Browsing by Author "Knio, Omar M"

Polynomial chaos expansions provide an efficient and robust framework to analyze and quantify uncertainty in computational models. This dissertation explores the use of adaptive sparse grids to reduce the computational cost of determining a polynomial model surrogate while examining and implementing new adaptive techniques.

Determination of chaos coefficients using traditional tensor product quadrature suffers the so-called curse of dimensionality, where the number of model evaluations scales exponentially with dimension. Previous work used a sparse Smolyak quadrature to temper this dimensional scaling, and was applied successfully to an expensive Ocean General Circulation Model, HYCOM during the September 2004 passing of Hurricane Ivan through the Gulf of Mexico. Results from this investigation suggested that adaptivity could yield great gains in efficiency. However, efforts at adaptivity are hampered by quadrature accuracy requirements.

We explore the implementation of a novel adaptive strategy to design sparse ensembles of oceanic simulations suitable for constructing polynomial chaos surrogates. We use a recently developed adaptive pseudo-spectral projection (aPSP) algorithm that is based on a direct application of Smolyak's sparse grid formula, and that allows for the use of arbitrary admissible sparse grids. Such a construction ameliorates the severe restrictions posed by insufficient quadrature accuracy. The adaptive algorithm is tested using an existing simulation database of the HYCOM model during Hurricane Ivan. The {\it a priori} tests demonstrate that sparse and adaptive pseudo-spectral constructions lead to substantial savings over isotropic sparse sampling.

In order to provide a finer degree of resolution control along two distinct subsets of model parameters, we investigate two methods to build polynomial approximations. The two approaches are based with pseudo-spectral projection (PSP) methods on adaptively constructed sparse grids. The control of the error along different subsets of parameters may be needed in the case of a model depending on uncertain parameters and deterministic design variables. We first consider a nested approach where an independent adaptive sparse grid pseudo-spectral projection is performed along the first set of directions only, and at each point a sparse grid is constructed adaptively in the second set of directions. We then consider the application of aPSP in the space of all parameters, and introduce directional refinement criteria to provide a tighter control of the projection error along individual dimensions. Specifically, we use a Sobol decomposition of the projection surpluses to tune the sparse grid adaptation. The behavior and performance of the two approaches are compared for a simple two-dimensional test problem and for a shock-tube ignition model involving 22 uncertain parameters and 3 design parameters. The numerical experiments indicate that whereas both methods provide effective means for tuning the quality of the representation along distinct subsets of parameters, adaptive PSP in the global parameter space generally requires fewer model evaluations than the nested approach to achieve similar projection error.

In order to increase efficiency even further, a subsampling technique is developed to allow for local adaptivity within the aPSP algorithm. The local refinement is achieved by exploiting the hierarchical nature of nested quadrature grids to determine regions of estimated convergence. In order to achieve global representations with local refinement, synthesized model data from a lower order projection is used for the final projection. The final subsampled grid was also tested with two more robust, sparse projection techniques including compressed sensing and hybrid least-angle-regression. These methods are evaluated on two sample test functions and then as an {\it a priori} analysis of the HYCOM simulations and the shock-tube ignition model investigated earlier. Small but non-trivial efficiency gains were found in some cases and in others, a large reduction in model evaluations with only a small loss of model fidelity was realized. Further extensions and capabilities are recommended for future investigations.

This dissertation focuses on the development and calibration of reaction models for multilayered nanocomposites. The nanocomposites comprise sputter deposited alternating layers of distinct metallic elements. Specifically, we focus on the equimolar Ni-Al and Zr-Al multilayered systems. Computational models are developed to capture the transient reaction phenomena as well as understand the dependence of reaction properties on the microstructure, composition and geometry of the multilayers. Together with the available experimental data, simulations are used to calibrate the models and enhance the accuracy of their predictions.

