Dissertations / Theses on the topic 'Numerical and computational mathematics'
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Baer, Lawrence H. "Numerical aspects of computational geometry." Thesis, McGill University, 1992. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=22507.
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This thesis is concerned with the numerical issues resulting from the implementation of geometric algorithms on finite precision digital computers. From an examination of the general problem and a survey of previous research, it appears that the central problem of numerical computational geometry is how to deal with degenerate and nearly degenerate input. For some applications, such as solid modeling, degeneracy is often intended but we cannot always ascertain its existence using finite precision. For other applications, degenerate input is unwanted but nearly degenerate input is unavoidable. Near degeneracy is associated with ill-conditioning of the input and can lead to a serious loss of accuracy and program failure. These observations lead us to a discussion of problem condition in the context of computational geometry. We use the Voronoi diagram construction problem as a case study and show that problem condition can also play a role in algorithm design.
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Djambazov, Georgi Stefanov. "Numerical techniques for computational aeroacoustics." Thesis, University of Greenwich, 1998. http://gala.gre.ac.uk/6149/.
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The problem of aerodynamic noise is considered following the Computational Aeroacoustics approach which is based on direct numerical simulation of the sound field. In the region of sound generation, the unsteady airflow is computed separately from the sound using Computational Fluid Dynamics (CFD) codes. Overlapping this region and extending further away is the acoustic domain where the linearised Euler equations governing the sound propagation in moving medium are solved numerically. After considering a finite volume technique of improved accuracy, preference is given to an optimised higher order finite difference scheme which is validated against analytical solutions of the governing equations. A coupling technique of two different CFD codes with the acoustic solver is demonstrated to capture the mechanism of sound generation by vortices hitting solid objects in the flow. Sub-grid turbulence and its effect 011sound generation has not been considered in this thesis. The contribution made to the knowledge of Computational Aeroacoustics can be summarised in the following: 1) Extending the order of accuracy of the staggered leap-frog method for the linearised Euler equations in both finite volume and finite difference formulations; 2) Heuristically determined optimal coefficients for the staggered dispersion relation preserving scheme; 3) A solution procedure for the linearised Euler equations involving mirroring at solid boundaries which combines the flexibility of the finite volume method with the higher accuracy of the finite difference schemes; 4) A method for identifying the sound sources in the CFD solution at solid walls and an expansion technique for sound sources inside the flow; 5) Better understanding of the three-level structure of the motions in air: mean flow, flow perturbations, and acoustic waves. It can be used, together with detailed simulation results, in the search for ways of reducing the aerodynamic noise generated by propellers, jets, wind turbines, tunnel exits, and wind-streamed buildings.
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Kuster, Christopher M. "Fast Numerical Methods for Evolving Interfaces." NCSU, 2006. http://www.lib.ncsu.edu/theses/available/etd-04262006-083221/.
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Free and/or moving boundary problems occur in a wide range of applications. These boundaries can obey either local or global conditions. In this dissertation, new numerical techniques for solving some of these problems are developed, analyzed, implemented and tested. The new techniques for free and moving boundary problems are 1) a second order method for solving moving boundary problems and 2) a hybrid level set/boundary element method for solving some free boundary problems. The main tool used in both is the Fast Marching method, a fast algorithm for solving the eikonal equation. An application using Fast Marching to solve a model for sand pile formation in domains with obstacles is shown. A new, second order Fast Marching scheme for domains with obstacles is introduced. We look at the stability and accuracy of discretizations commonly used with Fast Marching. The performance of Fast Marching is compared that of Fast Sweeping, another eikonal solver. The second order method for solving moving boundary problems is applied to some simple examples. Finally, a globally defined free boundary problem inspired by fluid dynamics, the Bernoulli problem, is solved using the hybrid method.
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Lindgren, Jonas. "Numerical modelling of district heating networks." Thesis, Umeå universitet, Institutionen för fysik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-143896.
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District heating is today, in Sweden, the most common method used for heating buildings in cities. More than half of all the buildings, both commercial and residential, are heated using district heating. The load on the district heating networks are affected by, among other things, the time of the day and different external conditions, such as temperature differences. One has to be able to simulate the heat and pressure losses in the network in order to deliver the amount of heat demanded by the customers. Expansions of district heating networks and disrupted pipes also demand good simulations of the networks. To cope with this, energy companies use simulation software. These software need to contain numerical methods that provide accurate and stable results and at the same time be fast and efficient. At the moment there are available software packages that works but these have some limitations. Among other things you may need to divide the whole network into smaller loops or try to guess how the distribution of pressure and flow in the network looks like. The development in recent years makes it possible to use better and more efficient algorithms for these types of problems. The purpose of this report is therefore to introduce a better and more efficient method than that used in the current situation. This work is the first step in order to replace a current method used in a simulation software provided by Vitec energy. Therefore, we will in this report, stick to computing pressure and flow in the network. The method we will introduce in this report is called the gradient method and it is based on the Newton Raphson method. Unlike with older methods like Hardy Cross which is a relaxation method, you do not have to divide the network into loops. Instead you create a matrix representation of the network that is used in the computations. The idea is also that you should not need to make good initial guesses to get the method to converge quickly. We performed a number of test simulations in order to examine how the method performs. We tested how different initial guesses and how different sizes of the networks affected the number of iterations. The results shows that the model is capable of solving large networks within a reasonable number of iterations. The results also show that the initial guesses have little impact on the number of iterations. Changing the initial guess on the pressure does not affect the number at all but it turns out that changing the initial guess on the flow can affect the number of iterations a little, but not much.
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Engblom, Stefan. "Numerical methods for the chemical master equation." Licentiate thesis, Uppsala : Univ. : Dept. of Information Technology, Univ, 2006. http://www.it.uu.se/research/publications/lic/2006-007/2006-007.pdf.
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Eliasson, Bengt. "Numerical simulation of kinetic effects in ionospheric plasma." Licentiate thesis, Uppsala : Dept. of Information Technology, Univ, 2001. http://www.it.uu.se/research/reports/lic/2001-004/2001-004-nc.pdf.
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Mitrouli, Marilena Th. "Numerical issues and computational problems in algebraic control theory." Thesis, City University London, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.280573.
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Kormann, Katharina. "Numerical methods for quantum molecular dynamics." Licentiate thesis, Uppsala : Department of Information Technology, Uppsala University, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-108366.
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Bastounis, Alexander James. "On fundamental computational barriers in the mathematics of information." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/279086.
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This thesis is about computational theory in the setting of the mathematics of information. The first goal is to demonstrate that many commonly considered problems in optimisation theory cannot be solved with an algorithm if the input data is only known up to an arbitrarily small error (modelling the fact that most real numbers are not expressible to infinite precision with a floating point based computational device). This includes computing the minimisers to basis pursuit, linear programming, lasso and image deblurring as well as finding an optimal neural network given training data. These results are somewhat paradoxical given the success that existing algorithms exhibit when tackling these problems with real world datasets and a substantial portion of this thesis is dedicated to explaining the apparent disparity, particularly in the context of compressed sensing. To do so requires the introduction of a variety of new concepts, including that of a breakdown epsilon, which may have broader applicability to computational problems outside of the ones central to this thesis. We conclude with a discussion on future research directions opened up by this work.
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Berglund, André. "Numerical Simulations of Linear Stochastic Oscillators : driven by Wiener and Poisson processes." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-134800.
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The main component of this essay is the numerical analysis of stochastic differential equations driven by Wiener and Poisson processes. In order to do this, we focus on two model problems, the geometric Brownian motion and the linear stochastic oscillator, studied in the literature for stochastic differential equations only driven by a Wiener process. This essay covers theoretical as well as numerical investigations of jump - or more specifically, Poisson - processes and how they influence the above model problems.Den huvudsakliga komponenten av uppsatsen är en numerisk analys av stokastiska differentialekvationer drivna av Wiener- och Poisson-processer. För att göra det så fokuserar vi på två modellproblem, den geometriska Brownska rörelsen samt den linjära stokastiska oscillatorn, studerade i litteratur för stokastiska differentialekvationer som bara drivs av en Wiener-process.Den här uppsatsen täcker teoretiska samt numeriska undersökningar av hopp - eller mer specifikt, Poisson - processer och hur de påverkar de ovan nämnda modellproblemen.
