Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario

A variety of methods is available to quantify uncertainties arising with\-in the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods' advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond

Embedded multilevel Monte Carlo for uncertainty quantification in random domains

The multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for Uncertainty quantification in PDE models. It combines approximations at different levels of accuracy using a hierarchy of meshes in a similar way as multigrid. The generation of body-fitted mesh hierarchies is only possible for simple geometries. On top of that, MLMC for random domains involves the generation of a mesh for every sample. Instead, here we consider the use of embedded methods which make use of simple background meshes of an artificial domain (a bounding-box) for which it is easy to define a mesh hierarchy, thus eliminating the need of body-fitted unstructured meshes, but can produce ill-conditioned discrete problems. To avoid this complication, we consider the recent aggregated finite element method (AgFEM). In particular, we design an embedded MLMC framework for (geometrically and topologically) random domains implicitly defined through a random level-set function, which makes use of a set of hierarchical background meshes and the AgFEM. Performance predictions from existing theory are verified statistically in three numerical experiments, namely the solution of the Poisson equation on a circular domain of random radius, the solution of the Poisson equation on a topologically identical but more complex domain, and the solution of a heat-transfer problem in a domain that has geometric and topological uncertainties. Finally, the use of AgFE is statistically demonstrated to be crucial for complex and uncertain geometries in terms of robustness and computational cost. Date: November 28, 2019

Embedded multilevel monte carlo for uncertainty quantification in random domains

The multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for uncertainty quantification (UQ) in partial differential equation (PDE) models. It combines approximations at different levels of accuracy using a hierarchy of meshes whose generation is only possible for simple geometries. On top of that, MLMC and Monte Carlo (MC) for random domains involve the generation of a mesh for every sample. Here we consider the use of embedded methods which make use of simple background meshes of an artificial domain (a bounding-box) for which it is easy to define a mesh hierarchy. We use the recent aggregated finite element method (AgFEM) method, which permits to avoid ill-conditioning due to small cuts, to design an embedded MLMC (EMLMC) framework for (geometrically and topologically) random domains implicitly defined through a random level-set function. Predictions from existing theory are verified in numerical experiments and the use of AgFEM is statistically demonstrated to be crucial for complex and uncertain geometries in terms of robustness and computational cost.Peer ReviewedPostprint (author's final draft

B-Spline based uncertainty quantification for stochastic analysis

The consideration of uncertainties has become inevitable in state-of-the-art science and technology. Research in the field of uncertainty quantification has gained much importance in the last decades. The main focus of scientists is the identification of uncertain sources, the determination and hierarchization of uncertainties, and the investigation of their influences on system responses. Polynomial chaos expansion, among others, is suitable for this purpose, and has asserted itself as a versatile and powerful tool in various applications. In the last years, its combination with any kind of dimension reduction methods has been intensively pursued, providing support for the processing of high-dimensional input variables up to now. Indeed, this is also referred to as the curse of dimensionality and its abolishment would be considered as a milestone in uncertainty quantification. At this point, the present thesis starts and investigates spline spaces, as a natural extension of polynomials, in the field of uncertainty quantification. The newly developed method 'spline chaos', aims to employ the more complex, but thereby more flexible, structure of splines to counter harder real-world applications where polynomial chaos fails. Ordinarily, the bases of polynomial chaos expansions are orthogonal polynomials, which are replaced by B-spline basis functions in this work. Convergence of the new method is proved and emphasized by numerical examples, which are extended to an accuracy analysis with multi-dimensional input. Moreover, by solving several stochastic differential equations, it is shown that the spline chaos is a generalization of multi-element Legendre chaos and superior to it. Finally, the spline chaos accounts for solving partial differential equations and results in a stochastic Galerkin isogeometric analysis that contributes to the efficient uncertainty quantification of elliptic partial differential equations. A general framework in combination with an a priori error estimation of the expected solution is provided