Denna rapport är skriven som en del av ett kandidatarbete inom civilingenjörsprogrammet i teknisk fysik på Uppsala universitet, vårterminen 2022. Målet med arbetet var att jämföra två olika matematiska verktyg, Monte Carlo- och Kvasi-Monte Carlo-metoder och se vilken ut av dessa som var effektivast när det kommer till att beräkna svårlösta integraler. Monte Carlo-simuleringar har använts frekvent inom matematiken sedan 1940-talet och är ett samlingsbegrepp för statistiska simuleringar där pseudoslumptalssekvenser används för att beräkna komplicerade numeriska problem. Kvasi-Monte Carlo-metodens algoritm är den samma som för Monte Carlo, med undantaget att i stället för att använda pseudoslumptalssekvenser vid lösningen, används sekvenser med låg diskrepans, t.ex. Sobolsekvensen. Genom att använda programmeringsprogrammet Matlab kunde den högdimensionella integralen av Rosenbrocks funktion, den endimensionella integralen som används för att prissätta europeiska köp- och säljoptioner samt den högdimensionella integralen som används för att prissätta asiatiska köpoptioner beräknas med hjälp av Monte Carlo- och Kvasi-Monte Carlo-metoden. Således kunde dessa metoders konvergens undersökas. Det visade sig, som tidigare forskning också gett stöd för, att Kvasi-Monte Carlo-metoden konvergerade betydligt snabbare mot det analytiska värdet än vad Monte Carlo-metoden gjorde, för samtliga tre problem.  För framtida bruk av Monte Carlo-metoder när integraler skall beräknas rekommenderas det att i stället för att använda pseudoslumptalssekvenser, använda tal från sekvenser med låg diskrepans, gärna en sobolsekvens. Detta då lösningen både konvergerar snabbare mot det analytiska värdet och är mer exakt, trots höga dimensioner.

It is widely recognized that various biotic and abiotic factors cause changes in the size of a population and its age distribution. Population structure, intra-specific competition, temporal variability and spatial heterogeneity are identified as the most important factors that, alone or in combination, influence population dynamics. Despite being well-known, these factors are difficult to study, both theoretically and empirically. However, in an increasingly variable world, permanence of a growing number of species is threatened by climate changes, habitat fragmentation or reduced habitat quality. For purposes of conservation of species and land management, it is crucially important to have a good analysis of population dynamics, which will increase our theoretical knowledge and provide practical guidelines. One way to address the problem of population dynamics is to use mathematical models. The choice of a model depends on what we want to study or what we aim to achieve. For an extensive theoretical study of population processes and for obtaining qualitative results about population growth or decline, analytical models with various level of complexity are used. The competing interests of realism and solvability of the model are always present. This means that, on one hand, we always aim to make a model that will truthfully reflect reality, while on the other hand, we need to keep the model mathematically solvable. This prompts us to carefully choose the most prominent ecological factors relevant to the problem at hand and to incorporate them into a model. Ideally, the results give new insights into population processes and complex interactions between the mentioned factors and population dynamics. The objective of the thesis is to formulate, analyze, and apply various mathematical models of population dynamics. We begin with a classical linear age-structured model and gradually add temporal variability, intra-specific competition and spatial heterogeneity. In this way, every subsequent model is more realistic and complex than the previous one. We prove existence and uniqueness of a nonnegative solution to each boundary-initial problem, and continue with investigation of the large time behavior of the solution. In the ecological terms, we are establishing conditions under which a population can persist in a certain environment. Since our aim is a qualitative analysis of a solution, we often examine upper and lower bounds of a solution. Their importance is in the fact that they are obtained analytically and parameters in their expression have biological meaning. Thus, instead of analyzing an exact solution (which often proves to be difficult), we analyze the corresponding upper and lower solutions. We apply our models to demonstrate the influence of seasonal changes (or some other periodic temporal variation) and spatial structure of the habitat on population persistence. This is particularly important in explaining behavior of migratory birds or populations that inhabits several patches, some of which are of low quality. Our results extend the previously obtained results in some aspects and point out that all factors (age structure, density dependence, spatio-temporal variability) need to be considered when setting up a population model and predicting population growth.

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.

In the thesis, we introduce linear regression models such as Simple Linear Regression, Multiple Regression, and Polynomial Regression. We explain basic methods of the model parameters estimation, Ordinary Least Squares (OLS) and Maximum Likelihood Estimation (MLE). The properties of the estimates, and what assumptions need to be made for the model for the estimates to be the Best Linear Unbiased Estimates (BLUE) are given. The basic Bootstrap methods are introduced. The real world problem is simulated in order to see how measurement error affects the quotient of two estimated models.

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.

A proper understanding of biological processes requires an understanding of genetics and evolutionary mechanisms. The vast amounts of genetical information that can routinely be extracted with modern technology have so far not been accompanied by an equally extended understanding of the corresponding processes. The relationship between a single gene and the resulting properties, phenotype of an individual is rarely clear. This thesis addresses several computational challenges regarding identifying and assessing the effects of quantitative trait loci (QTL), genomic positions where variation is affecting a trait. The genetic information available for each individual is rarely complete, meaning that the unknown variable of the genotype in the loci modelled also needs to be addressed. This thesis contains the presentation of new tools for employing the information that is available in a way that maximizes the information used, by using hidden Markov models (HMMs), resulting in a change in algorithm runtime complexity from exponential to log-linear, in terms of the number of markers. It also proposes the introduction of inferred haplotypes to further increase the power to assess these unknown variables for pedigrees of related genetically diverse individuals. Modelling consequences of partial genetic information are also treated. Furthermore, genes are not directly affecting traits, but are rather expressed in the environment of and in concordance with other genes. Therefore, significant interactions can be expected within genes, where some combination of genetic variation gives a pronounced, or even opposite, effect, compared to when occurring separately. This thesis addresses how to perform efficient scans for multiple interacting loci, as well as how to derive highly accurate empirical significance tests in these settings. This is done by analyzing the mathematical properties of the objective function describing the quality of model fits, and reformulating it through a simple transformation. Combined with the presented prototype of a problem-solving environment, these developments can make multi-dimensional searches for QTL routine, allowing the pursuit of new biological insight. / eSSENCE

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.

