance. Les trois méthodes reposent sur un modèle mathématique de la métrique de performance du circuit qui est construit à partir d'un développement de Taylor à l'ordre un. Les résultats théoriques et expérimentaux obtenus démontrent la supériorité des méthodes proposées par rapport aux méthodes existantes, à la fois en terme de précision de l'estimateur et en terme de réduction du nombre de simulations de circuits. / Semiconductor device fabrication is a complex process which is subject to various sources of variability. These variations can impact the functionality and performance of analog integrated circuits, which leads to yield loss, potential chip modifications, delayed time to market and reduced profit. Statistical circuit simulation methods enable to estimate the parametric yield of the circuit early in the design stage so that corrections can be done before manufacturing. However, traditional methods such as Monte Carlo method and corner simulation have limitations. Therefore an accurate analog yield estimate based on a small number of circuit simulations is needed. In this thesis, existing statistical methods from electronics and non-Electronics publications are first described. However, these methods suffer from sever drawbacks such as the need of initial time-Consuming circuit simulations, or a poor scaling with the number of random variables. Second, three novel statistical methods are proposed to accurately estimate the parametric yield of analog/RF integrated circuits based on a moderate number of circuit simulations: An automatically sorted quasi-Monte Carlo method, a kernel-Based control variates method and an importance sampling method. The three methods rely on a mathematical model of the circuit performance metric which is constructed based on a truncated first-Order Taylor expansion. This modeling technique is selected as it requires a minimal number of SPICE-Like circuit simulations. Both theoretical and simulation results show that the proposed methods lead to significant speedup or improvement in accuracy compared to other existing methods.

Random fields are frequently used in computational simulations of real-life processes. In particular, in this work they are used in modeling of flow and transport in porous media. Porous media as they arise in geological formations are intrinsically deterministic but there is significant uncertainty involved in determination of their properties such as permeability, porosity and diffusivity. In many situations description of properties of the porous media is aided by a limited number of observations at fixed points. These observations constrain the randomness of the field and lead to conditional simulations. In this work we propose a method of simulating the random fields which respect the observed data. An advantage of our method is that in the case that additional data becomes available it can be easily incorporated into subsequent representations. The proposed method is based on infinite series representations of random fields. We provide truncation error estimates which bound the discrepancy between the truncated series and the random field. We additionally provide the expansions for some processes that have not yet appeared in the literature. There are several approaches to efficient numerical computations for partial differential equations with random parameters. In this work we compare the solutions of flow and transport equations obtained by conditional simulations with Monte Carlo (MC) and stochastic collocation (SC) methods. Due to its simplicity MC method is one of the most popular methods used for the solution of stochastic equations. However, it is computationally expensive. The SC method is functionally similar to the MC method but it provides the faster convergence of the statistical moments of the solutions through the use of the carefully chosen collocation points at which the flow and transport equations are solved. We show that for both methods the conditioning on measurements helps to reduce the uncertainty of the solutions of the flow and transport equations. This especially holds in the neighborhood of the conditioning points. Conditioning reduces the variances of solutions helping to quantify the uncertainty in the output of the flow and transport equations. / Graduation date: 2013

