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This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuron model.

This work develops a methodology for creating a data-driven digital twin from a library of physics-based models representing various asset states. The digital twin is updated using interpretable machine learning. Specifically, we use optimal trees---a recently developed scalable machine learning method---to train an interpretable data-driven classifier. Training data for the classifier are generated offline using simulated scenarios solved by the library of physics-based models. These data can be further augmented using experimental or other historical data. In operation, the classifier uses observational data from the asset to infer which physics-based models in the model library are the best candidates for the updated digital twin. The approach is demonstrated through the development of a structural digital twin for a 12ft wingspan unmanned aerial vehicle. This digital twin is built from a library of reduced-order models of the vehicle in a range of structural states. The data-driven digital twin dynamically updates in response to structural damage or degradation and enables the aircraft to replan a safe mission accordingly. Within this context, we study the performance of the optimal tree classifiers and demonstrate how their interpretability enables explainable structural assessments from sparse sensor measurements, and also informs optimal sensor placement., Comment: 20 pages, 13 figures, submitted to AIAA Journal

The flexibility of student pathways in today’s university programs can make it difficult for students to perceive the relationships between the courses they take and to integrate their knowledge and skills in a useful way. In this paper we describe a tool we have developed to help students cope with this challenge.

We introduce an algorithm to locate contours of functions that are expensive to evaluate. The problem of locating contours arises in many applications, including classification, constrained optimization, and performance analysis of mechanical and dynamical systems (reliability, probability of failure, stability, etc.). Our algorithm locates contours using information from multiple sources, which are available in the form of relatively inexpensive, biased, and possibly noisy approximations to the original function. Considering multiple information sources can lead to significant cost savings. We also introduce the concept of contour entropy, a formal measure of uncertainty about the location of the zero contour of a function approximated by a statistical surrogate model. Our algorithm locates contours efficiently by maximizing the reduction of contour entropy per unit cost.

Mapping Outcomes in an Undergraduate Aerospace Engineering Program Student learning outcomes are well known to be an important component of developing subject content, communicating expectations to students, and designing effective assessments. This project focuses on mapping the relationships among outcomes across an entire undergraduate aerospace engineering program. In many domains, outcomes are presented simply as lists of capabilities students should gain by the end of a learning activity. This project expands upon the standard presentation by placing all outcomes from the undergraduate program in an interconnected contextual map. To bring further structure to the map, we group outcomes within modules, where we define a module as a learning unit comprised of a set of outcomes.This new outcomes framework provides a structured process for reevaluation and assessment of a curriculum, specifically with regard to intra and interconnectedness of subjects. Several specific motivations drove this project: first, to inspire discussion among faculty about the current and ideal shape of the undergraduate curriculum and potential gaps or overlap among classes; second, to highlight modularity inherent in the curriculum; third, to provide an educational roadmap to students so that they are able to view their progress through the program and self-assess as necessary; and fourth, to provide a framework for dynamic, integrated assessment within subjects.As a part of this project, faculty revised and expanded upon existing outcomes or generated new outcomes. In addition, faculty identified the relationships among outcomes using prerequisite structures and they grouped outcomes in modules. This led to a set of over 700 outcomes for subjects taught within the Aerospace department, over 150 outcomes for prerequisite mathematics and physics subjects taught outside the department, and almost 1000 prerequisite links between outcomes. These outcomes and links, along with their associated subjects and modules, have been entered into a database for which a custom visualization has been developed.The figure on the following page shows connections among outcomes across subjects taught within the aerospace engineering department (showing all subjects that contribute to meeting departmental undergraduate degree requirements).One specific example of how these outcome networks have been utilized is within a junior/senior-level computational methods subject that utilizes a blended learning model. The outcomes map provides an explicit structure that integrates pre-class online readings with in-class active learning activities. Each section of the online readings and the associated embedded assessments are tagged explicitly with the specific outcomes to which they relate. The online platform also provides an integrated linkage of all outcomes across the subject, permitting students to navigate through the online resources to find other lessons and assessment activities that relate to each outcome.The final paper will present the process used to collect and organize the outcomes mapping, the development of the database and visualization tool, conclusions drawn from assessment data collected via student feedback, and specific case studies that demonstrate the utilization of the mapping.

