Stochastic Modeling of Parametric and Model-Form Uncertainties in Computational Mechanics: Applications to Multiscale and Multimodel Predictive Frameworks

Uncertainty quantification (UQ) plays a critical role in computational science and engineering. The representation of uncertainties stands at the core of UQ frameworks and encompasses the modeling of parametric uncertainties --- which are uncertainties affecting parameters in a well-known model --- and model-form uncertainties --- which are uncertainties defined at the operator level. Past contributions in the field have primarily focused on parametric uncertainties in idealized environments involving simple state spaces and index sets. On the other hand, the consideration of model-form uncertainties (beyond model error correction) is still in its infancy. In this context, this dissertation aims to develop stochastic modeling approaches to represent these two forms of uncertainties in multiscale and multimodel settings.

The case of spatially-varying geometrical perturbations on nonregular index sets is first addressed. We propose an information-theoretic formulation where a push-forward map is used to induce bounded variations and the latent Gaussian random field is implicitly defined through a stochastic partial differential equation on the manifold defining the surface of interest. Applications to a gyroid structure and patient-specific brain interfaces are presented. We then address operator learning in a multiscale setting where we propose a data-free training method, applied to Fourier neural operators. We investigate the homogenization of random media defined at microscopic and mesoscopic scales. Next, we develop a Riemannian probabilistic framework to capture operator-form uncertainties in the multimodel setting (i.e., when a family of model candidates is available). The proposed methodology combines a proper-orthogonal-decomposition reduced-order model with Riemannian operators ensuring admissibility in the almost sure sense. The framework exhibits several key advantages, including the ability to generate a model ensemble within the convex hull defined by model proposals and to constrain the mean in the Fréchet sense, as well as ease of implementation. The method is deployed to investigate model-form uncertainties in various molecular dynamics simulations on graphene sheets. We finally propose an extension of this framework to systems described by coupled partial differential equations, with emphasis on the phase-field approach to brittle fracture.