In this thesis, we bring forward the study of the spectral properties of graphs and we extend this theory for chemical hypergraphs, a new class of hypergraphs that model chemical reaction networks.

As a first aspect the thesis treats existence results of the perturbed eigenvalue problem of the 1-Laplace operator. This is done with the aid of a quite general critical point theory results with the genus as topological index. Moreover we show that the eigenvalues of the perturbed 1-Laplace operator converge to the eigenvalues of the unperturebed 1-Laplace operator when the perturbation goes to zero. As a second aspect we treat the eigenvalue problems of the vectorial 1-Laplace operator and the symmetrized 1-Laplace operator. And as a third aspect certain related parabolic problems are considered.

Experimental design can be vital when experiments are resource-exhaustive and time-consuming. In this work, we carry out experimental design in the Bayesian framework. To measure the amount of information that can be extracted from the data in an experiment, we use the expected information gain as the utility function, which specifically is the expected logarithmic ratio between the posterior and prior distributions. Optimizing this utility function enables us to design experiments that yield the most informative data about the model parameters. One of the major difficulties in evaluating the expected information gain is that it naturally involves nested integration over a possibly high dimensional domain. We use the Multilevel Monte Carlo (MLMC) method to accelerate the computation of the nested high dimensional integral. The advantages are twofold. First, MLMC can significantly reduce the cost of the nested integral for a given tolerance, by using an optimal sample distribution among different sample averages of the inner integrals. Second, the MLMC method imposes fewer assumptions, such as the asymptotic concentration of posterior measures, required for instance by the Laplace approximation (LA). We test the MLMC method using two numerical examples. The first example is the design of sensor deployment for a Darcy flow problem governed by a one-dimensional Poisson equation. We place the sensors in the locations where the pressure is measured, and we model the conductivity field as a piecewise constant random vector with two parameters. The second one is chemical Enhanced Oil Recovery (EOR) core flooding experiment assuming homogeneous permeability. We measure the cumulative oil recovery, from a horizontal core flooded by water, surfactant and polymer, for different injection rates. The model parameters consist of the endpoint relative permeabilities, the residual saturations and the relative permeability exponents for the three phases: water, oil and microemulsions. We also compare the performance of the MLMC to the LA and the direct Double Loop Monte Carlo (DLMC). In fact, we show that, in the case of the aforementioned examples, MLMC combined with LA turns to be the best method in terms of computational cost.

Bayesian inference is particularly challenging on hierarchical statistical models as computational complexity becomes a significant issue. Sampling-based methods like the popular Markov Chain Monte Carlo (MCMC) can provide accurate solutions, but they likely suffer a high computational burden. An attractive alternative is the Integrated Nested Laplace Approximations (INLA) approach, which is faster when applied to the broad class of Latent Gaussian Models (LGMs). The method computes fast and empirically accurate deterministic posterior marginal approximations of the model's unknown parameters. In the first part of this thesis, we discuss how to extend the software's applicability to a joint posterior inference by constructing a new class of joint posterior approximations, which also add marginal corrections for location and skewness. As these approximations result from a combination of a Gaussian Copula and internally pre-computed accurate Gaussian Approximations, we name this class Skew Gaussian Copula (SGC). By computing moments and correlation structure of a mixture representation of these distributions, we achieve new fast and accurate deterministic approximations for linear combinations in a subset of the model's latent field. The same mixture approximates a full joint posterior density through a Monte Carlo sampling on the hyperparameter set. We set highly skewed examples based on Poisson and Binomial hierarchical models and verify these new approximations using INLA and MCMC. The new skewness correction from the Skew Gaussian Copula is more consistent with the outcomes provided by the default INLA strategies. In the last part, we propose an extension of the parametric fit employed by the Simplified Laplace Approximation strategy in INLA when approximating posterior marginals. By default, the strategy matches log derivatives from a third-order Taylor expansion of each Laplace Approximation marginal with those derived from Skew Normal distributions. We consider a fourth-order term and adapt an Extended Skew Normal distribution to produce a more accurate approximation fit when skewness is large. We set similarly skewed data simulations with Poisson and Binomial likelihoods and show that the posterior marginal results from the new extended strategy are more accurate and coherent with the MCMC ones than its original version.