Mathematical models are used extensively for diverse tasks including analysis, optimization, and decision making. Frequently, those models are principled but imperfect representations of reality. This is either due to incomplete physical description of the underlying phenomenon (simplified governing equations, defective boundary conditions, etc.), or due to numerical approximations (discretization, linearization, round-off error, etc.). Model misspecification can lead to erroneous model predictions, and respectively suboptimal decisions associated with the intended end-goal task. To mitigate this effect, one can amend the available model using limited data produced by experiments or higher fidelity models. A large body of research has focused on estimating explicit model parameters. This work takes a different perspective and targets the construction of a correction model operator with implicit attributes. We investigate the case where the end-goal is inversion and illustrate how appropriate choices of properties imposed upon the correction and corrected operator lead to improved end-goal insights.

This paper introduces a computational strategy to determine optimal sets of sensor locations to support real-time operational decisions. We exploit unsupervised learning strategies (specifically self-organizing maps) to identify the most informative locations to place sensors. The sensor placement procedure is then combined with a Multi-Step-Reduced Order Modeling approach that exploits the low-dimensional map between the sparse sensed data and the decisions at hand. The approach is demonstrated for the real-time assessment of an unmanned aircraft wing panel undergoing structural degradation. For this application, we compare the optimal sets of sensor locations with random placements for a variety of sensor availabilities. By adopting our placement strategy, we achieve improvements in accuracy and robustness of capability predictions, even when measured data are sparse and cover less than 10% of the reference data., United States. Air Force. Office of Scientific Research (Grant FA9550-16-1-0108), SUTD-MIT International Design Centre (IDC)

Design of complex engineering systems requires coupled analyses of the multiple disciplines affecting system performance. The coupling among disciplines typically contributes significantly to the computational cost of analyzing the system, and can become particularly burdensome when coupled analyses are embedded within a design or optimization loop. In many cases, disciplines may be weakly coupled, so that some of the coupling or interaction terms can be neglected without significantly impacting the accuracy of the system output. However, typical practice derives such approximations in an ad hoc manner using expert opinion and domain experience. This paper proposes a new approach that formulates an optimization problem to find a model that optimally balances accuracy of the model outputs with the sparsity of the discipline couplings. An adaptive sequential Monte Carlo sampling-based technique is used to efficiently search the combinatorial model space of different discipline couplings. Finally, an algorithm for optimal model selection is presented and applied to identify the important discipline couplings in a fire detection satellite model and a turbine engine cycle analysis model., United States. Air Force. Office of Scientific Research. Multidisciplinary University Research Initiative (Award FA9550-15-1-0038)

An inverse problem seeks to infer unknown model parameters using observed data. We consider a goal-oriented inverse problem, where the goal of inferring parameters is to use them in predicting a quantity of interest (QoI). Recognizing that multiple models of varying fidelity and computational cost may be available to describe the physical system, we formulate a goal-oriented model adaptivity approach that leverages multiple models while controlling the error in the QoI prediction. In particular, we adaptively form a mixed-fidelity model by using models of different levels of fidelity in different subregions of the domain. Taking the solution of the inverse problem with the highest-fidelity model as our reference QoI prediction, we derive an adjoint-based third-order estimate for the QoI error from using a lower-fidelity model. Localization of this error then guides the formation of mixed-fidelity models. We demonstrate the method for example problems described by convection–diffusion–reaction models. For these examples, our mixed-fidelity models use the high-fidelity model over only a small portion of the domain, but result in QoI estimates with small relative errors. We also demonstrate that the mixed-fidelity inverse problems can be cheaper to solve and less sensitive to the initial guess than the high-fidelity inverse problems. Keyword: Inference; Goal-oriented adaptive modeling; A posteriori error estimation; Multi-fidelity modeling; Adjoints, United States. Department of Energy ( DE-FC02-13ER26129/DE-21 SC000929)

