Probabilistic Godunov-type hydrodynamic modelling under multiple uncertainties: robust wavelet-based formulations

Intrusive stochastic Galerkin methods propagate uncertainties in a single model run, eliminating repeated sampling required by conventional Monte Carlo methods. However, an intrusive formulation has yet to be developed for probabilistic hydrodynamic modelling incorporating robust wetting-and-drying and stable friction integration under joint uncertainties in topography, roughness, and inflow. Robustness measures are well-developed in deterministic models, but rely on local, nonlinear operations that can introduce additional stochastic errors that destabilise an intrusive model. This paper formulates an intrusive hydrodynamic model using a multidimensional tensor product of Haar wavelets to capture fine-scale variations in joint probability distributions and extend the validity of robustness measures from the underlying deterministic discretisation. Probabilistic numerical tests are designed to verify intrusive model robustness, and compare accuracy and efficiency against a conventional Monte Carlo approach and two other alternatives: a nonintrusive stochastic collocation formulation sharing the same tensor product wavelet basis, and an intrusive formulation that truncates the basis to gain efficiency under multiple uncertainties. Tests reveal that: (i) a full tensor product basis is required to preserve intrusive model robustness, while the nonintrusive counterpart achieves identically accurate results at a reduced computational cost; and, (ii) Haar wavelets basis requires at least three levels of refinements per uncertainty dimension to reliably capture complex probability distributions. Accompanying model software and simulation data are openly available online

Advanced numerical and statistical techniques to assess erosion and flood risk in coastal zones

Throughout history, coastal zones have been vulnerable to the dual risks of erosion and flooding. With climate change likely to exacerbate these risks in the coming decades, coasts are becoming an ever more critical location on which to apply hydro-morphodynamic models blended with advanced numerical and statistical techniques, to assess risk. We implement a novel depth-averaged hydro-morphodynamic model using a discontinuous Galerkin based finite element discretisation within the coastal ocean model {\em Thetis}. Our model is the first with this discretisation to simulate both bedload and suspended sediment transport, and is validated for test cases in fully wet and wet-dry domains. These test cases show our model is more accurate, efficient and robust than industry-standard models. Additionally, we use our model to implement the first fully flexible and freely available adjoint hydro-morphodynamic model framework which we then successfully use for sensitivity analysis, inversion and calibration of uncertain parameters. Furthermore, we implement the first moving mesh framework with a depth-averaged hydro-morphodynamic model, and show that mesh movement can help resolve the multi-scale issues often present in hydro-morphodynamic problems, improving their accuracy and efficiency. We present the first application of the multilevel Monte Carlo method (MLMC) and multilevel multifidelity Monte Carlo method (MLMF) to industry-standard hydro-morphodynamic models as a tool to quantify uncertainty in erosion and flood risk. We use these methods to estimate expected values and cumulative distributions of variables which are of interest to decision makers. MLMC, and more notably MLMF, significantly reduce computational cost compared to the standard Monte Carlo method whilst retaining the same level of accuracy, enabling in-depth statistical analysis of complex test cases that was previously unfeasible. The comprehensive toolkit of techniques we develop provides a crucial foundation for researchers and stakeholders seeking to assess and mitigate coastal risks in an accurate and efficient manner.Open Acces

Multilevel Monte-Carlo methods applied to the stochastic analysis of aerodynamic problems

This paper demonstrates the capabilities of the Multi-Level Monte Carlo Methods (MLMC) for the stochastic analysis of CFD aeronautical problems with uncertainties. These capabilities are compared with the classical Monte Carlo Methods in terms of accuracy and computational cost through a set of benchmark test cases. The real possibilities of solving CFD aeronautical analysis with uncertainties by using MLMC methods with a reasonable computational cost are demonstrated.Postprint (published version

Uncertainty quantification in littoral erosion

International audienceWe aim at quantifying the impact of flow state uncertainties in lit-toral erosion to provide confidence bounds on deterministic predictions of bottom morphodynamics. Two constructions of the bathymetry standard deviation are discussed. The first construction involves directional quantile-based extreme scenarios using what is known on the flow state Probability Density Function (PDF) from on site observations. We compare this construction to a second cumulative one using the gradient by adjoint of a functional involving the energy of the system. These ingredients are illustrated for two models for the interaction between a soft bed and a flow in a shallow domain. Our aim is to keep the computational complexity comparable to the deterministic simulations taking advantage of what already available in our simulation toolbox

Numerical simulation of the shallow water equations coupled with a precipitation system driven by random forcing

Quantification of flood risk and flood inundation requires accurate numerical simulations, both in terms of the mathematical theory that underpins the methods used and the manner in which the meteorological phenomena that cause flooding are coupled to such systems. Through our research, we have demonstrated how rainfall and infiltration effects can be incorporated into existing flood models in a rigorous and mathematically consistent manner; this approach departs from preceding methods, which neglect terms representing such phenomena in the conservation or balancing of momentum. We demonstrate how the omission of these terms means the solution derived from such models cannot a priori be assumed to be the correct one, which is in contrast to solutions from the extended system we have developed which respect the energetic consistency of the problem. The second issue we address is determining how we can model these meteorological phenomena that lead to flooding, with a specific interest in how existing observation data from rain gauges can be incorporated into our modelling approach. To capture the random nature of the precipitation, we use stochastic processes to model the complex meteorological interactions, and demonstrate how an accurate representation of the precipitation can be built. Given the specific industrial applications we have mind in regards to flood modelling and prediction, there will be a high computational cost associated with any such simulations, and so we consider techniques which can be used to reduce the computational cost whilst maintaining the accuracy of our solutions. Having such an accurate flood model, coupled with a stochastic weather model designed for efficient computational modelling, will enable us to make useful predictions on how future climate change and weather patterns will impact flood risk and flood damage

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

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

A Massively Parallel Implementation of Multilevel Monte Carlo for Finite Element Models

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, combining model computations at different levels to create an accurate estimate. Still, the computational complexity of the resulting method is extremely high, particularly for 3D models, which requires advanced algorithms for the efficient exploitation of High Performance Computing (HPC). In this article we present a new implementation of the MLMC in massively parallel computer architectures, exploiting parallelism within and between each level of the hierarchy. The numerical approximation of the PDE is performed using the finite element method but the algorithm is quite general and could be applied to other discretization methods as well, although the focus is on parallel sampling. The two key ingredients of an efficient parallel implementation are a good processor partition scheme together with a good scheduling algorithm to assign work to different processors. We introduce a multiple partition of the set of processors that permits the simultaneous execution of different levels and we develop a dynamic scheduling algorithm to exploit it. The problem of finding the optimal scheduling of distributed tasks in a parallel computer is an NP-complete problem. We propose and analyze a new greedy scheduling algorithm to assign samples and we show that it is a 2-approximation, which is the best that may be expected under general assumptions. On top of this result we design a distributed memory implementation using the Message Passing Interface (MPI) standard. Finally we present a set of numerical experiments illustrating its scalability properties.Comment: 21 pages, 13 figure