PhD. (Department of Chemical Engineering, Faculty of Engineering and Technology), Vaal University of Technology. / The use of solar photocatalysis for the treatment of aromatic chemicals like phenol in wastewater has attracted significant attention due to the low cost of sunlight. However, sunlight is unreliable since its intensity fluctuates during the day. This drawback can be addressed by supplementing sunlight with artificial UV lamps when the solar intensity reduces. In this work, such a hybrid solar/UV lamp reactor, internally illuminated by the UV lamp and externally by sunlight, was designed. Phenol was used as the model pollutant and the nanophase Aeroxide P25 TiO2 was employed as the photocatalyst and fluidized by compressed air. The catalyst and bubble distribution in the reactor was analysed using computational fluid dynamics (CFD) while the Monte Carlo (MC) method was used to model the light distribution and reaction kinetics. Finally, a lamp controller was designed to specify the required UV lamp output as a function of the solar intensity. The CFD simulation using ANSYS CFX 17 showed that a fairly homogeneous distribution of the catalyst was achieved in the reactor. Consequently, accurate simulations of the light distribution could be achieved without considering the hydrodynamics. The MC models revealed that bubbles did not significantly influence light absorption at the optimum catalyst loading. This showed that air was a good medium for fluidization as it could provide good mixing and oxygen electron acceptor without negatively affecting light absorption. The forward scattering behaviour of the P25 TiO2 and the increase in light attenuation with catalyst loading was confirmed in this work. The optimum catalyst loading in the different reactor configurations was 0.15 g/L (tubular solar), 0.2 g/L (annular solar), 0.4 g/L (annular UV lamp), and 0.4 g/L (hybrid light). This resulted in experimental reaction rates of 0.337 mgL-1min-1 (tubular solar), 0.584 mgL-1min-1 (annular UV lamp), and 0.93 mgL-1min-1 (hybrid light). An analysis of the local volumetric rate of energy absorption (LVREA) and reaction rate profiles along the radial coordinate showed a non-uniformity which worsened with an increase in catalyst loading. The reaction order with respect to the volumetric rate of energy absorption (VREA) indicated that solar illumination resulted in a higher electron-hole recombination as compared to UV illumination. This, combined with the higher intensity of the UV lamp, resulted in a higher reaction rate under UV light as compared to sunlight, demonstrating that the UV lamp could be used to supplement sunlight. For a typical sunny day, a lamp controller was designed that could adjust the UV lamp output as a function of the solar intensity to maintain the reaction rate at a reference level while ensuring less energy consumption than an ON/OFF lamp controller. This work demonstrated the feasibility of hybrid solar/UV lamp photocatalysis reactor which could maintain the advantages of solar photocatalysis while mitigating its drawbacks.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Grain regression analysis forms an essential part of solid rocket motor simulation. In this thesis a numerical grain regression analysis module is developed as an alternative to cumbersome and time consuming analytical methods. The surface regression is performed by the level-set method, a numerical interface advancement scheme. A novel approach to the integration of the surface area and volume of a numerical interface, as defined implicitly in a level-set framework, by means of Monte-Carlo integration is proposed. The grain regression module is directly coupled to a quasi -1D internal ballistics solver in an on-line fashion, in order to take into account the effects of spatially varying burn rate distributions. A multi-timescale approach is proposed for the direct coupling of the two solvers. / AFRIKAANSE OPSOMMING: Gryn regressie analise vorm ’n integrale deel van soliede vuurpylmotor simulasie. In hierdie tesis word ’n numeriese gryn regressie analise model, as ’n alternatief tot dikwels omslagtige en tydrowende analitiese metodes, ontwikkel. Die oppervlak regressie word deur die vlak-set metode, ’n numeriese koppelvlak beweging skema uitgevoer. ’n Nuwe benadering tot die integrasie van die buite-oppervlakte en volume van ’n implisiete numeriese koppelvlak in ’n vlakset raamwerk, deur middel van Monte Carlo-integrasie word voorgestel. Die gryn regressie model word direk en aanlyn aan ’n kwasi-1D interne ballistiek model gekoppel, ten einde die uitwerking van ruimtelik-wisselende brand-koers in ag te neem. ’n Multi-tydskaal benadering word voorgestel vir die direkte koppeling van die twee modelle.