Recent modeling efforts for the Ni-Al system have investigated the nature of self-propagating reactions in the multilayers. Model fidelity was enhanced by incorporating melting effects due to aluminum [Besnoin et al. (2002)]. Salloum and Knio formulated a reduced model to mitigate computational costs associated with multi-dimensional reaction simulations [Salloum and Knio (2010a)]. However, exist- ing formulations relied on a single Arrhenius correlation for diffusivity, estimated for the self-propagating reactions, and cannot be used to quantify mixing rates at lower temperatures within reasonable accuracy [Fritz (2011)]. We thus develop a thermal model for a multilayer stack comprising a reactive Ni-Al bilayer (nanocalorimeter) and exploit temperature evolution measurements to calibrate the diffusion parameters associated with solid state mixing (720 K - 860 K) in the bilayer.

The equimolar Zr-Al multilayered system when reacted aerobically is shown to

exhibit slow aerobic oxidation of zirconium (in the intermetallic), sustained for about 2-10 seconds after completion of the formation reaction. In a collaborative effort, we aim to exploit the sustained heat release for bio-agent defeat application. A simplified computational model is developed to capture the extended reaction regime characterized by oxidation of Zr-Al multilayers. Simulations provide insight into the growth phenomenon for the zirconia layer during the oxidation process. It is observed that the growth of zirconia is predominantly governed by surface-reaction. However, once the layer thickens, the growth is controlled by the diffusion of oxygen in zirconia.

A computational model is developed for formation reactions in Zr-Al multilayers. We estimate Arrhenius diffusivity correlations for a low temperature mixing regime characterized by homogeneous ignition in the multilayers, and a high temperature mixing regime characterized by self-propagating reactions in the multilayers. Experimental measurements for temperature and reaction velocity are used for this purpose. Diffusivity estimates for the two regimes are first inferred using regression analysis and full posterior distributions are then estimated for the diffusion parameters using Bayesian statistical approaches. A tight bound on posteriors is observed in the ignition regime whereas estimates for the self-propagating regime exhibit large levels of uncertainty. We further discuss a framework for optimal design of experiments to assess and optimize the utility of a set of experimental measurements for application to reaction models.

This dissertation focuses on acceleration techniques for Uncertainty Quantification (UQ). The manuscript is divided into five chapters. Chapter 1 provides an introduction and a brief summary of Chapters 2, 3, and 4. Chapter 2 introduces a model reduction strategy that is used in the context of elasticity imaging to infer the presence of an inclusion embedded in a soft matrix, mimicking tumors in soft tissues. The method relies on Polynomial Chaos (PC) expansions to build a dictionary of surrogates models, where each surrogate is constructed using a different geometrical configuration of the potential inclusion. A model selection approach is used to discriminate against the different models and eventually select the most appropriate to estimate the likelihood that an inclusion is present in the domain. In Chapter 3, we use a Domain Decomposition (DD) approach to compute the Karhunen-Loeve (KL) modes of a random process through the use of local KL expansions at the subdomain level. Furthermore, we analyze the relationship between the local random variables associated to the local KL expansions and the global random variables associated to the global KL expansions. In Chapter 4, we take advantage of these local random variables and use DD techniques to reduce the computational cost of solving a Stochastic Elliptic Equation (SEE) via a Monte Carlo sampling method. The approach takes advantage of a lower stochastic dimension at the subdomain level to construct a PC expansion of a reduced linear system that is later used to compute samples of the solution. Thus, the approach consists of two main stages: 1) a preprocessing stage in which PC expansions of a condensed problem are computed and 2) a Monte Carlo sampling stage where samples of the solution are computed in order to solve the SEE. Finally, in Chapter 5 some brief concluding remarks are provided.

This work explores stochastic responses of various earth system models to different random sources, using polynomial chaos (PC) approaches. The following earth systems are considered, namely the HYbrid Coordinate Ocean Model (HYCOM, an ocean general circulation model (OGCM)) for the study of ocean circulation in the Gulf of Mexico (GoM); the Unified Wave INterface - Coupled Model (UWIN-CM, a dynamically coupled atmosphere-wave-ocean system) for Hurricane Earl (2010) modeling; and the earthquake seismology model for Bayesian inference of fault plane configurations.

