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In this work, we discuss elasticity equations on a two-dimensional domain with random boundaries and we apply these equations to modelling human corneas. References R. C. Augustyn, D. Nankivil, A. Mohamed, B. Maceo, F. Pierre, and J.-M. Parel. Human ocular biometry. Exp. Eye Res. 102 (2012), pp. 70–75. doi: 10.1016/j.exer.2012.06.009. F. Ballarin, A. Manzoni, G. Rozza, and S. Salsa. Shape optimization by free-form deformation: Existence results and numerical solution for Stokes flows. J. Sci. Comput. 60.3 (2014), pp. 537–563. doi: 10.1007/s10915-013-9807-8. S. C. Brenner and L.-Y. Sung. Linear finite element methods for planar linear elasticity. Math. Comp. 59 (1992), pp. 321–338. doi: 10.2307/2153060. M. C. Delfour and J.-P. Zolesio. Shapes and geometries. Advances in Design and Control. SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898719826. J. Dick. Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Statist. 39.3 (2011), pp. 1372–1398. doi: 10.1214/11-AOS880. J. Dick, F. Y. Kuo, Q. T. Le Gia, and Ch. Schwab. Multilevel higher order QMC Petrov–Galerkin discretization for affine parametric operator equations. SIAM J. Num. Anal. 4.54 (2015), pp. 2541–2568. doi: 10.1137/16M1078690 A. Eilaghi, J. G. Flanagan, I. Tertinegg, C. A. Simmons, G. W. Brodland, and C. R. Ethier. Biaxial testing of human sclera. J. Biomech. 43 (2010), pp. 1696–1701. doi: 10.1016/j.jbiomech.2010.02.031. U. Fares, A. M Otri, M. A. Al-Aqaba, and H. S. Dua. Correlation of central and peripheral corneal thickness in healthy corneas. Cont. Lens Anterior Eye 35 (2012), pp. 39–45. doi: 10.1016/j.clae.2011.07.004. R. N. Gantner and Ch. Schwab. Computational higher order quasi-Monte Carlo integration. Monte Carlo and quasi-Monte Carlo methods. Vol. 163. Springer Proc. Math. Stat. Springer, 2016, pp. 271–288. doi: 10.1007/978-3-319-33507-0_12. on p. C129). H. Harbrecht. Second moment analysis for Robin boundary value problems on random d

Inference for Bayesian models often require one to simulate from some non-standard multivariate probability distributions. In the first part of the thesis, we successfully simulate exactly from certain Bayesian posteriors (the Tobit, the constrained linear regression, smoothing spline, and the Lasso) by applying rejection sampling using exponentially tilted sequential proposal distributions. This technique is typically efficient for posteriors which have the form of truncated multivariate normal/student. In this manner, we are able to simulate exactly from the posterior in hundreds of dimensions, which has until now being unattainable. Due to the curse of dimensionality, these rejection schemes are unfortunately bound to fail as the dimensions of the problems grow. In such cases, one ultimately has to resort to approximate MCMC schemes. It is known that the sampling error of a Markov chain can be a lot easier if we can identify the regeneration times for the Markov chain. In particular, the convergence rate of a geometrically ergodic Markov chain can be estimated if one can identify the underlying regeneration events. While the idea of using regeneration in the error analysis of MCMC is not new, our contribution in the second part of the thesis is to provide simpler estimates of the total variation error, and a new graphical diagnostic with strong theoretical justification. Finally, in the third part of the thesis, we consider the exponentially tilted sequential distributions in part one as proposal distributions for the MCMC samplers in part two. We introduce a novel Reject-Regenerate sampler, which combines the lessons learned about exact sampling and regenerative MCMC into a single framework. The resulting MCMC algorithm is a Markov chain with clearly demarcated regeneration events. Moreover, in the event of a regeneration, the Markov chain achieves a perfect draw with some probability.

Bulk defects in silicon wafers are key contributors to solar cell efficiency loss. The identification and characterization of these defects are critical steps in the process of improving the efficiency and reliability of solar cells. In this study, we present the first successful application of machine learning for extraction of defect parameters from temperature- and injection-dependent lifetime spectroscopy (TIDLS). With approximately half a million simulated TIDLS curves, random forest regressors are trained to obtain the defect’s energy level and both capture cross-sections. The high correlation coefficient between predicted and simulated values highlights the usefulness of this novel approach. With no prior knowledge of the physical model at work, the regressor learns the physical limitations of that model. This work pioneers the use of machine learning for lifetime spectroscopy, bringing the newest prowess of artificial intelligence to material quality inspection. It opens a new era in the area of defect characterization, as it has the potential to overcome the limitations of current methods, such as dealing with the temperature dependence of various electrical parameters and provide new insights in the theory of lifetime spectroscopy.