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Lindholm, Love. "Numerical methods for the calibration problem in finance and mean field game equations." Doctoral thesis, KTH, Numerisk analys, NA, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-214082.
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This thesis contains five papers and an introduction. The first four of the included papers are related to financial mathematics and the fifth paper studies a case of mean field game equations. The introduction thus provides background in financial mathematics relevant to the first four papers, and an introduction to mean field game equations related to the fifth paper. In Paper I, we use theory from optimal control to calibrate the so called local volatility process given market data on options. Optimality conditions are in this case given by the solution to a Hamiltonian system of differential equations. Regularization is added by mollifying the Hamiltonian in this system and we solve the resulting equation using a trust region Newton method. We find that our resulting algorithm for the calibration problem is both accurate and robust. In Paper II, we solve the local volatility calibration problem using a technique that is related to - but also different from - the Hamiltonian framework in Paper I. We formulate the optimization problem by means of a Lagrangian multiplier and add a Tikhonov type regularization directly on the parameter we are trying to estimate. The resulting equations are solved with the same trust region Newton method as in Paper II, and again we obtain an accurate and robust algorithm for the calibration problem. Paper III formulates the problem of calibrating a local volatility process to option prices in a way that differs entirely from what is done in the first two papers. We exploit the linearity of the Dupire equation governing the prices to write the optimization problem as a quadratic programming problem. We illustrate by a numerical example that method can indeed be used to find a local volatility that gives good match between model prices and observed market prices on options. Paper IV deals with the hedging problem in finance. We investigate if so called quadratic hedging strategies formulated for a stochastic volatility model can generate smaller hedging errors than obtained when hedging with the standard Black-Scholes framework. We thus apply the quadratic hedging technique as well as the Black-Scholes hedging to observed option prices written on an equity index and calculate the empirical errors in the two cases. Our results indicate that smaller errors can be obtained with quadratic hedging in the models used than with hedging in the Black-Scholes framework. Paper V describes a model of an electricity market consisting of households that try to minimize their electricity cost by dynamic battery usage. We assume that the price process of electricity depends on the aggregated momentaneous electricity consumption. With this assumption, the cost minimization problem of each household is governed by a system of mean field game equations. We also provide an existence and uniqueness result for these mean field game equations. The equations are regularized and the approximate equations are solved numerically. We illustrate how the battery usage affects the electricity price.Den här avhandlingen innehåller fyra artiklar och en introduktion. De första fyra av de inkluderade artiklarna är relaterade till finansmatematik och den femte artikeln studerar ett fall av medelfältsekvationer. Introduktionen ger bakgrund i finansmatematik som har relevans för de fyra första artiklarna och en introduktion till medelfältsekvationer relaterad till den femte artikeln. I Artikel I använder vi teori från optimal styrning för att kalibrera den så kallade lokala volatilitetsprocessen givet marknadsdata för optionspriser. Optimalitetsvillkor ges i det här fallet av lösningen till ett Hamiltonskt system av differentialekvationer. Vi regulariserar problemet genom att släta ut systemets Hamiltonian och vi löser den resulterande ekvationen med en trust region Newtonmetod. Den resulterande algoritmen är både noggrann och robust i att lösa kalibreringsproblemet. I Artikel II löser vi kalibreringsproblemet för lokal volatilitet med en teknik som är besläktad med - men också skiljer sig från - det Hamiltonska ramverket i Artikel I. Vi formulerar optimeringsproblemet med en Lagrangemultiplikator och använder en Tikhonovregularisering direkt på den parameter vi försöker uppskatta. De resulterande ekvationerna löses med samma trust region Newtonmetod som i Artikel II. Även i detta fall erhåller vi en noggrann och robust algoritm för kalibreringsproblemet. Artikel III formulerar problemet att kalibrera en lokal volatilitet till optionspriser på att sätt som skiljer sig helt från vad som görs i de två första artiklarna. Vi utnyttjar linjäriteten hos Dupires ekvation som ger optionspriserna och kan skriva optimieringsproblemet som ett kvadratiskt programmeringsproblem. Vi illusterar genom ett numeriskt exempel att metoden kan användas för att hitta en lokal volatilitet som ger en bra anpassning av modellpriser till observerade marknadspriser på optioner. Artikel IV behandlar hedgingproblemet i finans. Vi undersöker om så kallad kvadratiska hedgingstrategier formulerade för en stokastisk volatilitetsmodell kan generera mindre hedgingfel än vad som erhålls med hedging i den standardmässiga Black-Scholes modellen. Vi tillämpar således teorin för kvadratisk hedging så väl som hedging med Black-Scholes modell på observerade priser för optioner skrivna på ett aktieindex, och beräknar de empiriska felen i båda fallen. Våra resultat indikerar att mindre fel kan erhållas med kvadratisk hedging med de använda modellerna än med hedging genom Black-Scholes modell. Artikel V beskriver en modell av en elmarknad som består av hushåll som försöker minimera sin elkostnad genom dynamisk batterianvändning. Vi antar att prisprocessen för el beror på den aggregerade momentana elkonsumtionen. Med detta antagande kommer kostnadsminimeringen för varje hushåll att styras av ett system av medelfältsekvationer. Vi ger också ett existens- och entydighetsresultat för dessa medelfältsekvationer. Ekvationerna regulariseras och de approximerade ekvationerna löses numeriskt. Vi illustrerar hur batterianvändningen påverkar elpriset.
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Mendoza-Smith, Rodrigo. "Numerical algorithms for the mathematics of information." Thesis, University of Oxford, 2017. http://ora.ox.ac.uk/objects/uuid:451a418b-eca0-454f-8b54-7b6476056969.
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This thesis presents a series of algorithmic innovations in Combinatorial Compressed Sensing and Persistent Homology. The unifying strategy across these contributions is in translating structural patterns in the underlying data into specific algorithmic designs in order to achieve: better guarantees in computational complexity, the ability to operate on more complex data, highly efficient parallelisations, or any combination of these.
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Shaw, Jeremy A. "Computational Algorithms for Improved Representation of the Model Error Covariance in Weak-Constraint 4D-Var." PDXScholar, 2017. https://pdxscholar.library.pdx.edu/open_access_etds/3473.
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Four-dimensional variational data assimilation (4D-Var) provides an estimate to the state of a dynamical system through the minimization of a cost functional that measures the distance to a prior state (background) estimate and observations over a time window. The analysis fit to each information input component is determined by the specification of the error covariance matrices in the data assimilation system (DAS). Weak-constraint 4D-Var (w4D-Var) provides a theoretical framework to account for modeling errors in the analysis scheme. In addition to the specification of the background error covariance matrix, the w4D-Var formulation requires information on the model error statistics and specification of the model error covariance. Up to now, the increased computational cost associated with w4D-Var has prevented its practical implementation. Various simplifications to reduce the computational burden have been considered, including writing the model error covariance as a scalar multiple of the background error covariance and modeling the model error.
In this thesis, the main objective is the development of computationally feasible techniques for the improved representation of the model error statistics in a data assimilation system. Three new approaches are considered. A Monte Carlo method that uses an ensemble of w4D-Var systems to obtain flow-dependent estimates to the model error statistics. The evaluation of statistical diagnostic equations involving observation residuals to estimate the model error covariance matrix. An adaptive tuning procedure based on the sensitivity of a short-range forecast error measure to the model error DAS parametrization.
The validity and benefits of these approaches are shown in two stages of numerical experiments. A proof-of-concept is shown using the Lorenz multi-scale model and the shallow water equations for a one-dimensional domain. The results show the potential of these methodologies to produce improved state estimates, as compared to other approaches in data assimilation. It is expected that the techniques presented will find an extended range of applications to assess and improve the performance of a w4D-Var system.
