On September 26th, 2018, Zaid Sawlan successfully defended
his PhD thesis entitled "Statistical analysis and Bayesian methods for
fatigue life prediction and inverse problems in linear time
dependent PDEs with uncertainties"
Committee Members:
Prof. Marco Scavino, UdelaR, Montevideo, Uruguay
Prof. Serge Prudhomme, Poly Montreal, Canada
Prof. Olivier LeMaitre, LIMSI, Paris, France
Prof. Haavard Rue, KAUST
Prof. Slim Alouini, KAUST
Prof. Fabio Nobile, EPFL, Lausanne, Switzerland External Examiner
Prof. Raul Tempone (KAUST, CEMSE, AMCS) Thesis Advisor
Abstract:
This work employs
statistical and Bayesian techniques to analyze mathematical forward
models with several sources of uncertainty. The forward models usually
arise from phenomenological and physical phenomena and are expressed
through regression-based models or partial differential equations (PDEs)
associated with uncertain parameters and input data. One of the
critical challenges in real-world applications is, for any proposed
model, to estimate and quantify uncertainties of its unknown parameters
using the available observations. To this purpose, methods based on the
likelihood function, and Bayesian techniques constitute the two main
statistical inferential approaches considered here, the latter providing
a full probability distribution for the parameters of interest, named
the posterior distribution. In the classical approach, bootstrap
confidence intervals procedures are used to assess the variability of
the estimates robustly.
Two problems are
studied in this thesis. The first problem is the prediction of fatigue
life of metallic specimens. The second part is related to inverse
problems in linear PDEs. Both problems require the inference of unknown
parameters given some measurements. We first estimate the parameters by
means of maximum likelihood approach. Next, we seek a more comprehensive
Bayesian inference using analytical asymptotic approximations, such as
those based on the Laplace method, or computational techniques, such as
Markov chain Monte Carlo (MCMC) methods, or combinations of the
mentioned approaches. Each problem has also its own difficulties and
challenges that need to be addressed.
In the fatigue
life prediction, there are several plausible probabilistic
stress-lifetime (S-N) models. These models are calibrated given uniaxial
fatigue experiments. To generate accurate fatigue life predictions,
competing S-N models are ranked according to several classical
information-based measures, such as the Akaike information criterion
(AIC). A different set of predictive information criteria, relying on
cross-validation techniques, is then used to compare the candidate
Bayesian models. Moreover, we propose a spatial stochastic model to
generalize S-N models to fatigue crack initiation in general geometries.
The model is based on a spatial Poisson process with intensity function
that combines the S-N curves with an averaged effective stress that is
computed on the surface of the specimens from the solution of the linear
elasticity equations.
For the inverse
problems in linear PDEs, we assume that, besides the unknown physical
parameters, the boundary conditions are unknown exactly. However, noisy
measurements of the boundary conditions are available. Here, we develop a
novel marginalization method using a hierarchical Bayesian framework.
This method accounts for uncertainties in the boundary conditions and
therefore reduces the bias error in the estimated parameters. We apply
the marginalization technique to the real-world problem of estimating
thermal properties of building walls. Furthermore, we generalize the
marginalization technique to a sequential framework by deriving and
implementing an ensemble marginalized Kalman filter (EnMKF).
Biography:
Zaid Sawlan is a
Ph.D. candidate in applied mathematics and computational science at King
Abdullah University of Science and Technology (KAUST). He earned his
master degree in Applied mathematics from KAUST in 2012. Before joining
KAUST, he graduated from King Saud University with first class honor in
mathematics. His main research interests are uncertainty quantification,
inverse problems, and data assimilation.