One-Step Recurrences for Stationary Random Fields on the Sphere

Recurrences for positive definite functions in terms of the space dimension have been used in several fields of applications. Such recurrences typically relate to properties of the system of special functions characterizing the geometry of the underlying space. In the case of the sphere ${\mathbb S}^{d-1} \subset {\mathbb R}^d$ the (strict) positive definiteness of the zonal function $f(\cos \theta)$ is determined by the signs of the coefficients in the expansion of $f$ in terms of the Gegenbauer polynomials $\{C^\lambda_n\}$, with $\lambda=(d-2)/2$. Recent results show that classical differentiation and integration applied to $f$ have positive definiteness preserving properties in this context. However, in these results the space dimension changes in steps of two. This paper develops operators for zonal functions on the sphere which preserve (strict) positive definiteness while moving up and down in the ladder of dimensions by steps of one. These fractional operators are constructed to act appropriately on the Gegenbauer polynomials $\{C^\lambda_n\}$

Development of reduced polynomial chaos-Kriging metamodel for uncertainty quantification of computational aerodynamics

2018 Summer.Includes bibliographical references.Computational fluid dynamics (CFD) simulations are a critical component of the design and development of aerodynamic bodies. However, as engineers attempt to capture more detailed physics, the computational cost of simulations increases. This limits the ability of engineers to use robust or multidisciplinary design methodologies for practical engineering applications because the computational model is too expensive to evaluate for uncertainty quantification studies and off-design performance analysis. Metamodels (surrogate models) are a closed-form mathematical solution fit to only a few simulation responses which can be used to remedy this situation by estimating off-design performance and stochastic responses of the CFD simulation for far less computational cost. The development of a reduced polynomial chaos-Kriging (RPC-K) metamodel is another step towards eliminating simulation gridlock by capturing the relevant physics of the problem in a cheap-to-evaluate metamodel using fewer CFD simulations. The RPC-K metamodel is superior to existing technologies because its model reduction methodology eliminates the design parameters which contribute little variance to the problem before fitting a high-fidelity metamodel to the remaining data. This metamodel can capture non-linear physics due to its inclusion of both the long-range trend information of a polynomial chaos expansion and local variations in the simulation data through Kriging. In this thesis, the RPC-K metamodel is developed, validated on a convection-diffusion-reaction problem, and applied to the NACA 4412 airfoil and aircraft engine nacelle problems. This research demonstrates the metamodel's effectiveness over existing polynomial chaos and Kriging metamodels for aerodynamics applications because of its ability to fit non-linear fluid flows with far fewer CFD simulations. This research will allow aerospace engineers to more effectively take advantage of detailed CFD simulations in the development of next-generation aerodynamic bodies through the use of the RPC-K metamodel to save computational cost

Multi-Resolution Functional ANOVA for Large-Scale, Many-Input Computer Experiments

The Gaussian process is a standard tool for building emulators for both deterministic and stochastic computer experiments. However, application of Gaussian process models is greatly limited in practice, particularly for large-scale and many-input computer experiments that have become typical. We propose a multi-resolution functional ANOVA model as a computationally feasible emulation alternative. More generally, this model can be used for large-scale and many-input non-linear regression problems. An overlapping group lasso approach is used for estimation, ensuring computational feasibility in a large-scale and many-input setting. New results on consistency and inference for the (potentially overlapping) group lasso in a high-dimensional setting are developed and applied to the proposed multi-resolution functional ANOVA model. Importantly, these results allow us to quantify the uncertainty in our predictions. Numerical examples demonstrate that the proposed model enjoys marked computational advantages. Data capabilities, both in terms of sample size and dimension, meet or exceed best available emulation tools while meeting or exceeding emulation accuracy

Multi-Fidelity Cost-Aware Bayesian Optimization

Bayesian optimization (BO) is increasingly employed in critical applications such as materials design and drug discovery. An increasingly popular strategy in BO is to forgo the sole reliance on high-fidelity data and instead use an ensemble of information sources which provide inexpensive low-fidelity data. The overall premise of this strategy is to reduce the overall sampling costs by querying inexpensive low-fidelity sources whose data are correlated with high-fidelity samples. Here, we propose a multi-fidelity cost-aware BO framework that dramatically outperforms the state-of-the-art technologies in terms of efficiency, consistency, and robustness. We demonstrate the advantages of our framework on analytic and engineering problems and argue that these benefits stem from our two main contributions: (1) we develop a novel acquisition function for multi-fidelity cost-aware BO that safeguards the convergence against the biases of low-fidelity data, and (2) we tailor a newly developed emulator for multi-fidelity BO which enables us to not only simultaneously learn from an ensemble of multi-fidelity datasets, but also identify the severely biased low-fidelity sources that should be excluded from BO

Deriving probabilistic short-range forecasts from a deterministic high-resolution model

In order to take full advantage of short-range forecasts from deterministic high-resolution NWP models, the direct model output must be addressed in a probabilistic framework. A promising approach is mesoscale ensemble prediction. However, its operational use is still hampered by conceptual deficiencies and large computational costs. This study tackles two relevant issues: (1) the representation of model-related forecast uncertainty in mesoscale ensemble prediction systems and (2) the development of post-processing procedures that retrieve additional probabilistic information from a single model simulation. Special emphasis is laid on mesoscale forecast uncertainty of summer precipitation and 2m-temperature in Europe. Source of forecast guidance is the deterministic high-resolution model Lokal-Modell (LM) of the German Weather Service. This study gains more insight into the effect and usefulness of stochastic parametrisation schemes in the representation of short-range forecast uncertainty. A stochastic parametrisation scheme is implemented into the LM in an attempt to simulate the stochastic effect of sub-grid scale processes. Experimental ensembles show that the scheme has a substantial effect on the forecast of precipitation amount. However, objective verification reveals that the ensemble does not attain better forecast goodness than a single LM simulation. Urgent issues for future research are identified. In the context of statistical post-processing, two schemes are designed: the neighbourhood method and wavelet smoothing. Both approaches fall under the framework of estimating a large array of statistical parameters on the basis of a single realisation on each parameter. The neighbourhood method is based on the notion of spatio-temporal ergodicity including explicit corrections for enhanced predictability from topographic forcing. The neighbourhood method derives estimates of quantiles, exceedance probabilities and expected values at each grid point of the LM. If the post-processed precipitation forecast is formulated in terms of probabilities or quantiles, it attains clear superiority in comparison to the raw model output. Wavelet smoothing originates from the field of image denoising and includes concepts of multiresolution analysis and non-parametric regression. In this study, the method is used to produce estimates of the expected value, but it may be easily extended to the additional estimation of exceedance probabilities. Wavelet smoothing is not only computationally more efficient than the neighbourhood method, but automatically adapts the amount of spatial smoothing to local properties of the underlying data. The method apparently detects deterministically predictable temperature patterns on the basis of statistical guidance only