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Kronbichler, Martin. "Numerical methods for the Navier-Stokes equations applied to turbulent flow and to multi-phase flow /." Licentiate thesis, Uppsala : Uppsala University, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-110246.
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Elago, David. "Robust computational methods for two-parameter singular perturbation problems." Thesis, University of the Western Cape, 2010. http://etd.uwc.ac.za/index.php?module=etd&action=viewtitle&id=gen8Srv25Nme4_1693_1308039217.
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This thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.
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Setta, Mario. "Multiscale numerical approximation of morphology formation in ternary mixtures with evaporation : Discrete and continuum models for high-performance computing." Thesis, Karlstads universitet, Institutionen för matematik och datavetenskap (from 2013), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-85036.
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We propose three models to study morphology formations in interacting ternary mixtures with the evaporation of one component. Our models involve three distinct length scales: microscopic, mesoscopic, and respectively, macroscopic. The real-world application we have in mind concerns charge transport through the heterogeneous structures arising in the fabrication of organic solar cells. As first model, we propose a microscopic 3-spins lattice dynamics with short-range interactions between the considered species. This microscopic model is approximated numerically via a Monte Carlo Metropolis-based algorithm. We explore the effect of the model parameters (volatility of the solvent, system's temperature, and interaction strengths) on the structure of the formed morphologies. Our second model is built upon the first one, by introducing a new mesoscale corresponding to the size of block spins. The link between these two models as well as between the effects of the model parameters and formed morphologies are studied in detail. These two models offer insight into cross-sections of the modeling box. Our third model encodes a macroscopic view of the evaporating mixture. We investigate its capability to lead to internal coherent structures. We propose a macroscopic system of nonlinearly coupled Cahn-Hilliard equations to capture numerical results for a top view of the modeling box. Effects of effective evaporation rates, effective interaction energy parameters, and degree of polymerization on the wanted morphology formation are explored via the computational platform FEniCS using a FEM approximation of a suitably linearized system. High-performance computing resources and Python-based parallel implementations have been used to facilitate the numerical approximation of the three models.
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Lundgren, Lukas. "Efficient numerical methods for the shallow water equations." Thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-354689.
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In this thesis a high order finite difference scheme is derived and implemented solving the shallow water equations using the SBP-SAT method. This method was tested against various benchmark problems were convergence was verified. The shallow water equations were also solved on a multi-block setup representing a tsunami approaching a shoreline from the ocean. Experiments show that a bottom topography with many spikes provides a dispersing effect on the incoming tsunami wave. Higher order convergence is not guaranteed for the multi-block simulations and could be investigated further in a future study.
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Qirezi, Fatmir. "Discrete schemes for thermoviscoelasticity with thermorheologically-simple nonlinear coupling." Thesis, Brunel University, 2014. http://bura.brunel.ac.uk/handle/2438/13456.
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Zerroukat, Mohamed. "Numerical computation of moving boundary phenomena." Thesis, University of Glasgow, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.285256.
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Broni-Mensah, Edwin. "Numerical solutions of weather derivatives and other incomplete market problems." Thesis, University of Manchester, 2012. https://www.research.manchester.ac.uk/portal/en/theses/numerical-solutions-of-weather-derivatives-and-other-incomplete-market-problems(26fdd9c6-c5dd-4fea-87fe-11537c353ee7).html.
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The valuation of weather derivatives is complex since the underlying temperature process has no negotiable price. This thesis introduces a selection of models for the valuation of weather derivative contracts, governed by a stochastic underlying temperature process. We then present a new weather pricing model, which is used to determine the fair hedging price of a weather derivative under the assumptions of mean self-financing. This model is then extended to incorporate a compensation (or market price of risk) awarded to investors who hold undiversifiable risks. This results in the derivation of a non-linear two-dimensional PDE, for which the numerical evaluation cannot be performed using standard finite-difference techniques. The numerical techniques applied in this thesis are based on a broad range of lattice based schemes, including enhancements to finite-differences, quadrature methods and binomial trees. Furthermore simulations of temperature processes are undertaken that involves the development of Monte Carlo based methods.
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Eriksson, Gustav. "A Numerical Solution to the Incompressible Navier-Stokes Equations." Thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-387386.
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A finite difference based solution method is derived for the velocity-pressure formulation of the two-dimensional incompressible Navier-Stokes equations. The method is proven stable using the energy method, facilitated by SBP operators, for characteristic and Dirichlet boundary condition implemented using the SAT technique. The numerical experiments show the utility of high-order finite difference methods as well as emphasize the problem of pressure boundary conditions. Furthermore, we demonstrate that a discretely divergence free solution can be obtained by use of the projection method.
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Taylor, Simon. "Design environment and anisotropic adaptive meshing in computational magnetics." Thesis, University of Southampton, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.301211.
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Wik, Niklas, David Niemelä, and Zethrin Valter Wagner. "Numerical simulations of the Dynamic Beam Equation in discontinuous media." Thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-416818.
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The study examines the Projection method and the simultaneousapproximation-term (SAT) method as boundary treatment for the dynamic beam equation using summation-by-parts (SBP) operators for handling the inner domain. The methods are examined for both the homogeneous constant coefficient case, and the inhomogeneous piecewise constant coefficient case with a coupled interface. The outer boundaries are handled by SAT or Projection, the coupled interfaced is handled by Projection or a mix between Projection and SAT. Solutions are integrated in time with finite central difference schemes and compared to analytical solutions. A convergence study is conducted with respect to the spatial discretization to measure the accuracy, and the stability is examined by numerical simulations of the CFL-condition. The study shows that Projection has the same accuracy as SAT for most boundary conditions while allowing for a larger timestep. A discontinuity in the medium is found to be handled equally accurate by Projection and the Projection and SAT mixture for all but one case studied, where the mixture was slightly more accurate.
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Yu, Yang. "A Numerical Approach for Interfacial Motion and its Application to viscous effects in the Benjamin-Feir instability." The Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=osu1252600763.
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Sjöberg, Paul. "Numerical solution of the Fokker–Planck approximation of the chemical master equation." Licentiate thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-86354.
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The chemical master equation (CME) describes the probability for the discrete molecular copy numbers that define the state of a chemical system. Each molecular species in the chemical model adds a dimension to the state space. The CME is a difference-differential equation which can be solved numerically if the state space is truncated at an upper limit of the copy number in each dimension. The size of the truncated CME suffers from an exponential growth for an increasing number of chemical species. In this thesis the chemical master equation is approximated by a continuous Fokker-Planck equation (FPE) which makes it possible to use sparser computational grids than for CME. FPE on conservative form is used to compute steady state solutions by computation of an extremal eigenvalue and the corresponding eigenvector as well as time-dependent solutions by an implicit time-stepping scheme. The performance of the numerical solution is compared to a standard Monte Carlo algorithm. The computational work for a solutions with the same estimated error is compared for the two methods. Depending on the problem, FPE or the Monte Carlo algorithm will be more efficient. FPE is well suited for problems in low dimensions, especially if high accuracy is desirable.
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Leonard, Katherine H. L. "Mathematical and computational modelling of tissue engineered bone in a hydrostatic bioreactor." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:05845740-1a74-4e19-95ea-6b5229d1af27.