In the OGCM study, we aim at analyzing the combined impact of uncertainties in initial conditions and wind forcing fields on ocean circulation using PC expansions. Empirical Orthogonal Functions (EOF) are used to represent both spatial perturbations of initial condition and space-time wind forcing fields, namely in the form of a superposition of modal components with uniformly distributed random amplitudes. The forward deterministic HYCOM simulations are used to propagate input uncertainties in ocean circulation in the GoM during the 2010 Deepwater Horizon (DWH) oil spill, and to generate a realization ensemble based on which PC surrogate models are constructed for both localized and field quantities of interest (QoIs), focusing specifically on Sea Surface Height (SSH) and Mixed Layer Depth (MLD). These PC surrogate models are constructed using Basis Pursuit DeNoising (BPDN) methodology, and their performance is assessed through various statistical measures. A global sensitivity analysis is then performed to quantify the impact of individual random sources as well as their interactions on ocean circulation. At the basin scale, SSH in the deep GoM is mostly sensitive to initial condition perturbations, while over the shelf it is sensitive to wind forcing perturbations. On the other hand, the basin MLD is almost exclusively sensitive to wind perturbations. For both quantities, the two random sources (initial condition and wind forcing) of uncertainties have limited interactions. Finally, computations indicate that whereas local quantities can exhibit complex behavior that necessitates a large number of realizations to build PC surrogate models, the modal analysis of field sensitivities can be suitably achieved with a moderate size ensemble.

It is noted that HYCOM simulations in the aforementioned OGCM study only focus on the ocean circulation, and ignore the oceanic feedback (e.g. momentum, energy, humidity etc) to the atmosphere. A more elaborated analysis is consequently performed to understand the atmosphere dynamics in a fully-coupled atmosphere-wave-ocean system. In particular, we explore the stochastic evolution of Hurricane Earl (2010) in response to uncertainties stemming from random perturbations in the storm's initial size, strength and rotational stretch. To this end, the UWIN-CM framework is employed as the forecasting system, which is used to propagate input uncertainties and generate a realization ensemble. PC surrogate models for time evolutions of both maximum wind speed and minimum sea level pressure (SLP) are constructed. These PC surrogates provide statistical insights on probability distributions of model responses throughout the simulation time span. Statistical analysis of rapid intensification (RI) process suggests that storms with enhanced initial intensity and counter-clockwise rotation perturbations are more likely to undergo a RI process. In addition, the RI process seems mostly sensitive to the mean wind strength and rotational stretch, rather than storm size and asymmetric wind amplitude perturbations. This is consistent with global sensitivity analysis of PC surrogate models. Finally we combine parametric storm perturbations with global stochastic kinetic energy backscatter (SKEBS) forcing in UWIN-CM simulations, and conclude that whereas the storm track is substantially influenced by global perturbations, it is weakly influenced by the properties of the initial storm.

The PC framework not only provides easy access to traditional statistical insights and global sensitivity indices, but also reduces the computational burden of sampling the system response, as performed for instance in Bayesian inference. These two advantages of PC approaches are well demonstrated in the study of earthquake seismology model response to random fault plane configurations. The PC statistical analysis suggests that the hypocenter location plays a dominant role in earthquake ground motion responses (in terms of peak ground velocities, PGVs), while elliptical patch properties only show secondary influence. In addition, Our PC based Bayesian analysis successfully identified the most `likely' fault plane configuration with respect to the chosen ground motion prediction equation (GMPE) curve, i.e. the hypocenter is more likely to be in the bottom right quadrant of the fault plane and the elliptical patch centers at the bottom left quadrant. To incorporate additional physical restrictions on fault plane configurations, a novel restricted sampling methodology is introduced. The results indicate that the restricted inference is more physically sensible, while retaining plausible consistency with findings from unrestricted inference.