Fixed point iteration is a common strategy to handle interdisciplinary coupling within a feedback-coupled multidisciplinary analysis. For each coupled analysis, this requires a large number of disciplinary high-fidelity simulations to resolve the interactions between different disciplines. When embedded within an uncertainty analysis loop (e.g., with Monte Carlo sampling over uncertain parameters), the number of high-fidelity disciplinary simulations quickly becomes prohibitive, because each sample requires a fixed point iteration and the uncertainty analysis typically involves thousands or even millions of samples. This paper develops a method for uncertainty quantification in feedback-coupled systems that leverage adaptive surrogates to reduce the number of cases forwhichfixedpoint iteration is needed. The multifidelity coupled uncertainty propagation method is an iterative process that uses surrogates for approximating the coupling variables and adaptive sampling strategies to refine the surrogates. The adaptive sampling strategies explored in this work are residual error, information gain, and weighted information gain. The surrogate models are adapted in a way that does not compromise the accuracy of the uncertainty analysis relative to the original coupled high-fidelity problem as shown through a rigorous convergence analysis., United States. Army Research Office. Multidisciplinary University Research Initiative (Award FA9550-15-1-0038)

This paper presents methodology to enable path planning for an unmanned aerial vehicle that uses dynamic data-driven flight capability estimation. The main contribution of the work is a general mathematical approach that leverages offline vehicle analysis and design data together with onboard sensor measurements to achieve dynamic path planning. The mathematical framework, expressed as a Constrained Partially Observable Markov Decision Process, accounts for vehicle capability constraints and is robust to modeling error and disturbances in both the vehicle process and measurement models. Vehicle capability constraints are incorporated using Probabilistic Support Vector Machine surrogates of high-fidelity physics-based models that adequately capture the richness of the vehicle dynamics. Sensor measurements are treated in a general manner and can include combinations of multiple modalities such as GPS/IMU data as well as structural strain data of the airframe. Results are presented for a simulated 3-D environment and point-mass airplane model. The vehicle can dynamically adjust its trajectory according to the observations it receives about its current state of health, thereby retaining a high probability of survival and mission success.

Two major bottlenecks to the solution of large-scale Bayesian inverse problems are the scaling of posterior sampling algorithms to high-dimensional parameter spaces and the computational cost of forward model evaluations. Yet incomplete or noisy data, the state variation and parameter dependence of the forward model, and correlations in the prior collectively provide useful structure that can be exploited for dimension reduction in this setting-both in the parameter space of the inverse problem and in the state space of the forward model. To this end, we show how to jointly construct low-dimensional subspaces of the parameter space and the state space in order to accelerate the Bayesian solution of the inverse problem. As a byproduct of state dimension reduction, we also show how to identify low-dimensional subspaces of the data in problems with high-dimensional observations. These subspaces enable approximation of the posterior as a product of two factors: (i) a projection of the posterior onto a low-dimensional parameter subspace, wherein the original likelihood is replaced by an approximation involving a reduced model; and (ii) the marginal prior distribution on the high-dimensional complement of the parameter subspace. We present and compare several strategies for constructing these subspaces using only a limited number of forward and adjoint model simulations. The resulting posterior approximations can rapidly be characterized using standard sampling techniques, e.g., Markov chain Monte Carlo. Two numerical examples demonstrate the accuracy and efficiency of our approach: inversion of an integral equation in atmospheric remote sensing, where the data dimension is very high; and the inference of a h eterogeneous transmissivity field in a groundwater system, which involves a partial differential equation forward model with high dimensional state and parameters., United States. Department of Energy. Office of Advanced Scientific Computing Research (Grant DE-SC0009297)