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In vitro tissue engineering is a method for developing living and functional tissues external to the body, often within a device called a bioreactor to control the chemical and mechanical environment. However, the quality of bone tissue engineered products is currently inadequate for clinical use as the implant cannot bear weight. In an effort to improve the quality of the construct, hydrostatic pressure, the pressure in a fluid at equilibrium that is required to balance the force exerted by the weight of the fluid above, has been investigated as a mechanical stimulus for promoting extracellular matrix deposition and mineralisation within bone tissue. Thus far, little research has been performed into understanding the response of bone tissue cells to mechanical stimulation. In this thesis we investigate an in vitro bone tissue engineering experimental setup, whereby human mesenchymal stem cells are seeded within a collagen gel and cultured in a hydrostatic pressure bioreactor. In collaboration with experimentalists a suite of mathematical models of increasing complexity is developed and appropriate numerical methods are used to simulate these models. Each of the models investigates different aspects of the experimental setup, from focusing on global quantities of interest through to investigating their detailed local spatial distribution. The aim of this work is to increase understanding of the underlying physical processes which drive the growth and development of the construct, and identify which factors contribute to the highly heterogeneous spatial distribution of the mineralised extracellular matrix seen experimentally. The first model considered is a purely temporal model, where the evolution of cells, solid substrate, which accounts for the initial collagen scaffold and deposited extracellular matrix along with attendant mineralisation, and fluid in response to the applied pressure are examined. We demonstrate that including the history of the mechanical loading of cells is important in determining the quantity of deposited substrate. The second and third models extend this non-spatial model, and examine biochemically and biomechanically-induced spatial patterning separately. The first of these spatial models demonstrates that nutrient diffusion along with nutrient-dependent mass transfer terms qualitatively reproduces the heterogeneous spatial effects seen experimentally. The second multiphase model is used to investigate whether the magnitude of the shear stresses generated by fluid flow, can qualitatively explain the heterogeneous mineralisation seen in the experiments. Numerical simulations reveal that the spatial distribution of the fluid shear stress magnitude is highly heterogeneous, which could be related to the spatial heterogeneity in the mineralisation seen experimentally.
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Hellander, Andreas. "Numerical simulation of well stirred biochemical reaction networks governed by the master equation." Licentiate thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-85856.
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Numerical simulation of stochastic biochemical reaction networks has received much attention in the growing field of computational systems biology. Systems are frequently modeled as a continuous-time discrete space Markov chain, and the governing equation for the probability density of the system is the (chemical) master equation. The direct numerical solution of this equation suffers from an exponential growth in computational time and memory with the number of reacting species in the model. As a consequence, Monte Carlo simulation methods play an important role in the study of stochastic chemical networks. The stochastic simulation algorithm (SSA) due to Gillespie has been available for more than three decades, but due to the multi-scale property of the chemical systems and the slow convergence of Monte Carlo methods, much work is currently being done in order to devise more efficient approximate schemes. In this thesis we review recent work for the solution of the chemical master equation by direct methods, by exact Monte Carlo methods and by approximate and hybrid methods. We also describe two conceptually different numerical methods to reduce the computational time when studying models using the SSA. A hybrid method is proposed, which is based on the separation of species into two subsets based on the variance of the copy numbers. This method yields a significant speed-up when the system permits such a splitting of the state space. A different approach is taken in an algorithm that makes use of low-discrepancy sequences and the method of uniformization to reduce variance in the computed density function.
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Machado, Tavares Rodrigo. "The use of numerical optimisation techniques in computational fire engineering models : a study through evacuation modelling analyses." Thesis, University of Greenwich, 2011. http://gala.gre.ac.uk/7659/.
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Evacuation models have been playing an important function in the transition process from prescriptive fire safety codes to performance-based ones over the last three decades. In fact, such models became also useful tools in different tasks within fire safety engineering field, such as fire risks assessment and fire investigation. However, there are some difficulties in this process when using these models. For instance, during the evacuation modeling analysis, a common problem faced by fire safety engineers concerns the number of simulations which needs to be performed. In other terms, which fire designs (i.e., scenarios) should be investigated using the evacuation models? This type of question becomes more complex when specific issues such as the optimal positioning of exits within an arbitrarily structure needs to be addressed. In the other hand, numerical optimisation techniques have been applied to a range of different fields such as structural analysis. These techniques have shown to be a powerful tool for designers, saving their time and consequently reducing costs during the process. For this reason, the emphasis, throughout this study, is to develop a methodology that enables the optimisation of fire safety analysis of structural designs. In other words, the current research was primarily intended to demonstrate and develop this combination of fire engineering tools and techniques such as the Design of Experiments (DoE) and numerical optimisation techniques. For this purpose, a Computational Fire Engineering (CFE) tool combined with Numerical Optimisation Techniques and associated statistical methods (i.e., Design of Experiments (DoE) and Response Surface Models (RSM)) are used. The study is focused on evacuation modelling; nevertheless the methodology proposed here could equally be applied to CFD-based fire simulation tools. While the approach that has been developed is intended to be generally applicable, the techniques have been explored and demonstrated using the buildingEXODUS computational package. This fire engineering simulation tool is used worldwide, to improve the fire safety in building designs. This study therefore intended, besides to develop a numerical methodology to allow the efficient optimisation of fire safety aspects of structural designs, to understand how the core variables impact the evacuation efficiency. For instance, a common problem faced by fire safety engineers, in the field of evacuation analysis, is the optimal positioning of exits within an arbitrarily complex structure. This problem is usually addressed through time consuming and expensive trial and error. While a solution is usually found, to this problem, it is seldom the optimal solution, resulting in a compromise in building performance and safety. The methodology explored in this thesis, as applied to CFE, was initially based around a relatively small set of physical variables. This approach evolved and was subsequently expanded to include more complex behavioural, procedural and environmental parameters. The methodology has also been further developed and applied to evacuation simulation. This integrated approach is intended to help fire safety engineers and designers to develop optimal designs (i.e., safe designs) in an optimised manner. In reality, this was the motivation of this study: to introduce numerical optimisation techniques and associated concepts, well known within the operational research field, as an approach for a more efficient and systematic procedure when developing and/or improving fire safety designs. Post comparisons between the outputs obtained, using these different DoE techniques, have been also performed in order to analyze which technique is most suitable for the optimisation of structural designs. This thesis describes a number of analyses (of a variety of structural designs) that have been used to evaluate the application of optimisation techniques into the CFE context. This included the use of the buildingEXODUS simulation tool, as mentioned previously, followed by the application of a variety of optimisation techniques (both gradient and non-gradient based numerical optimisation techniques) as well as different types of DoE (such as Latin Hypercube, Central Composite Design (CCD) and also a Random approach) in order to improve the designs according to a number of different variables. These variables have initially included physical modifications to the geometry. The proposed methodological approach developed in this thesis is demonstrated on a variety of practical problems. These problems are represented by 4 case studies which vary from complexity to the nature of the variables. These case studies involved both types of problems, namely: unconstrained and constrained. The results obtained have shown to be satisfactory, i.e., global minima and local minima closest to the global minima region were found. For all the cases, a gradient-based algorithm (i.e., the Fletcher-Reeves numerical optimisation technique) and non gradient-based algorithm (i.e., the Particle Swarm Optimisation numerical optimisation technique) were used to find the optimal solution. And as mentioned before, different DoE techniques were also applied. Important issues within building fire safety design were found and discussed in this thesis. For example, it was shown that the positioning of the exits can have a stronger impact on the design evacuation time rather than the exits' widths for some scenarios. Furthermore, the level of life safety for buildings should also consider the exits' locations within enclosures. The analysis revealed that this methodology seems to be a very powerful tool for evacuation modelling analysis. This systematic methodology to efficiently optimise evacuation safety aspects of structural designs should be also extended to more complex designs, such as larger enclosures and open spaces. This methodology is also intended to be applied to problems found in the field of fire simulation, such as: the sizing and positioning of smoke extraction vents and the modelling of cable fires.
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Norton, Richard. "Numerical computation of band gaps in photonic crystal fibres." Thesis, University of Bath, 2008. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.501623.
