Data Compression, Uncertainty Quantification, and Prediction Using Low-Rank ApproximationZamani Ashtiani, Shaghayegh (2024) Data Compression, Uncertainty Quantification, and Prediction Using Low-Rank Approximation. Doctoral Dissertation, University of Pittsburgh. (Unpublished)
AbstractDimension reduction techniques are valuable for both data-rich and data-poor problems. For applications involving massive high-dimensional data, dimension reduction can be utilized for data compression and data-driven discovery. In the data-poor regime, low-rank subspaces enable field reconstruction with only a few sparse measurements. Moreover, reduced-order modeling effectively propagates parametric uncertainty in high-dimensional partial differential equations. This work aims to develop dimension reduction techniques based on spatiotemporal subspaces for applications across the data-availability spectrum as well as performing uncertainty quantification in high-dimensional dynamical systems. First, we develop a low-rank approximation that compresses the size of transient simulation data in real-time, which helps with storage and input/output limitations. These limitations also restrict data analysis and visualization in large-scale simulations. To address this issue, we present an in-situ dimension reduction technique that decomposes the streaming data into a set of time-dependent bases and a core tensor in real-time. This method is adaptive and controls the compression error through the addition or removal of modes. We then develop dimension-reduction methodologies for prediction in data-poor regimes. While it is possible to predict blood flow using machine learning models, clinical measurements, such as Transcranial Doppler ultrasound, may be insufficient or too low-resolution for the training process. Therefore, developing a computational model that provides predictions based on sparse data is crucial. To this end, we present a physics-informed regression framework based on Gaussian process regression to predict blood flow properties using very few sparse measurements. Lastly, we extend the application of low-rank approximation to uncertainty quantification in blood flow simulations. In clinical settings, measurements are often intrusive and inherently uncertain. Numerical simulations could aid in developing non-invasive assessments. However, physiological variability introduces uncertainties in simulation parameters, necessitating a large number of computationally expensive simulations. To address these challenges, we explore implementing a low-rank approximation approach that reduces computational costs while maintaining accuracy. Share
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