Probabilistic numerics methods are a novel approach to quantifying the approximation errors in numerical computations as probabilistic uncertainties. A recent method that was developed is the Bayesian Finite Element Method, which aims to determine the discretization errors along a coarse mesh probabilistically. This work analyzes the use of priors in this method. It is shown that the priors in the right-hand side or forcing term of the partial differential equation are superior to applying them directly in the solution. It is demonstrated that the maximum log-likelihood is not an appropriate estimator of hyperparameters, and as solutions, hyperparameters are optimized using objective functions to capture the discretization error with the posterior deviation. The advantages of non-stationary priors are studied in order to have standard deviations that show the error along the mesh. Additionally, this work examines how the optimal hyperparameters change for mesh refinements and different arrangements of elements in the same problem. Finally, the research delves into determining the approximate number of samples necessary for ensembling covariance matrices and obtaining similar results.

A common problem in wireless communication is the existence of multipath propagation. This means that a transmitted signal is received multiple times because of reflections caused by the environment. We present two ways of modeling multipath propagation of an acoustic underwater signal. We discretise these models to solve them numerically. During the solving process, we are presented with inconsistent, overdetermined systems of linear equations. We investigate two methods to go about these systems: the ordinary least squares method and the total least squares method. We reconstruct a signal using both of these methods and compare their results. The method of least squares reconstructs the signal moderately well. For the total least squares method this is not the case. It turns out that it is not straightforward to formulate a total least squares problem in the corresponding model. We suspect that, in part, this is why the signal reconstruction does not work well for the total least squares method.

Integrated circuits are vital in the modern world. Testing these circuits is often a months long process involving measurements at multiple times during long stress tests. In this work, final measurements from such tests are predicted based on early measurements, potentially reducing the time needed for such tests and giving preliminary results. In addition to this problem, research is done into the benefits of model stacking ensambles, and around the performance impact of using a low bit precision. From our experiments, we observe a significant performance improvement when using model stacking variations. We also find that model stacking retains its performance better than all other tested models when using a lower bit precision.

In anti-cancer therapy, ntiangiogenic treatments are applied and take effect on the vascularization of tissue. To evaluate the efficacy of treatments, we adopt two methods to solve the physiological pharmacokinetic model’s parameter estimation problem, providing discrete, partial, and noisy observations of stochastic differential equations. One is to compute the exact likelihood using the Kalman filter recursion and implement numerical maximization. The other is a novel Markov Chain Monte Carlo algorithm to estimate parameters using guided proposals in a Bayesian setup, namely Backward Filtering with Forward Guiding algorithm. The identifiability of the model and parameters are established before parametric inference. We extend the BFFG algorithm to include an automatic optimal kernel finding scheme for the Metropolis-Hastings-within-Gibbs sampler. In comparison, a Conjugate Gradient algorithm is applied when employing the maximum likelihood method. Besides performing parameter estimation via different methods separately, joint estimation is performed using the Bayesian approach. After that, time delay and Arterial Input Function in the statistical model are estimated via change point detection and piecewise inference. We illustrate the goodness-of-fit of estimates and advantages of the bayesian approach towards the maximum likelihood method.