# Browsing by Subject "Dimension reduction"

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Item Essays on data-driven optimization(2019-06-20) Zhao, Long, Ph. D.; Muthuraman, Kumar; Chakrabarti, Deepayan; Tompaidis, Efstathios; Caramanis, ConstantineShow more The estimation of a data matrix contains two parts: the well estimated and the poorly estimated. The latter is usually throwing away because the estimations are off. As argued in this paper, ignoring is the wrong thing to do as the poorly estimated part is orthogonal to the well estimated. I will show how to use such orthogonality information via robust optimization and provide application in portfolio optimization, least-square regression, and dimension reduction. Across a large number of experiments, utilizing the orthogonality information consistently improves the performance.Show more Item Tensor dimensionality reduction and applications(2022-07-19) Jin, Ruhui; Ward, Rachel, 1983-; Krahmer , Felix; Martinsson, Per-Gunnar; Neeman, JosephShow more Dimensionality reduction is a fundamental idea in data science and machine learning. Tensor is ubiquitous in modern data science due to its representation power for complex data settings. In this thesis, we study to efficiently reduce the size of tensor-structured data while preserving the essential information. We generalize classical reduction methods for vectors, matrices, such as random projections and low-rank decompositions, to be suitable for higher-order tensors. Numerical experiments show the proposed reduction algorithms result in significant storage savings and computation speed-ups. As a trade-off, from the theoretical perspective, the approximation errors between the original and reduced data are rigorously analyzed. Finally, these tensor dimension reduction techniques find usages in solving inverse problems, anomaly detection and financial portfolio allocation.Show more Item Uncertainty quantification in stochastic models for extreme loads(2020-01-29) Nguyen, Phong The Truong; Manuel, Lance; Bui-Thanh, Tan; Gilbert, Robert; Kinnas, Spyridon A.; Sela, Polina; Veers, PaulShow more Many response parameters for offshore structures such as wave energy converters (WECs), wind turbines, oil and gas platforms, etc. can be modeled as stochastic processes. The extreme of such a response process over any selected interval of time is a random variable. Having accurate estimates of such extremes during a structure's life is crucial in structural design, but there are challenges in their estimation due to various sources of uncertainty. These include uncertainty from environmental conditions or the climate/weather as well as from short-term simulations of these stochastic processes at appropriate time and space resolution. Together, these uncertainty sources make up a high-dimension vector of random variables (that can be on the order of thousands). Many offshore structures must withstand many years of exposure and use return periods for design that are on order of 50 to 100 years. The focus of this study is on rare events or response levels that are associated with very low probabilities of exceedance (e.g., on the order of $10⁻⁶ over a typical 1-hour duration). Time-domain simulations of dynamic offshore structures can be computationally expensive even for a single simulation. Various approaches can be adopted in practice to account for uncertainties in extreme response prediction. Monte Carlo Simulation (MCS) is the most common for exhaustive prediction of the response for all conditions. Since MCS can be computationally very demanding, the development of efficient surrogate models is presented to more efficiently deal with these uncertainties. A proposed method, in this study, is based on the use of an ensemble of multiple polynomial chaos expansion (M-PCE) surrogate models to propagate the uncertainty from the environment through the stochastic input simulation to eventual design load prediction. In particular, each PCE model in the ensemble provides an approximate relationship between the structural response and the underlying environmental variables, while variability in the short-term simulations is accounted for by the multiple surrogates. M-PCE helps overcome the curse of dimensionality since, instead of dealing with development of a high-dimensional surrogate model, the M-PCE ensemble includes multiple low-dimensional PCE models, each defined in terms of only the long-term environmental variables, which are of low dimension. It is found that the M-PCE ensemble can efficiently predict long-term extreme loads associated at exceedance probability levels (in 1 hour) of $10⁻⁵ or higher. Next, by considering MCS and M-PCE as high-fidelity and low-fidelity models, respectively, this study proposes a bi-fidelity approach that combines M-PCE and MCS outputs so as to control, or even eliminate bias introduced by the use of the M-PCE ensemble alone. The approach takes advantage of the robustness of MCS on the one hand and the efficiency of M-PCE model on the other. The key idea is that many of the model simulations are carried out using the inexpensive M-PCE ensemble while a very small number of simulations use the costly high-fidelity model. In this way, the new method significantly enhances the efficiency of MCS and improves the accuracy of the M-PCE ensemble. Finally, this dissertation explores the use of a combination of sliced inverse regression (SIR) and polynomial chaos expansion in uncertainty quantification of response extremes. The SIR procedure is adopted to reduce the original high-dimensional problem to a low-dimensional one; then, the PCE model is employed as a surrogate in the reduced-dimension space in this SIR-PCE scheme. All the proposed approaches including the M-PCE ensemble, the bi-fidelity MPCE-MCS and the SIR-PCE scheme can help mitigate the curse of dimensionality issue; thus, they are all viable approaches for probabilistic assessment of high-dimensional stochastic models, especially when predicting very rare long-term extreme response levels for offshore structures. The proposed methods are validated using examples ranging from benchmarking analytical functions to offshore structures that include studies on a maximum wave elevation, a linear single-degree-of-freedom system response, and a nonlinear wave energy converter. All the proposed methods are found to be efficient and need significantly less effort to achieve unbiased estimations of extreme response levels compared with MCS.Show more Item Uncertainty quantification in the dynamic analysis of offshore structures(2019-12) Lim, HyeongUk; Manuel, Lance; Kallivokas, Loukas F; Kinnas, Spyros A; Nagy, Gyorzy Zoltan; Haberman, Michael RShow more Consideration for uncertainty is critical in problems associated with structural dynamics, especially in the offshore environment. Deterministic solutions are often insufficient to achieve confidence in computational results and for use in design. In uncertainty quantification (UQ) for problems dealing with many random or stochastic sources, surrogate models are often developed to reduce costs associated with running a "truth" model to verify response levels associated with low probability. Polynomial chaos expansion (PCE) is one approach used in developing such surrogate models. However, conventional PCE relies on parametric families to define the polynomials for expansion. Also, for high-dimensional problems, PCE can be inefficient if appropriate dimension reduction is not employed. Arbitrary PCE (aPCE) approaches, based on Gram-Schmidt orthogonalization, can be used to define polynomials in terms of the uncertain variables and offers a non-parametric option for UQ. An aPCE approach that can systematically account for multivariate stochasticity is developed in this study. Dimension reduction can aid in developing computationally efficient surrogate models; in this study, a gradient-based active subspace approach that identifies dominant influences on a model output's variability is employed for dimension reduction. Both aPCE and active-subspace dimension reduction are employed in studies that are focused on the dynamics of offshore structures and on low-probability events of interest. This dissertation represents a collection of four papers that deal with PCE surrogate modeling and dimension reduction. Each paper, representing a separate chapter, contributes to the development of accurate and efficient UQ approaches for offshore structure dynamics problems involving various sources of uncertainty. We demonstrate the proposed approaches in different applications: I. estimation of fatigue damage in a marine riser due to vortex-induced vibration; II. prediction of the long-term extreme response of a moored floating structure; III. surrogate model development for structural reliability analysis; and IV. dimension reduction in the extreme response prediction of offshore structures.Show more