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: This study’s focus is on the determination and quantization of radiation damage on a cellular level due to the decay of the Auger electron-emitting 123I and the replication of this energy deposition using Geant4 Monte Carlo simulations. The relatively short half-life of 123I (13.2 hours) makes it ideal for studies of Auger electrons which induce biological damage similar to that of high linear energy transfer radiations, when permitted to deposit their energy in close proximity to DNA. Due to small cellular dimensions, direct dose measurements are impossible but estimates may be made from Monte Carlo simulations. In this investigation the thymidine analogue 5-[123I]-iodo-2-deoxyuridine (123IUdR) was used to incorporate the 123I into the cellular DNA of T-lymphocytes from two human donors. Radiation induced micronuclei were numerated in binucleated cells using fluorescence microscopy. The energy deposition per decay of 123I was calculated within a spherical geometry, having the same size and density as a human lymphocyte, using the open source Geant4 toolkit. The absorbed energy per disintegration was used to convert the incorporated 123I activity (Bq) into absorbed dose (Gy) values, in order to compare the biological damage caused by the radioactive iodine to 60Co γ-radiation. A linear relationship between micronuclei frequency and 123I activity could be established. The linear dose-response noted for Auger electrons in the study is indicative of the high-LET nature of these particles. Using the linear-quadratic dose-response curve for micronuclei frequencies following exposure to graded doses of 60Co γ-rays, the relative biological effectiveness (RBE) of the DNA incorporated 123I estimated in this work was found to range from 19 ± 10 to 32 ± 7 for lymphocyte donor 1 and 15 ± 6 to 42 ± 11 for donor 2. The dose limiting RBE (RBEM) for lymphocyte donor 1 and 2 are respectively 34 ± 8 and 50 ± 15 and follows the expected shift in terms of the inherent radiosensitivity of the donors. We also considered the inclusion of the S-phase fraction of the lymphocytes in the dosimetry calculations. The resultant RBEs of the dose points of lymphocyte donor 1 ranges from 4 ± 2 to 7 ± 2, and those of donor 2 ranges from 3 ± 1 to 9 ± 2. The RBEM for lymphocyte donor 1 and 2 are respectively 7 ± 2 and 11 ± 3. The inclusion of the S-phase fraction reduces the calculated RBEs significantly and these observed RBE values relate well to those obtained in studies with fibroblasts and 125IUdR. / AFRIKAANSE OPSOMMING: Hierdie studie fokus op die bepaling en kwantisering van stralingskade op 'n sellulêre vlak as gevolg van die verval van 123I wat Auger elektrone afgee, asook die simulering van hierdie energie afsetting met behulp van die Geant4 Monte Carlo program. Die relatiewe kort half-leeftyd van 123I (13.2 uur) maak dit ideaal vir studies van Auger elektrone wat biologiese skade soortgelyk aan dié van 'n hoë lineêre-energie-oordrag uitstraling veroorsaak, indien die energie van die elektrone naby sellulêre DNA geabsorbeer word. As gevolg van die klein sellulêre dimensies is direkte dosis metings egter onmoontlik, maar skattings kan gemaak word met behulp van Monte Carlo simulasies. Die timidien analoog 5-[123I]-jodo-2-deoxyuridien (123IUdR) was in hierdie ondersoek gebruik om die 123I in die DNA van menslike T-limfosiete in te bou. Mikrokerne in dubbel-kernige selle wat vorm as gevolg van die Auger elektrone was getel met behulp van fluoressensie mikroskopie. Die energie afsetting per 123I verval was bereken binne ‘n sferiese geometrie, met dieselfde grootte en digtheid as 'n menslike limfosiet, met behulp van die Geant4 sagteware. Die geabsorbeerde energie per verval was gebruik om die geïnkorporeerde 123I aktiwiteit (Bq) om te skakel na ‘n waarde van geabsorbeerde dosis (Gy), ten einde die biologiese skade wat veroorsaak word deur die radioaktiewe jodium-123 met kobalt-60 gamma straling te vergelyk. ‘n Lineêre verwantskap tussen die mikrokerne frekwensies en die 123I aktiwiteit is vasgestel. Hierdie verwantskap vir Auger elektrone is 'n aanduiding van die hoë lineêre-energie-oordrag van hierdie deeltjies. Die lineêr-kwadratiese dosis-effek krommes vir mikrokerne frekwensies na blootstelling aan 60Co γ-strale was gebruik om die relatiewe biologiese doeltreffendheid (RBE) van die DNA geïnkorporeerde 123I te beraam. RBE waardes wissel van 19 ± 10 tot 32 ± 7 vir limfosiete van skenker 1 en 15 ± 6 tot 42 ± 11 vir skenker 2. Die dosis beperkte RBE (RBEM) vir limfosiet skenker 1 en 2 is onderskeidelik 34 ± 8 en 50 ± 15 en volg die verwagte skuif in terme van die inherente radiogevoeligheid van die skenkers. Die fraksie van limfosiete wat in S-fase was tydens die blootstelling aan 125IUdR was ingesluit in verdere dosimetrie berekeninge. Die gevolglike RBEs van die dosispunte van limfosiete van skenker 1 wissel van 4 ± 2 tot 7 ± 2 en dié van skenker 2 wissel van 3 ± 1 tot 9 ± 2. Die RBEM vir limfosiet skenker 1 en 2 is onderskeidelik 7 ± 2 en 11 ± 3. Die insluiting van die S-fase fraksie verminder die berekende RBEs aansienlik en die RBE waardes waargeneem hou goed verband met die wat in studies met fibroblaste en 125IUdR verkry is.