A set of crude oil samples with different composition and characteristics is studied by means of Fast Field Cycling (FFC) 1H relaxometry, which probes the distribution of longitudinal relaxation times T1 as a function of the Larmor frequency. Investigation of T1 profiles at different temperatures is able to provide an insight into the dynamics and structural changes of oil components, with our particular interest being the high temperature behaviour of asphaltene. It is well-known that asphaltenes tend to form porous clusters in crude oils, which can cause severe problems for the process of oil extraction. Therefore, FFC experiments are conducted on Stelar Spinmaster FFC2000 in the temperature range 203K < T <443K, where the upper limit of 443K is aimed at approximating the typical maximal in-situ well temperatures. FFC relaxometry data of crude oils at such a high temperature are obtained for the first time with the use of a specially modified NMR probe. Inverse Laplace transformation is applied to the longitudinal agnetization decays, yielding T1 distributions at different frequencies. A comparative analysis of these distributions for different Larmor frequencies and temperatures showed that there is a systematic variation of the frequency dependence of T1 correlating with the asphaltene content in the samples, at temperatures similar to the well conditions.

We propose a simple adjustment to a Wald confidence interval to estimate a linear function of binomial proportions. This method is an extension to the adjusted Wald confidence intervals for proportions and their differences that have recently been proposed.

In this theses, an iterativ linear solver for the non-linear p-Poisson problem is introduced. After the theoretical convergence results some numerical examples of a fully adaptive solver are presented.

Climate change is likely to disrupt the timing of developmental events (phenology) in insect populations in which development time is largely determined by temperature. Shifting phenology puts insects at risk of being exposed to seasonal weather extremes during sensitive life stages and losing synchrony with biotic resources. Additionally, warming may result in loss of developmental synchronization within a population, making it difficult to find mates or mount mass attacks against well-defended resources at low population densities. It is unknown whether genetic evolution of development time can occur rapidly enough to moderate these effects. The work presented here is largely motivated by the need to understand how mountain pine beetle (MPB) populations will respond to climate change. MPB is an important forest pest from both an economic and ecological perspective, because MPB outbreaks often result in massive timber loss. Recent MPB range expansion and increased outbreak frequency have been linked to warming temperatures. We present a novel approach to modeling the evolution of phenology by allowing the parameters of a phenology model to evolve in response to selection on emergence time and density. We also develop a temperature-dependent phenology model for MPB that accounts for multiple types of developmental variation: variation that persists throughout a life stage, random variation, and variation due to the MPB oviposition mechanism. This model is parameterized using MPB development time data from constant temperature laboratory experiments. We use Laplace's method to approximate steady distributions of the evolution model under stable temperatures. Here the mean phenotype allows for parents and offspring to be oviposited at exactly the same time of year in consecutive generations. These results are verified numerically for both MPB and a two-stage model insect. The evolution model is also applied to investigate the evolution of phenology for MPB and the two-stage model insect under warming temperatures. The model predicts that local populations can only adapt to climate change if development time can adapt so that individuals can complete exactly one generation per year and if the rate of temperature change is moderate.

We give an Ansatz to construct pairs of locally homogeneous nearly Kaehler manifolds that are isospectral for the Dirac and the Hodge Laplace operator in dimensions higher than six and investigate the existence of generic isospectral pairs in dimension six.

Since the start of the COVID-19 pandemic in December 2019, much research has been done to develop the spatial-temporal methods to track it and to predict the spread of the virus. In this thesis, a COVID-19 dataset containing the number of biweekly infected cases registered in Ontario since the start of the pandemic to the end of June 2021 is analysed using Bayesian Spatial-temporal models and Area-to-area (Area-to-point) Poisson Kriging models. With the Bayesian models, spatial-temporal effects on infected risk will be checked and ATP Poisson Kriging models will show how the virus spreads over the space and the spatial clustering feature. According to these models, a Shinyapp website https://mujingrui.shinyapps.io/covid19 is developed to present the results.