Publications

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Control of nonlinear dynamical systems is a complex and multifaceted process. Essential elements of many engineering systems include high-fidelity physics-based modeling, offline trajectory planning, feedback control design, and data acquisition strategies to reduce uncertainties. This article proposes an optimization-centric perspective which couples these elements in a cohesive framework. We introduce a novel use of hyper-differential sensitivity analysis to understand the sensitivity of feedback controllers to parametric uncertainty in physics-based models used for trajectory planning. These sensitivities provide a foundation to define an optimal experimental design which seeks to acquire data most relevant in reducing demand on the feedback controller. Our proposed framework is illustrated on the Zermelo navigation problem and a hypersonic trajectory control problem using data from NASA’s X-43 hypersonic flight tests.

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Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model that must be estimated. However, high dimensionality of the parameters and computational complexity of the PDE solves make such problems challenging. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to estimate the other parameters. In this article, hyper-differential sensitivity analysis (HDSA) is used to assess the sensitivity of the solution of the PDE-constrained optimization problem to changes in the auxiliary parameters. Foundational assumptions for HDSA require satisfaction of the optimality conditions which are not always practically feasible as a result of ill-posedness in the inverse problem. We introduce novel theoretical and computational approaches to justify and enable HDSA for ill-posed inverse problems by projecting the sensitivities on likelihood informed subspaces and defining a posteriori updates. Our proposed framework is demonstrated on a nonlinear multiphysics inverse problem motivated by estimation of spatially heterogeneous material properties in the presence of spatially distributed parametric modeling uncertainties.

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Deep neural networks (DNNs) have achieved state-of-the-art performance across a variety of traditional machine learning tasks, e.g., speech recognition, image classification, and segmentation. The ability of DNNs to efficiently approximate high-dimensional functions has also motivated their use in scientific applications, e.g., to solve partial differential equations and to generate surrogate models. In this paper, we consider the supervised training of DNNs, which arises in many of the above applications. We focus on the central problem of optimizing the weights of the given DNN such that it accurately approximates the relation between observed input and target data. Devising effective solvers for this optimization problem is notoriously challenging due to the large number of weights, nonconvexity, data sparsity, and nontrivial choice of hyperparameters. To solve the optimization problem more efficiently, we propose the use of variable projection (VarPro), a method originally designed for separable nonlinear least-squares problems. Our main contribution is the Gauss--Newton VarPro method (GNvpro) that extends the reach of the VarPro idea to nonquadratic objective functions, most notably cross-entropy loss functions arising in classification. These extensions make GNvpro applicable to all training problems that involve a DNN whose last layer is an affine mapping, which is common in many state-of-the-art architectures. In our four numerical experiments from surrogate modeling, segmentation, and classification, GNvpro solves the optimization problem more efficiently than commonly used stochastic gradient descent (SGD) schemes. Finally, GNvpro finds solutions that generalize well, and in all but one example better than well-tuned SGD methods, to unseen data points.

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The generalized singular value decomposition (GSVD) is a valuable tool that has many applications in computational science. However, computing the GSVD for large-scale problems is challenging. Motivated by applications in hyper-differential sensitivity analysis (HDSA), we propose new randomized algorithms for computing the GSVD which use randomized subspace iteration and weighted QR factorization. Detailed error analysis is given which provides insight into the accuracy of the algorithms and the choice of the algorithmic parameters. We demonstrate the performance of our algorithms on test matrices and a large-scale model problem where HDSA is used to study subsurface flow.

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Imbalance in the autonomic nervous system can lead to orthostatic intolerance manifested by dizziness, lightheadedness, and a sudden loss of consciousness (syncope); these are common conditions, but they are challenging to diagnose correctly. Uncertainties about the triggering mechanisms and the underlying pathophysiology have led to variations in their classification. This study uses machine learning to categorize patients with orthostatic intolerance. We use random forest classification trees to identify a small number of markers in blood pressure, and heart rate time-series data measured during head-up tilt to (a) distinguish patients with a single pathology and (b) examine data from patients with a mixed pathophysiology. Next, we use Kmeans to cluster the markers representing the time-series data. We apply the proposed method analyzing clinical data from 186 subjects identified as control or suffering from one of four conditions: postural orthostatic tachycardia (POTS), cardioinhibition, vasodepression, and mixed cardioinhibition and vasodepression. Classification results confirm the use of supervised machine learning. We were able to categorize more than 95% of patients with a single condition and were able to subgroup all patients with mixed cardioinhibitory and vasodepressor syncope. Clustering results confirm the disease groups and identify two distinct subgroups within the control and mixed groups. The proposed study demonstrates how to use machine learning to discover structure in blood pressure and heart rate time-series data. The methodology is used in classification of patients with orthostatic intolerance. Diagnosing orthostatic intolerance is challenging, and full characterization of the pathophysiological mechanisms remains a topic of ongoing research. This study provides a step toward leveraging machine learning to assist clinicians and researchers in addressing these challenges. [Figure not available: see fulltext.].

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