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Photonic crystal fibres are capable of special light guiding properties that ordinary optical fibres do not possess, and efforts have been made to numerically model these properties. The plane wave expansion method is one of the numerical methods that has been used. Unfortunately, the function that describes the material in the fibre n(x) is discontinuous, and convergence of the plane wave expansion method is adversely affected by this. For this reason, the plane wave expansion method may not be every applied mathematician’s first choice method but we will show that it is comparable in implementation and convergence to the standard finite element method. In particular,an optimal preconditioner for the system matrix A can easily be obtained and matrixvector products with A can be computed in O(N logN) operations (where N is the size of A) using the Fast Fourier Transform. Although we are always interested in the efficiency of the method, the main contribution of this thesis is the development of convergence analysis for the plane wave expansion method applied to 4 different 2nd-order elliptic eigenvalue problems in R and R2 with discontinuous coefficients. To obtain the convergence analysis three issues must be confronted: regularity of the eigenfunctions; approximation error with respect to plane waves; and stability of the plane wave expansion method. We successfully tackle the regularity and approximation error issues but proving stability relies on showing that the plane wave expansion method is equivalent to a spectral Galerkin method, and not all of our problems allow this. However, stability is observed in all of our numerical computations. It has been proposed in [40], [53], [63] and [64] that replacing the discontinuous coefficients in the problem with smooth coefficients will improve the plane wave expansion method, despite the additional error. Our convergence analysis for the method in[63] and [64] shows that the overall rate of convergence is no faster than before. To define A we need the Fourier coefficients of n(x), and sometimes these must be approximated, thus adding an additional error. We analyse the errors for a method where n(x) is sampled on a uniform grid and the Fourier coefficients are computed with the Fast Fourier Transform. We then devise a strategy for setting the grid-spacing that will recover the convergence rate of the plane wave expansion method with exact Fourier coefficients.
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Sidahmed, Abdelmgid Osman Mohammed. "Mesh free methods for differential models in financial mathematics." Thesis, University of the Western Cape, 2011. http://etd.uwc.ac.za/index.php?module=etd&action=viewtitle&id=gen8Srv25Nme4_3917_1319185202.
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Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.
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Mautner, Karin. "Numerical treatment of the Black-Scholes variational inequality in computational finance." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2007. http://dx.doi.org/10.18452/15595.
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In der Finanzmathematik hat der Besitzer einer amerikanische Option das Recht aber nicht die Pflicht, eine Aktie innerhalb eines bestimmten Zeitraums, für einen bestimmten Preis zu kaufen oder zu verkaufen. Die Bewertung einer amerikanische Option wird als so genanntes optimale stopping Problem formuliert. Erfolgt die Modellierung des Aktienkurses durch eine geometrische Brownsche Bewegung, wird der Wert einer amerikanischen Option durch ein deterministisches freies Randwertproblem (FRWP), oder einer äquivalenten Variationsungleichung (VU) auf ganz R in gewichteten Sobolev Räumen gegeben. Um Standardmethoden der Numerischen Mathematik anzuwenden, wird das unbeschränkte Gebiet zu einem beschränkten Gebiet abgeschnitten. Mit Hilfe der Fourier-Transformation wird eine Integraldarstellung der Lösung die den freien Rand explizit beinhaltet, hergeleitet. Mittels dieser Integraldarstellung werden Abschneidefehlerschranken bewiesen. Danach werden gewichtete Poincare Ungleichungen mit expliziten Konstanten bewiesen. Der Abschneidefehler und die gewichtete Poincare Ungleichung ermöglichen, einen zuverlässigen a posteriori Fehlerschätzer zwischen der exakten Lösung der VU und der semidiskreten Lösung des penalisierten Problems auf R herzuleiten. Eine hinreichend glatte Lösung der VU garantiert die Konvergenz der Lösung des penaltisierten Problems zur Lösung der VU. Ein a priori Fehlerschätzer für den Fehler zwischen der exakten Lösung der VU und der semidiskreten Lösung des penaltisierten Problems beendet die numerische Analysis. Die eingeführten aposteriori Fehlerschätzer motivieren einen Algorithmus für adaptive Netzverfeinerung. Numerische Experimente zeigen die verbesserte Konvergenz des adaptiven Verfahrens gegenüber der uniformen Verfeinerung. Der zuverlässige a posteriori Fehlerschätzer ermöglicht es, den Abschneidepunkt so zu wählen, dass der Gesamtfehler (Diskretisierungsfehler plus Abschneidefehler) kleiner als eine gegebenen Toleranz ist.Among the central concerns in mathematical finance is the evaluation of American options. An American option gives the holder the right but not the obligation to buy or sell a certain financial asset within a certain time-frame, for a certain strike price. The valuation of American options is formulated as an optimal stopping problem. If the stock price is modelled by a geometric Brownian motion, the value of an American option is given by a deterministic parabolic free boundary value problem (FBVP) or equivalently a non-symmetric variational inequality (VI) on weighted Sobolev spaces on R. To apply standard numerical methods, the unbounded domain R is truncated to a bounded one. Applying the Fourier transform to the FBVP yields an integral representation of the solution including the free boundary explicitely. This integral representation allows to prove explicit truncation errors. Since the VI is formulated within the framework of weighted Sobolev spaces, we establish a weighted Poincare inequality with explicit determined constants. The truncation error estimate and the weighted Poncare inequality enable a reliable a posteriori error estimate between the exact solution of the VI and the semi-discrete solution of the penalised problem on R. A sufficient regular solution provides the convergence of the solution of the penalised problem to the solution of the VI. An a priori error estimate for the error between the exact solution of the VI and the semi-discrete solution of the penalised problem concludes the numerical analysis. The established a posteriori error estimates motivates an algorithm for adaptive mesh refinement. Numerical experiments show the improved convergence of the adaptive algorithm compared to uniform mesh refinement. The reliable a posteriori error estimate including explicit truncation errors allows to determine a truncation point such that the total error (discretisation and truncation error) is below a given error tolerance.
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Al-Awadi, Huda. "Efficient numerical computation of the dynamics of a spherical bubble." Thesis, University of Brighton, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.341282.
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Avrutin, Viktor [Verfasser]. "Bifurcation structures within robust chaos: computational aspects, numerical investigation and analytical explanation / Viktor Avrutin." Aachen : Shaker, 2011. http://d-nb.info/1075437423/34.
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Fosso-Tande, Jacob. "A Computational Chemistry Study of Spin Traps." Digital Commons @ East Tennessee State University, 2007. https://dc.etsu.edu/etd/2127.
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Many defects in physiological processes are due to free radical damage: reactive oxygen species, nitric oxide, and hydroxyl radicals have been implicated in the parthenogenesis of cancer, diabetes mellitus, and rheumatoid arthritis. We herein characterize the phenyl-N-ter-butyl nitrone (PBN) type spin traps in conjunction with the most studied dimethyl-1-pyrroline-N-oxide (DMPO) type spin traps using the hydroxyl radical. In this study, theoretical calculations are carried out on the two main types of spin traps (DMPO and PBN) at the density functional theory level (DFT). The energies of the optimized structures, hyperfine calculations in gaseous and aqueous phases of the spin traps and the hydroxyl radical adduct are calculated at the B3LYP correlation and at the 6-31G (d) and 6-311G (2df, p) basis sets respectively. The dielectric effect on the performance of the spin trap is determined using the polarized continuum model. Calculations show a localization of spin densities in both cases. However, DMPO spin traps are shown to be more stable and more interactive in aqueous environment.
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Chen, Weitao. "Fast Sweeping Methods for Steady State Hyperbolic Conservation Problems and Numerical Applications for Shape Optimization and Computational Cell Biology." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1366279632.
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Nyqvist, Robert. "Algebraic Dynamical Systems, Analytical Results and Numerical Simulations." Doctoral thesis, Växjö : Växjö University Press, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:vxu:diva-1142.
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Reis, Francisco das Chagas Azevedo dos. "Mathematical and computational modeling of contamination of aquifers with the use of numerical methods without mesh." Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=13670.
