[ Moved from MW HS 2235 ]
The analysis and comparison of dynamic objects and deforming shapes is important in many real-world applications. Examples include wildfire front-tracking problems, impulse propagation in cardiac tissues, tumor growth, oil reservoir and spill simulations, and pollutant plume dispersion, just to name a few. There are several difficulties that can make the analysis a daunting task and hence need to be addressed: 1) the problem is subjected to uncertainty in the location of structures due to numerical errors, measurement noise, and/or intrinsic variations in the system; 2) strong shape deformations and topological changes may not be well captured at all scales; and 3) the notion of distance or similarity between objects can be characterized in various ways.
This situation has fostered a recent body of work focused on both analytical and computational developments in metric spaces. As an example, the Wasserstein metric has become an increasingly popular tool in such diverse fields as image processing, optimization, neural networks, seismic imaging, and numerical conservation laws. It opens up promising avenues for uncertainty quantification, Bayesian inference and data assimilation, where robust comparisons and mappings between different probability measures are often needed.
This MS will review recent advances, applications and remaining challenges of tailored metric spaces and similarity measures for structure-sensitive uncertainty quantification and inference problems.
Model misspecification and transport-Lagrangian distances in seismic inversion
Andrea Scarinci | Massachusets Institute of Technology (MIT) | United States
The Wasserstein distance has recently emerged as a powerful metric to assess similarity between functional data (i.e., curves, surfaces, or any type of data where each sample can be considered a function). In this talk, we present an application of a Wasserstein-derived metric, known as Transport-Lagrangian distance (TL-p), in the context of full waveform inversion (FWI). The objective is to infer the characteristics of an earthquake and the nearby earth structure, by solving an inverse problem based on recorded seismograms (i.e., waveform traces). Given the complexity of the Earth’s physics and inherent computational difficulties in specifying an appropriate model for inference, we propose a Bayesian framework where the TL-p distance allows for more robust uncertainty quantification in the presence of model misspecification. In fact, we argue that this kind of distance can help discern how much of the model-data misfit is meaningful in terms of model calibration (inference under a well-specified model) and how much is instead due to misrepresentation of the physics (inference under a misspecified model). Computational and statistical aspects of the TL-p distance, including its integration into a Bayesian inference framework, will also be discussed.
Sensitivity Analysis in Wildland Fire Modeling for Front Data Assimilation
Mélanie C. Rochoux | CERFACS | France
At regional scales, wildland fire spread models represent the fire as a propagating front.
However, they are subject to multiple uncertainties. First, there is a modeling challenge
associated with providing accurate mathematical representations of the physical processes, in
particular the interactions between the fire and the atmosphere. One way to better represent
near-surface wind conditions is to couple the wildland fire spread model with an atmospheric
model. Second, there is a data challenge associated with providing accurate estimates of the
initial fire state and of the physical parameters. One way to reduce parametric uncertainties is
to integrate fire modeling and fire sensing technologies using data assimilation. A front data
assimilation method based on the Chan-Vese contour fitting functional has been recently
designed and evaluated in wildland fire problems for state and parameter estimation. A shape
similarity measure has been introduced within an ensemble Kalman filter to measure position
errors between simulated and observed fronts. To ensure a meaningful feedback is achieved,
it is of primary importance to analyze the sensitivity of the shape similarity measure to changes
in the model parameters. This talk will present the shape similarity measure and its application
to the coupled fire-atmosphere model ForeFire-MesoNH to identify which are the most influent
parameters to include in the estimation process.
Wasserstein Convergence of Finite Volume Schemes for Hyperbolic Conservation Laws
Kjetil Lye | ETH Zurich | Switzerland
Hyperbolic conservation laws model an abundance of problems in physics and engineering. However, the question of well-posedness for general systems of conservation law is still open. Newly developed theory, along with strong numerical evidence, suggest that the framework of statistical solutions is a suitable framework for this class of equations. The relevant topology for statistical solutions is metrized by the Wasserstein distance. In this talk, we review the theory of statistical solutions for hyperbolic conservation laws. We furthermore show convergence in the topology metrized by the Wasserstein distance for conservation laws, given some mild assumptions the scaling of the structure functions, which are verified using numerical experiments. In the case of a scalar equation, we prove convergence in the Wasserstein with a rate for a large class of initial data.
Using the Wasserstein distance to compare fields of pollutants: application to the radionuclide atmospheric dispersion of the Fukushima-Daiichi accident
Alban Farchi | CEREA (Ecole des Ponts ParisTech and EDF R&D) | France
The verification of simulations against data and the comparison of model simulation of pollutant fields rely on the critical choice of statistical indicators. Most of the scores are based on point-wise, that is, local, value comparison. Such indicators are impacted by the so-called double penalty effect. Typically, a misplaced blob of pollutants will doubly penalise such a score because it is predicted where it should not be and is not predicted where it should be. The effect is acute in plume simulations where the concentrations gradient can be sharp. A non-local metric that would match concentration fields by displacement, for example the Wasserstein distance, would avoid such double penalty.
In this presentation, we show how the Wasserstein distance can be used to compare fields of pollutants. The test case study is the dispersion of cesium-137 after the Fukushima-Daiichi nuclear power plant accident. The Wasserstein distance is used for model-to-model comparison but also for the verification of model simulation against a map of observed deposited cesium-137 over Japan. As hoped for, the Wasserstein distance is less penalising, and yet retains some of the key discriminating properties of the root mean square error indicator.