# Browsing by Subject "Uncertainty quantification"

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Item Open Access ADAPTIVE LOCAL REDUCED BASIS METHOD FOR RISK-AVERSE PDE CONSTRAINED OPTIMIZATION AND INVERSE PROBLEMS(2018) Zou, ZilongMany physical systems are modeled using partial dierential equations (PDEs) with uncertain or random inputs. For such systems, naively propagating a xed number of samples of the input probability law (or an approximation thereof) through the PDE is often inadequate to accurately quantify the risk associated with critical system responses. In addition, to manage the risk associated with system response and devise risk-averse controls for such PDEs, one must obtain the numerical solution of a risk-averse PDE-constrained optimization problem, which requires substantial computational eorts resulting from the discretization of the underlying PDE in both the physical and stochastic dimensions.

Bayesian Inverse problem, where unknown system parameters need to be inferred from some noisy data of the system response, is another important class of problems that suffer from excessive computational cost due to the discretization of the underlying PDE. To accurately characterize the inverse solution and quantify its uncertainty, tremendous computational eorts are typically required to sample from the posterior distribution of the system parameters given the data. Surrogate approximation of the PDE model is an important technique to expedite the inference process and tractably solve such problems.

In this thesis, we develop a goal-oriented, adaptive sampling and local reduced basis approximation for PDEs with random inputs. The method, which we denote by local RB, determines a set of samples and an associated (implicit) Voronoi partition of the parameter domain on which we build local reduced basis approximations of the PDE solution. The local basis in a Voronoi cell is composed of the solutions at a xed number of closest samples as well as the gradient information in that cell. Thanks to the local nature of the method, computational cost of the approximation does not increase as more samples are included in the local RB model. We select the local RB samples in an adaptive and greedy manner using an a posteriori error indicator based on the residual of the approximation.

Additionally, we modify our adaptive sampling process using an error indicator that is specifically targeted for the approximation of coherent risk measures evaluated at quantities of interest depending on PDE solutions. This allow us to tailor our method to efficiently quantify the risk associated with the system responses. We then combine our local RB method with an inexact trust region method to eciently solve risk-averse optimization problems with PDE constraints. We propose a numerical framework for systematically constructing surrogate models for the trust-region subproblem and the objective function using local RB approximations.

Finally, we extend our local RB method to eciently approximate the Gibbs posterior distribution for inverse problems under uncertainty. The local RB method is employed to construct a cheap surrogate model for the loss function in the Gibbs posterior formula. To improve the accuracy of the surrogate approximation, we adopt a Sequential Monte Carlo framework to guide the progressive and adaptive construction of the local RB surrogate. The resulted method provides subjective and ecient inference of unknown system parameters under general distribution and noise assumptions.

We provide theoretical error bounds for our proposed local RB method and its extensions, and numerically demonstrate the performance of our methods through various examples.

Item Open Access On Uncertainty Quantification for Systems of Computer Models(2017) Kyzyurova, KseniaScientific inquiry about natural phenomena and processes are increasingly relying on the use of computer models as simulators of such processes. The challenge of using computer models for scientific investigation is that they are expensive in terms of computational cost and resources. However, the core methodology of fast statistical emulation (approximation) of a computer model overcomes this computational problem.

Complex phenomena and processes are often described not by a single computer model, but by a system of computer models or simulators. Direct emulation of a system of simulators may be infeasible for computational and logistical reasons.

This thesis proposes a statistical framework for fast emulation of systems of computer models and demonstrates its potential for inferential and predictive scientific goals.

The first chapter of the thesis introduces the Gaussian stochastic process (GaSP) emulator of a single simulator and summarizes ideas and findings in the rest of the thesis. The second chapter investigates the possibility of using independent GaSP emulators of computer models for fast construction of emulators of systems of computer models. The resulting approximation to a system of computer models is called the linked emulator. The third chapter discusses the irrelevance of attempting to model multivariate output of a computer model, for the purpose of emulation of that model. The linear model of coregionalization (LMC) is used to demonstrate this irrelevance, from both a theoretical perspective and from simulation studies. The fourth chapter introduces a framework for calibration of a system of computer models, using its linked emulator. The linked emulator allows for development of independent emulators of submodels on their own separately constructed design spaces, thus leading to effective dimension reduction in explored parameter space. The fifth chapter addresses the use of some non-Gaussian emulators, in particular censored and truncated GaSP emulators. The censored emulator is constructed to appropriately account for zero-inflated output of a computer model, arising when there are large regions of the input space for which the computer model output is zero. The truncated GaSP accommodates computer model output that is constrained to appear in a certain region. The linked emulator, for systems of computer models whose individual subemulators are either censored or truncated, is also presented. The last chapter concludes with an exposition of further research directions based on the ideas explored in the thesis.

The methodology developed in this thesis is illustrated by an application to quantification of the hazard from pyroclastic flow from the Soufri\`{e}re Hills Volcano on the island of Montserrat; a case study on prediction of volcanic ash transport and dispersal from the Eyjafjallaj{\"o}kull volcano, Iceland in April 14-16, 2010; and calibration of a vapour-liquid equilibrium model, a submodel of the Aspen Plus \textcopyright~chemical process software for design and deployment of amine-based $\mathrm{CO_2}$ capture systems.