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Em muitos problemas da natureza e em uma diversidade enorme de Ãreas do conhecimento, existe a necessidade real de modelarmos fenÃmenos existentes. Em CiÃncias como MatemÃtica, FÃsica, QuÃmica, Biologia, Economia e nas Engenharias, de uma maneira geral, à comum por parte dos pesquisadores, o uso de modelos e simulaÃÃes, Ãs quais, quase sempre, envolvem taxas, princÃpios e leis, regidos por EquaÃÃes Diferenciais. Problemas envolvendo movimento de fluidos, intensidade de corrente elÃtrica, propagaÃÃo de calor, crescimento populacional, entre muitos outros, sÃo exemplos clÃssicos de aplicaÃÃes de modelos regidos por EquaÃÃes Diferencias, Ãs quais, podem ser diferenciadas quanto ao tipo em EquaÃÃes Diferenciais OrdinÃrias (EDO) e EquaÃÃes Diferenciais Parciais (EDP). Nas primeiras, a funÃÃo a ser determinada depende de uma Ãnica variÃvel independente, enquanto nas segundas, ocorre a dependÃncia de duas ou mais variÃveis independentes. Acontece à que em uma grande variedade de problemas da natureza, as equaÃÃes nÃo possuem soluÃÃes bem comportadas, analÃticas e, dessa maneira, faz-se necessÃrio o conhecimento de mÃtodos numÃricos, tais como, DiferenÃas Finitas, Elementos Finitos, Elementos de Contorno, entre outros, os quais necessitam da discretizaÃÃo do domÃnio e, portanto da criaÃÃo de uma malha (MESH), com fÃrmulas interativas para se estimar uma soluÃÃo e minimizar o erro da aproximaÃÃo. Nesse sentido, a proposta desse trabalho à utilizar um mÃtodo numÃrico bastante eficaz e independente de malha, denominado mÃtodo sem malhas (MESHLESS), mas especificamente o mÃtodo de Kansas, o qual lanÃa mÃo de FunÃÃes de Base Radial (Radial Basis Functions â RBF), ou simetria radial, da distÃncia entre um ponto central do domÃnio da funÃÃo e um ponto genÃrico do domÃnio. A funÃÃo interpoladora de base radial, tambÃm depende de um parÃmetro de forma âcâ a ser encontrado. Mas a questÃo preponderante Ã: como determinar um parÃmetro de forma âcâ Ãtimo, que possa oferecer uma soluÃÃo consistente, reduzindo o resÃduo e, portanto o erro existente? Para tanto, modelou-se um problema de contaminaÃÃo de aquÃfero fazendo uso da equaÃÃo de difusÃo, comparando o resultado de sua soluÃÃo analÃtica, com a soluÃÃo numÃrica obtida atravÃs do mÃtodo numÃrico sem malhas e com o parÃmetro de forma simulado e otimizado por meio da plataforma SCILABIn many problems of nature and a huge diversity of knowledge areas , there is a real need we model existing phenomena . Sciences like Mathematics , Physics , Chemistry, Biology , Economics and in Engineering , in general , is common among the researchers , the use of models and simulations , whi ch almost always involve fees , principles and laws , governed by Differential Equations . Problems involving fluid motion , intensity of electric current , heat propagation , population growth , among many others , are classic examples of applications of models g overned by Differential Equations , which can be differentiated as to type in Ordinary Differential Equations (ODE ) and Partial Differential Equations ( PDE). In the first , the function to be determined depends on a single variable, while in the second , the dependence of two or more independent variables occurs . Happens is that in a wide variety of problems of nature , the equations do not have well - behaved, analytic and thus solutions , it is necessary the knowledge of numerical methods such as Finite Differen ces , Finite Elements , Boundary Elements , among others, which require the discretization of the domain and therefore the creation of a mesh ( M ESH), with interactive formulas for estimating a solution and minimize the error of approximation . In this sense , t he purpose of this work is to use a very efficient and independent of mesh numerical method , called method without mesh ( MESHLESS), but specifically the method of Kansas , which makes use of Radial Basis Function ( Radial Basis Functions - RBF ) or radial sym metry , the distance between central point of the domain of the function and a generic point of the domain. The interpolating radial basis function also depends on a shape parameter " c" to be found . But the overriding question is how to determine a shape pa rameter " c" great, we can provide a consistent solution , reducing waste and therefore the existing error ? For both , modeled itself a problem of contamination of the aquifer by making use of the diffusion equation , comparing the results of its analytical so lution with the numerical solution obtained by numerical method without mesh and parameter simulated shape and optimized by SCILAB platform (version 5. 4 . 1 )
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Payne, Karl A. "Mathematical and Numerical Modeling of Hybrid Adsorption and Biological Treatment Systems for Enhanced Nitrogen Removal." Scholar Commons, 2018. https://scholarcommons.usf.edu/etd/7702.
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High nutrient loading into groundwater and surface water systems has deleterious impacts on the environment, such as eutrophication, decimation of fish populations, and oxygen depletion. Conventional onsite wastewater treatment systems (OWTS) and various waste streams with high ammonium (NH4+) concentrations present a challenge, due the inconsistent performance of environmental biotechnologies aimed at managing nutrients from these sources.
Biological nitrogen removal (BNR) is commonly used in batch or packed-bed reactor configurations for nitrogen removal from various waste streams. In recognition of the need for resource recovery, algal photobioreactors are another type of environmental biotechnology with the potential for simultaneously treating wastewater while recovering energy. However, irrespective of the technology adopted, outstanding issues remain that affect the consistent performance of environmental biotechnologies for nitrogen removal and resource recovery. In OWTS, transient loading can lead to inconsistent nitrogen removal efficiency, while the presence of high free ammonia (FA) can exert inhibitory effects on microorganisms that mediate transformation of nitrogen species as well as microalgae that utilize nitrogen. Therefore, to overcome these challenges there have been experimental studies investigating the addition of adsorption and ion exchange (IX) media that can temporarily take up specific nitrogen ions.
Bioreactors comprised of microorganisms and adsorption/IX media can attenuate transient loading as well as mitigate inhibitory effects on microorganisms and microalgae; however, the interplay between physicochemical and processes in these systems is not well understood. Therefore, the main objective of this dissertation was to develop theoretical and numerical models that elucidate the complex interactions that influence the fate of chemical species in the bioreactors.
To achieve this objective and address the issues related to improving the understanding of the underlying mechanisms occurring within the environmental biotechnologies investigated, the following three research studies were done: (i) experimental and theoretical modeling studies of an IX-assisted nitrification process for treatment of high NH4+ strength wastewater (Chapter 3), (ii) theoretical and numerical modeling of a hybrid algal photosynthesis and ion exchange (HAPIX) process for NH4+ removal and resource recovery (Chapter 4), and (iii) mathematical and numerical modeling of a mixotrophic denitrification process for nitrate (NO3-) removal under transient inflow conditions (Chapter 5).
The experimental results for the IX-assisted nitrification process showed that by amending the bioreactor with zeolite, there was a marked increase in the nitrification rate as evidenced by an increase in NO3– production from an initial concentration of 3.7 mg-N L-1 to 160 mg-N L-1. This increase is approximately an order of magnitude greater than the increase in the reactor without chabazite. Therefore, the experimental studies provided support for the hypothesis that IX enhances the nitrification process. To garner further support for the hypothesis and better understand the mechanisms in the bioreactor, a novel mathematical model was developed that mechanistically describes IX kinetics by surface diffusion coupled with a nitrification inhibition model described by the Andrews equation. The agreement between the model and data suggests that the mathematical model developed provides a theoretically sound conceptual understanding of IX-assisted nitrification.
A model based on the physics of Fickian diffusion, IX chemistry, and algal growth with co-limiting factors including NH4+, light irradiance, and temperature was developed to describe a batch reactor comprised of microalgae and zeolite. The model can reproduce the temporal history of NH4+ in the reactor as well as the growth of microalgae biomass. The mathematical model developed for the HAPIX process balances between simplicity and accuracy to provide a sound theoretical framework for mechanisms involved.