Thesis (MEng)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Water Quality Models are mathematical representations of ecological systems and they play a major role in the planning and management of water resources and aquatic environments. Important decisions concerning capital investment and environmental consequences often rely on the results of Water Quality Models and it is therefore very important that decision makers are aware and understand the uncertainty associated with these models. The focus of this study was on the use of Monte Carlo Simulation for the quantification of prediction uncertainty associated with Water Quality Models. Two types of uncertainty exist: Epistemic Uncertainty and Aleatory Uncertainty. Epistemic uncertainty is a result of a lack of knowledge and aleatory uncertainty is due to the natural variability of an environmental system. It is very important to distinguish between these two types of uncertainty because the analysis of a model’s uncertainty depends on it. Three different configurations of Monte Carlo Simulation in the analysis of uncertainty were discussed and illustrated: Single Phase Monte Carlo Simulation (SPMCS), Two Phase Monte Carlo Simulation (TPMCS) and Parameter Monte Carlo Simulation (PMCS). Each configuration of Monte Carlo Simulation has its own objective in the analysis of a model’s uncertainty and depends on the distinction between the types of uncertainty. As an experiment, a hypothetical river was modelled using the Streeter-Phelps model and synthetic data was generated for the system. The generation of the synthetic data allowed for the experiment to be performed under controlled conditions. The modelling protocol followed in the experiment included two uncertainty analyses. All three types of Monte Carlo Simulations were used in these uncertainty analyses to quantify the model’s prediction uncertainty in fulfilment of their different objectives. The first uncertainty analysis, known as the preliminary uncertainty analysis, was performed to take stock of the model’s situation concerning uncertainty before any effort was made to reduce the model’s prediction uncertainty. The idea behind the preliminary uncertainty analysis was that it would help in further modelling decisions with regards to calibration and parameter estimation experiments. Parameter uncertainty was reduced by the calibration of the model. Once parameter uncertainty was reduced, the second uncertainty analysis, known as the confirmatory uncertainty analysis, was performed to confirm that the uncertainty associated with the model was indeed reduced. The two uncertainty analyses were conducted in exactly the same way. In conclusion to the experiment, it was illustrated how the quantification of the model’s prediction uncertainty aided in the calculation of a Total Maximum Daily Load (TMDL). The Margin of Safety (MOS) included in the TMDL could be determined based on scientific information provided by the uncertainty analysis. The total MOS assigned to the TMDL was -35% of the mean load allocation for the point source. For the sake of simplicity load allocations from non-point sources were disregarded. / AFRIKAANSE OPSOMMING: Watergehalte modelle is wiskundige voorstellings van ekologiese sisteme en speel ’n belangrike rol in die beplanning en bestuur van waterhulpbronne en wateromgewings. Belangrike besluite rakende finansiële beleggings en besluite rakende die omgewing maak dikwels staat op die resultate van watergehalte modelle. Dit is dus baie belangrik dat besluitnemers bewus is van die onsekerhede verbonde met die modelle en dit verstaan. Die fokus van hierdie studie het berus op die gebruik van die Monte Carlo Simulasie om die voorspellingsonsekerhede van watergehalte modelle te kwantifiseer. Twee tipes onsekerhede bestaan: Epistemologiese onsekerheid en toeval afhangende onsekerheid. Epistemologiese onsekerheid is die oorsaak van ‘n gebrek aan kennis terwyl toeval afhangende onsekerheid die natuurlike wisselvalligheid in ’n natuurlike omgewing behels. Dit is belangrik om te onderskei tussen hierdie twee tipes onsekerhede aangesien die analise van ’n model se onsekerheid hiervan afhang. Drie verskillende rangskikkings van Monte Carlo Simulasies in die analise van die onsekerhede word bespreek en geïllustreer: Enkel Fase Monte Carlo Simulasie (SPMCS), Dubbel Fase Monte Carlo Simulasie (TPMCS) en