Estimating statistics of model outputs with the Monte Carlo method often requires a large number of model evaluations. This leads to long runtimes if the model is expensive to evaluate. Importance sampling is one approach that can lead to a reduction in the number of model evaluations. Importance sampling uses a biasing distribution to sample the model more efficiently, but generating such a biasing distribution can be difficult and usually also requires model evaluations. A different strategy to speed up Monte Carlo sampling is to replace the computationally expensive high-fidelity model with a computationally cheap surrogate model; however, because the surrogate model outputs are only approximations of the high-fidelity model outputs, the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high-fidelity model. We introduce a multifidelity importance sampling (MFIS) method, which combines evaluations of both the high-fidelity and a surrogate model. It uses a surrogate model to facilitate the construction of the biasing distribution, but relies on a small number of evaluations of the high-fidelity model to derive an unbiased estimate of the statistics of interest. We prove that the MFIS estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself. The MFIS method can be used with any type of surrogate model, such as projection-based reduced-order models and data-fit models. Furthermore, the MFIS method is applicable to black-box models, i.e., where only inputs and the corresponding outputs of the high-fidelity and the surrogate model are available but not the details of the models themselves. We demonstrate on nonlinear and time-dependent problems that our MFIS method achieves speedups of up to several orders of magnitude compared to Monte Carlo with importance sampling that uses the high-fidelity model only., United States. Department of Energy. Office of Advanced Scientific Computing Research. Applied Mathematics Program (award DE-FG02-08ER2585), United States. Department of Energy. Office of Advanced Scientific Computing Research. Applied Mathematics Program (award DE-SC0009297)

This paper presents a data-driven approach for the online updating of the flight envelope of an unmanned aerial vehicle subjected to structural degradation. The main contribution of the work is a general methodology that leverages both physics-based modeling and data to decompose tasks into two phases: expensive offline simulations to build an efficient characterization of the problem and rapid data-driven classification to support online decision making. In the approach, physics-based models at the wing and vehicle level run offline to generate libraries of information covering a range of damage scenarios. These libraries are queried online to estimate vehicle capability states. The state estimation and associated quantification of uncertainty are achieved by Bayesian classification using sensed strain data. The methodology is demonstrated on a conceptual unmanned aerial vehicle executing a pullup maneuver, in which the vehicle flight envelope is updated dynamically with onboard sensor information. During vehicle operation, the maximum maneuvering load factor is estimated using structural strain sensor measurements combined with physics-based information from precomputed damage scenarios that consider structural weakness. Compared to a baseline case that uses a static as-designed flight envelope, the self-aware vehicle achieves both an increase in probability of executing a successful maneuver and an increase in overall usage of the vehicle capability., United States. Air Force Office of Scientific Research. Dynamic Data-Driven Application Systems Program (Grant FA9550-11-1-0339)

Deciding how to represent and manage uncertainty is a vital part of designing complex systems. Widely used is a probabilistic approach—assigning a probability distribution to each uncertain variable. However, this presents the designer with the task of assuming or estimating these probability distributions from data; a task which is inevitably prone to error. This paper addresses this challenge by formulating a distributionally robust design optimization problem, and presents computationally ecient algorithms for solving the problem. In distributionally robust optimization (DRO) methods, the designer acknowledges that they are unable to exactly specify a probability distribution for the uncertain variables, and instead specifies a so-called ambiguity set of possible distributions. This paper uses an acoustic horn design problem to explore how the error incurred in estimating a probability distribution from data a↵ects the realized performance of designs found using a traditional multi-objective optimization under uncertainty. It is found that placing some importance on a risk reduction objective results in designs that are more robust to these errors, and thus have a better mean performance realized under the true distribution than if the designer were to focus all e↵orts on optimizing for mean performance alone. In contrast, the DRO approach is able to uncover designs that are not attainable using the multi-objective approach when given the same data. These DRO designs in some cases significantly outperform those designs found using the multi-objective approach., United States. Air Force. Office of Scientific Research (Grant FA9550-16-1-0108)

One important task of uncertainty quantification is propagating input uncertainties through a system of interest to quantify the uncertainties’ effects on the system outputs; however, numerical methods for uncertainty propagation are often based on Monte Carlo estimation, which can require large numbers of numerical simulations of the numerical model describing the system response to obtain estimates with acceptable accuracies. Thus, if the model is computationally expensive to evaluate, then Monte-Carlo-based uncertainty propagation methods can quickly become computationally intractable. We demonstrate that multifidelity methods can significantly speedup uncertainty propagation by leveraging low-cost low-fidelity models and establish accuracy guarantees by using occasional recourse to the expensive high-fidelity model. We focus on the multifidelity Monte Carlo method, which is a multifidelity approach that optimally distributes work among the models such that the mean-squared error of the multifidelity estimator is minimized for a given computational budget. The multifidelity Monte Carlo method is applicable to general types of low-fidelity models, including projection-based reduced models, data-fit surrogates, response surfaces, and simplified-physics models. We apply the multifidelity Monte Carlo method to a coupled aero-structural analysis of a wing and a flutter problem with a high-aspect-ratio wing. The low-fidelity models are data-fit surrogate models derived with standard procedures that are built in common software environments such as Matlab and numpy/scipy. Our results demonstrate speedups of orders of magnitude compared to using the high-fidelity model alone., United States. Air Force. Office of Scientific Research. Multidisciplinary University Research Initiative (Award FA9550-15-1-0038)