In OWTS, transient inflow conditions have an influence on the performance of environmental biotechnologies for nitrogen removal. Prior experiments have shown that for denitrification, a tire-sulfur hybrid adsorption and denitrification (T-SHAD) bioreactor consistently removes nitrogen under varying influent flow and concentration conditions. To enhance the understanding of the underlying mechanisms in the T-SHAD bioreactor, a mathematical model describing mass transport of NO3- and SO42- in the aqueous phase and mixotrophic denitrification was developed. Additionally, a numerical tool to solve the mathematical model was implemented and compared to previously conducted experiments. Results from the numerical simulations capture the trend of the experimental data showing approximately 90% NO3- -N removal under varying flow conditions. Moreover, the model describes the effluent characteristics of the process showing a transient response in correspondence the changes in fluid velocity.
The new tools developed provide new insight into the underlying mechanisms of physical, chemical, and biological processes within these bioreactors. The tools developed in this dissertation have a potential broad impact in environmental biotechnology for wastewater treatment in on-site systems, for treatment of high strength wastewater, and can be extended easily for stormwater management systems aimed at mitigating high nutrient loading to the environment.
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Giere, Swetlana [Verfasser]. "Numerical and Analytical Aspects of POD-Based Reduced-Order Modeling in Computational Fluid Dynamics / Swetlana Giere." Berlin : Freie Universität Berlin, 2016. http://d-nb.info/1119151341/34.
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Arthurs, Christopher J. "Efficient simulation of cardiac electrical propagation using adaptive high-order finite elements." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:ad31f06f-c4ed-4c48-b978-1ef3b12fe7a1.
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This thesis investigates the high-order hierarchical finite element method, also known as the finite element p-version, as a computationally-efficient technique for generating numerical solutions to the cardiac monodomain equation. We first present it as a uniform-order method, and through an a priori error bound we explain why the associated cardiac cell model must be thought of as a PDE and approximated to high-order in order to obtain the accuracy that the p-version is capable of. We perform simulations demonstrating that the achieved error agrees very well with the a priori error bound. Further, in terms of solution accuracy for time taken to solve the linear system that arises in the finite element discretisation, it is more efficient that the state-of-the-art piecewise linear finite element method. We show that piecewise linear FEM actually introduces quite significant amounts of error into the numerical approximations, particularly in the direction perpendicular to the cardiac fibres with physiological conductivity values, and that without resorting to extremely fine meshes with elements considerably smaller than 70 micrometres, we can not use it to obtain high-accuracy solutions. In contrast, the p-version can produce extremely high accuracy solutions on meshes with elements around 300 micrometres in diameter with these conductivities. Noting that most of the numerical error is due to under-resolving the wave-front in the transmembrane potential, we also construct an adaptive high-order scheme which controls the error locally in each element by adjusting the finite element polynomial basis degree using an analytically-derived a posteriori error estimation procedure. This naturally tracks the location of the wave-front, concentrating computational effort where it is needed most and increasing computational efficiency. The scheme can be controlled by a user-defined error tolerance parameter, which sets the target error within each element as a proportion of the local magnitude of the solution as measured in the H^1 norm. This numerical scheme is tested on a variety of problems in one, two and three dimensions, and is shown to provide excellent error control properties and to be likely capable of boosting efficiency in cardiac simulation by an order of magnitude. The thesis amounts to a proof-of-concept of the increased efficiency in solving the linear system using adaptive high-order finite elements when performing single-thread cardiac simulation, and indicates that the performance of the method should be investigated in parallel, where it can also be expected to provide considerable improvement. In general, the selection of a suitable preconditioner is key to ensuring efficiency; we make use of a variety of different possibilities, including one which can be expected to scale very well in parallel, meaning that this is an excellent candidate method for increasing the efficiency of cardiac simulation using high-performance computing facilities.
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Stary, Tomas. "Mathematical and computational study of Markovian models of ion channels in cardiac excitation." Thesis, University of Exeter, 2016. http://hdl.handle.net/10871/24166.
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This thesis studies numerical methods for integrating the master equations describing Markov chain models of cardiac ion channels. Such models describe the time evolution of the probability that ion channels are in a particular state. Numerical simulations of such models are often computationally demanding because many solvers require relatively small time steps to ensure numerical stability. The aim of this project is to analyse selected Markov chains and develop more efficient and accurate solvers. We separate a Markov chain model into fast and slow time-scales based on the speed of transitions between states. Eliminating the fast transitions, we find an asymptotic reduction of zeroth-order and first-order in a small parameter describing the time-scales separation. We apply the theory to a Markov chain model of the fast sodium channel INa. We consider several variants for classifying some transitions as fast in order to find reduced systems that yield a good accuracy. However, the time step size is still restricted by numerical instabilities. We adapt the Rush-Larsen technique originally developed for gate models. Assuming that a transition matrix can be considered constant during each time step, we solve the Markov chain model analytically. The solution provides a recipe for a stable exponential solver, which we call "Matrix Rush-Larsen" (MRL). Using operator splitting we design an even more flexible "hybrid" method that combines the MRL with other solvers. The resulting improvement in stability allows a large increase in the time step size. In some models, we obtain reasonably accurate results 27 times faster using a hybrid method than with the forward Euler method, even with the maximal time step allowed by the stability constraint. Finally, we extend the cardiac simulation package BeatBox by the developed exponential solvers. We upgrade a format of "ionic" modules which describe a cardiac cell, in order to allow for a specific definition of Markov chain models. We also modify a particular integrator for ionic modules to include the MRL and the hybrid method. To test the functionality of the code, we have converted a number of cellular models into the ionic format. The documented code is available in the official BeatBox package distribution.
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Saha, Suvash C. "Natural convection adjacent to an inclined flat plate and in an attic space under various thermal forcing conditions." Thesis, School of Engineering and Physical Sciences, 2009. https://eprints.qut.edu.au/44171/1/Master_phd.pdf.
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The effect of viscous dissipation on natural convection from a vertical plate placed in a thermally stratified environment has been investigated numerically. The reduced equations are integrated by employing the implicit finite difference scheme or Ke1ler-box method and obtained the effect of heat due to viscous dissipation on the local skin-friction and loca1 Nusselt number at various stratification levels, for fluids having Prandtl number equals 10, 50, and 100. Solutions are also obtained using the perturbation technique for small values of viscous dissipation parameters and compared with the Finite Difference solutions. Effect of the heat transfer due to viscous dissipation and the temperature stratification are also shown on the velocity and temperature distributions in the boundary layer region. A numerical study of laminar doubly diffusive free convection flows adjacent to a vertical surface in a stable thermally stratified medium is also considered for this study. Solutions are obtained using the implicit Finite Difference method and compared with the local non-similarity method. The velocity and temperature distributions for different values of stratification parameter are shown graphically. The results show many interesting aspects of complex interaction of the two buoyant mechanisms.
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Mechaik, Mehdi Mohamad. "Novel Theoretical And Numerical Methods For The Computation Of Electromagnetic Fields Due To Current Sources." Diss., The University of Arizona, 1994. http://hdl.handle.net/10150/186619.
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In this dissertation, series expansions are developed for the Incomplete Lipschitz-Hankel integrals (ILHIs) Je₀(a,z) and Ye₀(a,z). These expansions are obtained using the Laplace transform technique together with the theory of contour integration. These special functions are encountered in the solutions for numerous problems in electromagnetics. For example, ILHIs are used in this dissertation to obtain exact, closed-form field expressions for a semi-infinite traveling wave current filament in homogeneous space. They are also used together with the steepest descent technique to obtain expressions for the electromagnetic fields due to a semi-infinite traveling wave current filament above a half space. Superposition of these fields are used to obtain the fields due to a finite length wire carrying a traveling wave current. In addition, the ILHIs are also encountered when Prony's method is used to obtain field expressions for a vertical electric dipole source over earth.
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Hohn, Jennifer Lynn. "Generalized Probabilistic Bowling Distributions." TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/82.
