arXiv subjects

Steven Mattis

Publications and source records attributed to Steven Mattis.

5 recordsLinked to original sources

arXiv2020

Learning Quantities of Interest from Dynamical Systems for Observation-Consistent Inversion

Dynamical systems arise in a wide variety of mathematical models from science and engineering. A common challenge is to quantify uncertainties on model inputs (parameters) that correspond to a quantitative characterization of uncertainties on observable Quantities of Interest (QoI). To this end, we consider a stochastic inverse problem (SIP) with a solution described by a pullback probability measure. We call this an observation-consistent solution, as its subsequent push-forward through the QoI map matches the observed probability distribution on model outputs. A distinction is made between QoI useful for solving the SIP and arbitrary model output data. In dynamical systems, model output data are often given as a series of state variable responses recorded over a particular time window. Consequently, the dimension of output data can easily exceed $\mathcal{O}(1E4)$ or more due to the frequency of observations, and the correct choice or construction of a QoI from this data is not self-evident. We present a new framework, Learning Uncertain Quantities (LUQ), that facilitates the tractable solution of SIPs for dynamical systems. Given ensembles of predicted (simulated) time series and (noisy) observed data, LUQ provides routines for filtering data, unsupervised learning of the underlying dynamics, classifying observations, and feature extraction to learn the QoI map. Subsequently, time series data are transformed into samples of the underlying predicted and observed distributions associated with the QoI so that solutions to the SIP are computable. Following the introduction and demonstration of LUQ, numerical results from several SIPs are presented for a variety of dynamical systems arising in the life and physical sciences. For scientific reproducibility, we provide links to our Python implementation of LUQ and to all data and scripts required to reproduce the results in this manuscript.

math.NA↗

arXiv2019

Bayesian calibration and sensitivity analysis for a karst aquifer model using active subspaces

In this article, we perform a parameter study for a recently developed karst hydrological model. The study consists of a high-dimensional Bayesian inverse problem and a global sensitivity analysis. For the first time in karst hydrology, we use the active subspace method to find directions in the parameter space that dominate the Bayesian update from the prior to the posterior distribution in order to effectively reduce the dimension of the problem and for computational efficiency. Additionally, the calculated active subspace can be exploited to construct sensitivity metrics on each of the individual parameters and be used to construct a natural model surrogate. The model consists of 21 parameters to reproduce the hydrological behavior of spring discharge in a karst aquifer located in the Kerschbaum spring recharge area at Waidhofen a.d. Ybbs in Austria. The experimental spatial and time series data for the inference process were collected by the water works in Waidhofen. We show that this case study has implicit low-dimensionality, and we run an adjusted Markov chain Monte Carlo algorithm in a low-dimensional subspace to construct samples of the posterior distribution. The results are visualized and verified by plots of the posterior's push-forward distribution displaying the uncertainty in predicting discharge values due to the experimental noise in the data. Finally, a discussion provides hydrological interpretation of these results for the Kerschbaum area.

stat.CO↗

arXiv2018

On the relation between parameters and discharge data for a lumped karst aquifer model

Hydrological models of karst aquifers are often semi-distributed, and physical processes such as infiltration and spring discharge generation are described in a lumped way. Several works have previously addressed the problems associated with the calibration of such models, highlighting in particular the issue of model parameter estimation and model equifinality. In this work, we investigate the problem of model calibration using the active subspace (AS) method, a novel tool for model parameter dimension reduction. We apply the method to a newly proposed hydrological model for karst aquifers, LuKARS, to investigate if the AS framework identifies catchment-specific characteristics or if the results only depend on the chosen model structure. Therefore, we consider four different case studies, three synthetic and one real case (Kerschbaum springshed in Waidhofen a.d. Ybbs, Austria), with varying hydrotope distributions and properties. We find that both the hydrotope area coverage and the catchment characteristics have major impacts on parameter sensitivities. While model parameters are similarly informed in scenarios with less varying catchment characteristics, we find significant differences in parameter sensitivities when the applied hydrotopes were different from each other. Our results show that the AS method can be used to investigate the relation between the model structure, the area of a hydrotope, the physical properties of a catchment and the discharge data. Finally, we successfully effectively reduce the parameter dimensions of the LuKARS model for the Kerschbaum case study using the AS method. The model with reduced parameter dimensions is able to reproduce the observed impacts of land use changes in the Kerschbaum springshed, highlighting the robustness of the hydrotope-based modeling approach of LuKARS and its applicability for land use change impact studies in karstic systems.

physics.geo-ph↗

arXiv2018

Goal-oriented adaptive surrogate construction for stochastic inversion

Stochastic inverse problems are generally solved by some form of finite sampling of a space of uncertain parameters. For computationally expensive models, surrogate response surfaces are often employed to increase the number of samples used in approximating the solution. The result is generally a trade off in errors where the stochastic error is reduced at the cost of an increase in deterministic/discretization errors in the evaluation of the surrogate. Such stochastic errors pollute predictions based on the stochastic inverse. In this work, we formulate a method for adaptively creating a special class of surrogate response surfaces with this stochastic error in mind. Adjoint techniques are used to enhance the local approximation properties of the surrogate allowing the construction of a higher-level enhanced surrogate. Using these two levels of surrogates, appropriately derived local error indicators are computed and used to guide refinement of both levels of the surrogates. Three types of refinement strategies are presented and combined in an iterative adaptive surrogate construction algorithm. Numerical examples, including a complex vibroacoustics application, demonstrate how this adaptive strategy allows for accurate predictions under uncertainty for a much smaller computational cost than uniform refinement.

math.NA↗

arXiv2018

Efficient parameter estimation for a methane hydrate model with active subspaces

Methane gas hydrates have increasingly become a topic of interest because of their potential as a future energy resource. There are significant economical and environmental risks associated with extraction from hydrate reservoirs, so a variety of multiphysics models have been developed to analyze prospective risks and benefits. These models generally have a large number of empirical parameters which are not known a priori. Traditional optimization-based parameter estimation frameworks may be ill-posed or computationally prohibitive. Bayesian inference methods have increasingly been found effective for estimating parameters in complex geophysical systems. These methods often are not viable in cases of computationally expensive models and high-dimensional parameter spaces. Recently, methods have been developed to effectively reduce the dimension of Bayesian inverse problems by identifying low-dimensional structures that are most informed by data. Active subspaces is one of the most generally applicable methods of performing this dimension reduction. In this paper, Bayesian inference of the parameters of a state-of-the-art mathematical model for methane hydrates based on experimental data from a triaxial compression test with gas hydrate-bearing sand is performed in an efficient way by utilizing active subspaces. Active subspaces are used to identify low-dimensional structure in the parameter space which is exploited by generating a cheap regression-based surrogate model and implementing a modified Markov chain Monte Carlo algorithm. Posterior densities having means that match the experimental data are approximated in a computationally efficient way.

math.NA↗