# Browsing by Subject "Measure theory"

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Item Adaptive measure-theoretic parameter estimation for coastal ocean modeling(2015-08) Graham, Lindley Christin; Dawson, Clinton N.; Butler, Troy; Gamba, Irene; Ghattas, Omar; Moser, RobertShow more Since Hurricane Katrina (2005), there has been a marked increase in the quantity of field observations gathered during and after hurricanes. There has also been an increased effort to improve our ability to model hurricanes and other coastal ocean phenomena. The majority of death and destruction due to a hurricane is from storm surge. The primary controlling factor in storm surge is the balance between the surface stress due to the wind and bottom stress. Manning's formula can be used to model the bottom stress; the formula includes the Manning's n coefficient which accounts for momentum loss due to bottom roughness and is a spatially dependent field. It is impractical to measure Manning's n over large physical domains. Instead, given a computational storm surge model and a set of model observations, one may formulate and solve an inverse problem to determine probable Manning's n fields using observational data, which in turn can be used for predictive simulations. On land, Manning's n may be inferred from land cover classification maps. We leverage existing land cover classification data to determine the spatial distribution of land cover classifications which we consider certain. These classifications can be used to obtain a parameterized mesoscale representation of the Manning's n field. We seek to estimate the Manning's n coefficients for this parameterized field. The inverse problem we solve is formulated using a measure-theoretic approach; using the ADvanced CIRCulation model for coastal and estuarine waters as the forward model of storm surge. The uncertainty in observational data is described as a probability measure on the data space. The solution to the inverse problem is a non-parametric probability measure on the parameter space. The goal is to use this solution in order to measure the probability of arbitrary events in the parameter space. In the cases studied here the dimension of the data space is smaller than the dimension of the parameter space. Thus, the inverse of a fixed datum is generally a set of values in parameter space. The advantage of using the measure-theoretic approach is that it preserves the geometric relation between the data space and the parameter space within the probability measure. Solving an inverse problem often involves the exploration of a high-dimensional parameter space requiring numerous expensive forward model solves. We use adaptive algorithms for solving the stochastic inverse problem to reduce error in the estimated probability of implicitly defined parameter events while minimizing the number of forward model solves.Show more Item Data-driven uncertainty quantification for predictive subsurface flow and transport modeling(2019-02-06) He, Jiachuan; Dawson, Clinton N.; Landis, Chad; Bui, Tan; Ghattas, OmarShow more Specification of hydraulic conductivity as a model parameter in groundwater flow and transport equations is an essential step in predictive simulations. It is often infeasible in practice to characterize this model parameter at all points in space due to complex hydrogeological environments leading to significant parameter uncertainties. Quantifying these uncertainties requires the formulation and solution of an inverse problem using data corresponding to observable model responses. Several types of inverse problems may be formulated under various physical and statistical assumptions on the model parameters, model response, and the data. Solutions to most types of inverse problems require large numbers of model evaluations. In this study, we incorporate the use of surrogate models based on support vector machines to increase the number of samples used in approximating a solution to an inverse problem at a relatively low computational cost. To test the global capabilities of this type of surrogate model for quantifying uncertainties, we use a framework for constructing pullback and push-forward probability measures to study the data-to-parameter-to-prediction propagation of uncertainties under minimal statistical assumptions. Additionally, we demonstrate that it is possible to build a support vector machine using relatively low-dimensional representations of the hydraulic conductivity to propagate distributions. The numerical examples further demonstrate that we can make reliable probabilistic predictions of contaminant concentration at spatial locations corresponding to data not used in the solution to the inverse problem. This dissertation is based on the article entitled Data-driven uncertainty quantification for predictive flow and transport modeling using support vector machines by Jiachuan He, Steven Mattis, Troy Butler and Clint Dawson [32]. This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-SC0009286 as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center.Show more