Digital Thread offers the opportunity to use information generated across the product lifecycle to design the next generation of products. In this paper, we introduce a mathematical methodology that establishes the data-driven design and decision problem associated with Digital Thread. Our objectives are twofold: 1) Provide a mathematical definition of Digital Thread in the context of conceptual and preliminary design and establish a methodology for how information along the Digital Thread enters into the design problem as well how design decisions affect the Digital Thread. 2) Develop a data-driven design method that incorporates data from different sources from across the product life cycle. We illustrate aspects of our methodology through an example design of a structural fiber-steered composite component., United States. Air Force. Office of Scientific Research (Grant FA9550-16-1-0108), SUTD-MIT International Design Centre (IDC)

This paper presents a multifidelity method for optimization under uncertainty for aerospace problems. In this work, the effectiveness of the method is demonstrated for the robust optimization of a tailless aircraft that is based on the Boeing Insitu ScanEagle. Aircraft design is often affected by uncertainties in manufacturing and operating conditions. Accounting for uncertainties during optimization ensures a robust design that is more likely to meet performance requirements. Designing robust systems can be computationally prohibitive due to the numerous evaluations of expensive-to-evaluate high-fidelity numerical models required to estimate system-level statistics at each optimization iteration. This work uses a multifidelity Monte Carlo approach to estimate the mean and the variance of the system outputs for robust optimization. The method uses control variates to exploit multiple fidelities and optimally allocates resources to different fidelities to minimize the variance in the estimates for a given budget. The results for the ScanEagle application show that the proposed multifidelity method achieves substantial speed-ups as compared to a regular Monte-Carlo-based robust optimization., United States. Air Force. Office of Scientific Research. Multidisciplinary University Research Initiative (FA9550-15-1-0038)

This paper presents a new approach for optimization under uncertainty in the presence of probabilistic, interval, and mixed uncertainties, avoiding the need to specify probability distributions on uncertain parameters when such information is not readily available. Existing approaches for optimization under these types of uncertainty mostly rely on treating combinations of statistical moments as separate objectives, but this can give rise to stochastically dominated designs. Here, horsetail matching is extended for use with these types of uncertainties to overcome some of the limitations of existing approaches. The formulation delivers a single, differentiable metric as the objective function for optimization. It is demonstrated on algebraic test problems, the design of a wing using a low-fidelity coupled aerostructural code, and the aerodynamic shape optimization of a wing using computational fluid dynamics analysis.

We consider the task of dynamic capability estimation for an unmanned aerial vehicle, which is needed to provide the vehicle with the ability to dynamically and autonomously sense, plan, and act in real time. Our dynamic data-driven application systems framework employs reduced models to achieve rapid evaluation runtimes. Our reduced models must also adapt to underlying dynamic system changes, such as changes due to structural damage or degradation of the system. Our dynamic reduced models take into account changes in the underlying system by directly learning from the data provided by sensors, without requiring access to the original high-fidelity model. We present here an adaptivity indicator that detects a change in the underlying system and so allows the initiation of the dynamic reduced modeling adaptation if necessary. The adaptivity indicator monitors the error of the dynamic reduced model by comparing model predictions with sensor data, and signals a change if the error exceeds a given threshold. The indicator is demonstrated on a deflection model of a damaged plate in bending. Local damage of the plate is modeled by a change in the thickness of the plate. The numerical results show that in this example the adaptivity indicator detects all changes in the thickness and correctly initiates the adaptation of the reduced model., United States. Air Force Office of Scientific Research. Dynamic DataDriven Application Systems Program (Grant FA9550-11-1-0339)