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Have you ever wondered if you are better than the average bowler? If so, there are a variety of ways to compute the average score of a bowling game, including methods that account for a bowler’s skill level. In this thesis, we discuss several different ways to generate bowling scores randomly. For each distribution, we give results for the expected value and standard deviation of each frame's score, the expected value of the game’s final score, and the correlation coefficient between the score of the first and second roll of a single frame. Furthermore, we shall generalize the results in each distribution for an frame game on pins. Additionally, we shall generalize the number of possible games when bowling frames on pins. Then, we shall derive the frequency distribution of each frame’s scores and the arithmetic mean for frames on pins. Finally, to summarize the variety of distributions, we shall make tables that display the results obtained from each distribution used to model a particular bowler’s score. We evaluate the special case when bowling 10 frames on 10 pins, which represents a standard bowling game.
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PRUETT, CHARLES DAVID. "NUMERICAL SIMULATION OF NONLINEAR WAVES IN FREE SHEAR LAYERS (MIXING, COMPUTATIONAL, FLUID DYNAMICS, HYDRODYNAMIC STABILITY, SPATIAL, FLUID FLOW MODEL)." Diss., The University of Arizona, 1986. http://hdl.handle.net/10150/183869.
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A numerical model has been developed which simulates the three-dimensional stability and transition of a periodically forced free shear layer in an incompressible fluid. Unlike previous simulations of temporally evolving shear layers, the current simulations examine spatial stability. The spatial model accommodates features of free shear flow, observed in experiments, which in the temporal model are precluded by the assumption of streamwise periodicity; e.g., divergence of the mean flow and wave dispersion. The Navier-Stokes equations in vorticity-velocity form are integrated using a combination of numerical methods tailored to the physical problem. A spectral method is adopted in the spanwise dimension in which the flow variables, assumed to be periodic, are approximated by finite Fourier series. In complex Fourier space, the governing equations are spatially two-dimensional. Standard central finite differences are exploited in the remaining two spatial dimensions. For computational efficiency, time evolution is accomplished by a combination of implicit and explicit methods. Linear diffusion terms are advanced by an Alternating Direction Implicit/Crank-Nicolson scheme whereas the Adams-Bashforth method is applied to convection terms. Nonlinear terms are evaluated at each new time level by the pseudospectral (collocation) method. Solutions to the velocity equations, which are elliptic, are obtained iteratively by approximate factorization. The spatial model requires that inflow-outflow boundary conditions be prescribed. Inflow conditions are derived from a similarity solution for the mean inflow profile onto which periodic forcing is superimposed. Forcing functions are derived from inviscid linear stability theory. A numerical test case is selected which closely parallels a well-known physical experiment. Many of the aspects of forced shear layer behavior observed in the physical experiment are captured by the spatial simulation. These include initial linear growth of the fundamental, vorticity roll-up, fundamental saturation, eventual domination of the subharmonic, vortex pairing, emergence of streamwise vorticity, and temporary stabilization of the secondary instability. Moreover, the spatial simulation predicts the experimentally observed superlinear growth of harmonics at rates 1.5 times that of the fundamental. Superlinear growth rates suggest nonlinear resonances between fundamental and harmonic modes which are not captured by temporal simulations.
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Hellman, Fredrik. "Numerical Methods for Darcy Flow Problems with Rough and Uncertain Data." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-318589.
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We address two computational challenges for numerical simulations of Darcy flow problems: rough and uncertain data. The rapidly varying and possibly high contrast permeability coefficient for the pressure equation in Darcy flow problems generally leads to irregular solutions, which in turn make standard solution techniques perform poorly. We study methods for numerical homogenization based on localized computations. Regarding the challenge of uncertain data, we consider the problem of forward propagation of uncertainty through a numerical model. More specifically, we consider methods for estimating the failure probability, or a point estimate of the cumulative distribution function (cdf) of a scalar output from the model. The issue of rough coefficients is discussed in Papers I–III by analyzing three aspects of the localized orthogonal decomposition (LOD) method. In Paper I, we define an interpolation operator that makes the localization error independent of the contrast of the coefficient. The conditions for its applicability are studied. In Paper II, we consider time-dependent coefficients and derive computable error indicators that are used to adaptively update the multiscale space. In Paper III, we derive a priori error bounds for the LOD method based on the Raviart–Thomas finite element. The topic of uncertain data is discussed in Papers IV–VI. The main contribution is the selective refinement algorithm, proposed in Paper IV for estimating quantiles, and further developed in Paper V for point evaluation of the cdf. Selective refinement makes use of a hierarchy of numerical approximations of the model and exploits computable error bounds for the random model output to reduce the cost complexity. It is applied in combination with Monte Carlo and multilevel Monte Carlo methods to reduce the overall cost. In Paper VI we quantify the gains from applying selective refinement to a two-phase Darcy flow problem.
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Myers, Jeremy. "Computational Fluid Dynamics in a Terminal Alveolated Bronchiole Duct with Expanding Walls: Proof-of-Concept in OpenFOAM." VCU Scholars Compass, 2017. http://scholarscompass.vcu.edu/etd/5011.
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Mathematical Biology has found recent success applying Computational Fluid Dynamics (CFD) to model airflow in the human lung. Detailed modeling of flow patterns in the alveoli, where the oxygen-carbon dioxide gas exchange occurs, has provided data that is useful in treating illnesses and designing drug-delivery systems. Unfortunately, many CFD software packages have high licensing fees that are out of reach for independent researchers. This thesis uses three open-source software packages, Gmsh, OpenFOAM, and ParaView, to design a mesh, create a simulation, and visualize the results of an idealized terminal alveolar sac model. This model successfully demonstrates that OpenFOAM can be used to model airflow in the acinar region of the lung under biologically relevant conditions.
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Bujok, Karolina Edyta. "Numerical solutions to a class of stochastic partial differential equations arising in finance." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:d2e76713-607b-4f26-977a-ac4df56d54f2.
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We propose two alternative approaches to evaluate numerically credit basket derivatives in a N-name structural model where the number of entities, N, is large, and where the names are independent and identically distributed random variables conditional on common random factors. In the first framework, we treat a N-name model as a set of N Bernoulli random variables indicating a default or a survival. We show that certain expected functionals of the proportion LN of variables in a given state converge at rate 1/N as N [right arrow - infinity]. Based on these results, we propose a multi-level simulation algorithm using a family of sequences with increasing length, to obtain estimators for these expected functionals with a mean-square error of epsilon 2 and computational complexity of order epsilon−2, independent of N. In particular, this optimal complexity order also holds for the infinite-dimensional limit. Numerical examples are presented for tranche spreads of basket credit derivatives. In the second framework, we extend the approximation of Bush et al. [13] to a structural jump-diffusion model with discretely monitored defaults. Under this approach, a N-name model is represented as a system of particles with an absorbing boundary that is active in a discrete time set, and the loss of a portfolio is given as the function of empirical measure of the system. We show that, for the infinite system, the empirical measure has a density with respect to the Lebesgue measure that satisfies a stochastic partial differential equation. Then, we develop an algorithm to efficiently estimate CDO index and tranche spreads consistent with underlying credit default swaps, using a finite difference simulation for the resulting SPDE. We verify the validity of this approximation numerically by comparison with results obtained by direct Monte Carlo simulation of the basket constituents. A calibration exercise assesses the flexibility of the model and its extensions to match CDO spreads from precrisis and crisis periods.
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Skjerven, Brian M. "A parallel implementation of an agent-based brain tumor model." Link to electronic thesis, 2007. http://www.wpi.edu/Pubs/ETD/Available/etd-060507-172337/.
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Thesis (M.S.) -- Worcester Polytechnic Institute.Keywords: Visualization; Numerical analysis; Computational biology; Scientific computation; High-performance computing. Includes bibliographical references (p.19).
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Barnes, Gary James. "Computational modelling for type-II superconductivity and the investigation of high temperature superconducting electrical machines." Thesis, University of Oxford, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.365887.
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