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.

Numerical approximations using high order finite differences on summation-byparts (SBP) form are investigated for discontinuous and fully nonlinear systems of partial differential equations. Stability and conservation properties of the approximations are obtained through a weak imposition of interface and boundary conditions with the simultaneous-approximation-term (SAT) technique. The SBP-SAT approximations replicate the continuous integration by parts rule. From this property, well-posedness and integral properties of the continuous problem are mimicked, and energy estimates leading to stability are obtained. The first part of the thesis focuses on the simulations of discontinuous linear advection problems. An artificial interface is introduced, separating parts of the spatial domain characterized by different wave speeds. A set of flexible stability conditions at the interface are derived, which can be adapted to yield conservative or non-conservative approximations. This model can be interpreted as a simplified version of nonlinear problems involving jumps at shocks, or as a prototypical of wave propagation through different materials. In the second part of the thesis, the vorticity/stream function formulation of the nonlinear momentum equation for an incompressible inviscid fluid is considered. SBP operators are used to derive a new Arakawa-like Jacobian with mimetic properties by combining different consistent approximations of the convection terms. Energy and enstrophy conservation is obtained for periodic problems using schemes with arbitrarily high order of accuracy. These properties are crucial for long-term numerical calculations in climate and weather forecasts or ocean circulation predictions. The third and final contribution of the thesis is dedicated to the incompressible Navier-Stokes problem. First, different completely general formulations of energy bounding boundary conditions are derived for the nonlinear equations. The boundary conditions can be used at both far field and solid wall boundaries. The discretisation in time and space with weakly imposed initial and boundary conditions using the SBP-SAT framework is proved to be stable and the divergence free condition is approximated with the design order of the scheme. Next, the same formulations are considered in a linearised setting, whereupon the spectra associated with the initial boundary value problem and its SBP-SAT discretisation are derived using the Laplace-Fourier technique. The influence of different boundary conditions on the spectrum and in particular the convergence to steady state is studied. / Numeriska approximationer av ekvationer som styr fysikaliska lagar är avgörande i många tillämpningar. Förutom en matematisk modell som kan fånga huvuddragen i ett verkligt problem är det nödvändigt att kunna utföra tillförlitliga simuleringar. Denna avhandling behandlar numeriska approximationer som med hög noggrannhet bevarar både rent matematiska aspekter av ekvationerna så väl som viktiga egenskaper hos modellen. Dessutom ges särskild uppmärksamhet åt modeller med diskontinuiteter och icke-linjära beteenden. Den första delen av avhandlingen handlar om diskontinuerliga problem. Det fysiska rummet kan ha olika egenskaper i olika regioner, något som kan resultera i instabila lösningar. Tillvägagångssättet består av att införa artificiella gränssnitt som skiljer dessa regioneråt. På detta sätt kan varje region behandlas separat, men på liknande sätt. Exempel på naturliga tillämningsområden är vågutbredning genom olika material och jordbävningssimuleringar. I den andra delen av avhandlingen visar vi att om den numeriska approximationen imiterar partiell integration, då följer också de väsentliga egenskaperna hos modellen på ett naturligt sätt. Att fysikaliska egenskaper bevaras är nödvändigt för att bibehålla stabilitet under långa simuleringstider för bland annat geofysiska problem. Den sista delen av avhandlingen är ägnasåt en av de mest använda modellerna inom strömningsmekanik, nämligen Navier-Stokes ekvationer. Studien fokuserar på härledningen av randvillkor som garanterar att lösningen inte växer på ett oförutsett och okontrollerat vis. Slutligen visas att de härledda randvillkoren på ett korrekt och noggrant sätt återskapar den dissipativa mekanism som ger upphov till jämviktstillstånd.

Prioriteringsreglerna är en överenskommen konvention som beskriver räkneoperationers ordningsföljd. Reglerna har visat sig vara något som ställer till problem för elever. I många engelskspråkiga länder används så kallade minnesregler för att komma ihåg prioriteringsreglerna. Minnesregler används för att komma ihåg information med hjälp av akronymer. Trots att detta används frekvent i många läromedel har vetenskaplig forskning visat att det kan medföra missuppfattningar. Syftet med studien är att undersöka hur matematdidaktisk forskning beskriver elevers tillvägagångssätt vid beräkning av numeriska uttryck där prioriteringsregler behöver tillämpas. Denna litteraturstudie baseras på databassökning efter vetenskapliga artiklar. Forskningen som analyserats visar att elever upplever svårigheter vid beräkningar av numeriska uttryck där flera operationer förekommer. Forskning har även visat att det finns missuppfattningar bland elever vad gäller prioriteringsregler och struktur. Det kan beskrivas som att elever missbrukar, missuppfattar, ignorerar, glömmer bort eller inte har kunskap om de regler och konventioner som grundar strukturen för numeriska uttryck.