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1. High-Dimensional Uncertainty Quantification via Tensor Regression with Rank Determination and Adaptive Sampling

   https://doi.org/10.1109/TCPMT.2021.3093432  

   He, Zichang; Zhang, Zheng (June 2021, IEEE Transactions on Components, Packaging and Manufacturing Technology)

   null (Ed.)

   Fabrication process variations can significantly influence the performance and yield of nano-scale electronic and photonic circuits. Stochastic spectral methods have achieved great success in quantifying the impact of process variations, but they suffer from the curse of dimensionality. Recently, low-rank tensor methods have been developed to mitigate this issue, but two fundamental challenges remain open: how to automatically determine the tensor rank and how to adaptively pick the informative simulation samples. This paper proposes a novel tensor regression method to address these two challenges. We use a ℓq/ℓ2 group-sparsity regularization to determine the tensor rank. The resulting optimization problem can be efficiently solved via an alternating minimization solver. We also propose a two-stage adaptive sampling method to reduce the simulation cost. Our method considers both exploration and exploitation via the estimated Voronoi cell volume and nonlinearity measurement respectively. The proposed model is verified with synthetic and some realistic circuit benchmarks, on which our method can well capture the uncertainty caused by 19 to 100 random variables with only 100 to 600 simulation samples. 

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2. General-Purpose Bayesian Tensor Learning With Automatic Rank Determination and Uncertainty Quantification

   https://doi.org/10.3389/frai.2021.668353  

   Zhang, Kaiqi; Hawkins, Cole; Zhang, Zheng (January 2022, Frontiers in Artificial Intelligence)

   A major challenge in many machine learning tasks is that the model expressive power depends on model size. Low-rank tensor methods are an efficient tool for handling the curse of dimensionality in many large-scale machine learning models. The major challenges in training a tensor learning model include how to process the high-volume data, how to determine the tensor rank automatically, and how to estimate the uncertainty of the results. While existing tensor learning focuses on a specific task, this paper proposes a generic Bayesian framework that can be employed to solve a broad class of tensor learning problems such as tensor completion, tensor regression, and tensorized neural networks. We develop a low-rank tensor prior for automatic rank determination in nonlinear problems. Our method is implemented with both stochastic gradient Hamiltonian Monte Carlo (SGHMC) and Stein Variational Gradient Descent (SVGD). We compare the automatic rank determination and uncertainty quantification of these two solvers. We demonstrate that our proposed method can determine the tensor rank automatically and can quantify the uncertainty of the obtained results. We validate our framework on tensor completion tasks and tensorized neural network training tasks. 

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3. Bayesian Generalized Sparse Symmetric Tensor-on-Vector Regression

   https://doi.org/10.1080/00401706.2020.1784799  

   Guha, Sharmistha; Guhaniyogi, Rajarshi (June 2020, Technometrics)

   Motivated by brain connectome datasets acquired using diffusion weighted magnetic resonance imaging (DWI), this article proposes a novel generalized Bayesian linear modeling framework with a symmetric tensor response and scalar predictors. The symmetric tensor coefficients corresponding to the scalar predictors are embedded with two features: low-rankness and group sparsity within the low-rank structure. Besides offering computational efficiency and parsimony, these two features enable identification of important “tensor nodes” and “tensor cells” significantly associated with the predictors, with characterization of uncertainty. The proposed framework is empirically investigated under various simulation settings and with a real brain connectome dataset. Theoretically, we establish that the posterior predictive density from the proposed model is “close” to the true data generating density, the closeness being measured by the Hellinger distance between these two densities, which scales at a rate very close to the finite dimensional optimal rate, depending on how the number of tensor nodes grow with the sample size. 

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4. Graph-Assisted Tensor Disaggregation

   Mariana M Garcez Duarte, Evangelos E. (August 2022, 17th International Workshop on Mining and Learning with Graphs (MLG))

   Consider a multi-aspect tensor dataset which is only observed in multiple complementary aggregated versions, each one at a lower resolution than the highest available one. Recent work [2] has demonstrated that given two such tensors, which have been aggregated in lower resolutions in complementary dimensions, we can pose and solve the disaggregation as an instance of a coupled tensor decomposition. In this work, we are exploring the scenario in which, in addition to the two complementary aggregated views, we also have access to a graph where nodes correspond to samples of the tensor mode that has not been aggregated. Given this graph, we propose a graph-assisted tensor disaggregation method. In our experimental evaluation,we demonstrate that our proposed method performs on par with the state of the art when the rank of the underlying coupled tensor decomposition is low, and significantly outperforms the state of the art in cases where the rank increases, producing more robust and higher-quality disaggregation. 

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5. multiUQ: An intrusive uncertainty quantification tool for gas-liquid multiphase flows

   Turnquist, B.; Owkes, M. (July 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems)

   Assessing the effects of input uncertainty on simulation results for multiphase flows will allow for more robust engineering designs and improved devices. For example, in atomizing jets, surface tension plays a critical role in determining when and how coherent liquid structures break up. Uncertainty in the surface tension coefficient can lead to uncertainty in spray angle, drop size, and velocity distribution. Uncertainty quantification (UQ) determines how input uncertainties affect outputs, and the approach taken can be classified as non-intrusive or intrusive. A classical, non-intrusive approach is the Monte-Carlo scheme, which requires multiple simulation runs using samples from a distribution of inputs. Statistics on output variability are computed from the many simulation outputs. While non-intrusive schemes are straightforward to implement, they can quickly become cost prohibitive, suffer from convergence issues, and have problems with confounding factors, making it difficult to look at uncertainty in multiple variables at once. Alternatively, an intrusive scheme inserts stochastic (uncertain) variables into the governing equations, modifying the mathematics and numerical methods used, but possibly reducing computational cost. In this work, we extend UQ methods developed for single-phase flows to handle gas-liquid multiphase dynamics by developing a stochastic conservative level set approach and a stochastic continuous surface tension method. An oscillating droplet and a 2-D atomizing jet are used to test the method. In these test cases, uncertainty about the surface tension coefficient and initial starting position will be explored, including the impact on breaking/ merging interfaces. 

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