Parameter Monte Carlo Simulasie (PMCS). Elke rangskikking van Monte Carlo Simulasie het sy eie doelwit in die analise van ’n model se onsekerheid en hang af van die onderskeiding tussen die twee tipes onsekerhede. As eksperiment is ’n hipotetiese rivier gemodelleer deur gebruik te maak van die Streeter-Phelps teorie en sintetiese data is vir die rivier gegenereer. Die sintetiese data het gesorg dat die eksperiment onder beheerde toestande kon plaasvind. Die protokol in die eksperiment het twee onsekerheids analises ingesluit. Al drie die rangskikkings van die Monte Carlo Simulasie is gebruik in hierdie analises om die voorspellingsonsekerheid van die model te kwantifiseer en hul doelwitte te bereik. Die eerste analise, die voorlopige onsekerheidsanalise, is uitgevoer om die model se situasie met betrekking tot die onsekerheid op te som voor enige stappe geneem is om die model se voorspellings onsekerheid te probeer verminder. Die idee agter die voorlopige onsekerheidsanalise was dat dit sou help in verdere modelleringsbesluite ten opsigte van kalibrasie en die skatting van parameters. Onsekerhede binne die parameters is verminder deur die model te kalibreer, waarna die tweede onsekerheidsanalise uitgevoer is. Hierdie analise word die bevestigingsonsekerheidsanalise genoem en word uitgevoer met die doel om vas te stel of die onsekerheid geassosieer met die model wel verminder is. Die twee tipes analises word op presies dieselfde manier toegepas. In die afloop tot die eksperiment, is gewys hoe die resultate van ’n onsekerheidsanalise gebruik is in die berekening van ’n totale maksimum daaglikse belading (TMDL) vir die rivier. Die veiligheidgrens (MOS) ingesluit in die TMDL kon vasgestel word deur die gebruik van wetenskaplike kennis wat voorsien is deur die onsekerheidsanalise. Die MOS het bestaan uit -35% van die gemiddelde toegekende lading vir puntbelasting van besoedeling in die rivier. Om die eksperiment eenvoudig te hou is verspreide laste van besoedeling nie gemodelleer nie.

A risk-based approach for addressing several non-structural stream quality management objectives is presented. To estimate risk, the input process, the stream contaminant transport, and the consequence of contamination are modeled mathematically. The transport of soluble contaminant introduced at a point into a turbulent stream medium is modeled as a boundary value problem in which the contaminant satisfies the Kolmogorov forward equation within the medium. Observed properties of turbulence are used to justify the adoption of this equation. The fundamental solution, as the probabilistic response of the stream to an instantaneous unit flux input, is derived and used as the kernel in a stochastic integral representation of the transport problem. The bulk input is used as the forcing function in the integral equation. It is modeled as a sequence of independent pulses with random magnitude and duration and also with random interval between the incidence of adjacent pulses ,. Stochastic simulation is used to construct the moments and the probability distribution of stream concentration and those of several variables associated with the exceedance of the concentration above a specified threshold. The variables include the dosage and the time to the first exceedance. The probability that an observed stream concentration exceeds the threshold within a given interval of time is also constructed. Generalizations of the Chebyshev inequality are extended to the case of a stochastic process. Upper bounds on the constructed probability distributions are calculated using these extensions. Based on previous studies, a rectangular hyperbolic relationship is assumed between dosage and consequence. The relationship is combined with the empirical dosage density function to obtain estimates of value risk of stream concentration for various thresholds. Given an acceptable risk, the corresponding threshold may be used as the stream standard. The reliability function, defined as the complementary density function of exceedance times, may be used as a gauge of the effectiveness of pollution abatement measures. Other illustrated areas of application include the construction of a minimum cost contaminant discharge policy and the determination of the optimal sampling interval for stream surveillance.

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No description available.