To support effective decision making, engineers should comprehend and manage various uncertainties throughout the design process. Unfortunately, in today's modern systems, uncertainty analysis can become cumbersome and computationally intractable for one individual or group to manage. This is particularly true for systems comprised of a large number of components. In many cases, these components may be developed by different groups and even run on different computational platforms. This paper proposes an approach for decomposing the uncertainty analysis task among the various components comprising a feed-forward system and synthesizing the local uncertainty analyses into a system uncertainty analysis. Our proposed decomposition-based multicomponent uncertainty analysis approach is shown to be provably convergent in distribution under certain conditions. The proposed method is illustrated on quantification of uncertainty for a multidisciplinary gas turbine system and is compared to a traditional system-level Monte Carlo uncertainty analysis approach., SUTD-MIT International Design Centre, United States. Defense Advanced Research Projects Agency. META Program (United States. Air Force Research Laboratory Contract FA8650-10-C-7083), Vanderbilt University (Contract VU-DSR#21807-S7), United States. Federal Aviation Administration. Office of Environment and Energy (FAA Award 09-C-NE-MIT, Amendments 028, 033, and 038)

A self-aware aerospace vehicle can dynamically adapt the way it performs missions by gathering information about itself and its surroundings and responding intelligently. We consider the specific challenge of an unmanned aerial vehicle that can dynamically and autonomously sense its structural state and re-plan its mission according to its estimated current structural health. The challenge is to achieve each of these tasks in real time-executing online models and exploiting dynamic data streams-while also accounting for uncertainty. Our approach combines information from physics-based models, simulated offline to build a scenario library, together with dynamic sensor data in order to estimate current flight capability. Our physics-based models analyze the system at both the local panel level and the global vehicle level., United States. Air Force. Office of Scientific Research (Grant FA9550-11-1-0339)

A multifidelity approach to design and analysis for complex systems seeks to exploit optimally all available models and data. Existing multifidelity approaches generally attempt to calibrate low-fidelity models or replace low-fidelity analysis results using data from higher fidelity analyses. This paper proposes a fundamentally different approach that uses the tools of estimation theory to fuse together information from multifidelity analyses, resulting in a Bayesian-based approach to mitigating risk in complex system design and analysis. This approach is combined with maximum entropy characterizations of model discrepancy to represent epistemic uncertainties due to modeling limitations and model assumptions. Mathematical interrogation of the uncertainty in system output quantities of interest is achieved via a variance-based global sensitivity analysis, which identifies the primary contributors to output uncertainty and thus provides guidance for adaptation of model fidelity. The methodology is applied to multidisciplinary design optimization and demonstrated on a wing-sizing problem for a high altitude, long endurance vehicle., United States. Air Force Office of Scientific Research. Small Business Technology Transfer Program (Contract FA9550-09-C-0128)

Educational mapping is the process of analyzing an educational system to identify entities, relationships and attributes. This paper proposes a network modeling approach to educational mapping. Current mapping processes in education typically represent data in forms that do not support scalable learning analytics. For example, a curriculum map is usually a table, where relationships among curricular elements are represented implicitly in the rows of the table. The proposed network modeling approach overcomes this limitation through explicit modeling of these relationships in a graph structure, which in turn unlocks the ability to perform scalable analyses on the dataset. The paper presents network models for educational use cases, with concrete examples in curriculum mapping, accreditation mapping and concept mapping. Illustrative examples demonstrate how the formal modeling approach enables visualization and learning analytics. The analysis provides insight into learning pathways, supporting design of adaptive learning systems. It also permits gap analysis of curriculum coverage, supporting student advising, student degree planning and curricular design at scales ranging from an entire institution to an individual course.

As a measure to manage the climate impact of aviation, significant enhancements to aviation technologies and operations are necessary. When assessing these enhancements and their respective impacts on the climate, it is important that we also quantify the associated uncertainties. This is important to support an effective decision and policymaking process. However, such quantification of uncertainty is challenging, especially in a complex system that comprises multiple interacting components. The uncertainty quantification task can quickly become computationally intractable and cumbersome for one individual or group to manage. Recognizing the challenge of quantifying uncertainty in multicomponent systems, we utilize a divide-and-conquer approach, inspired by the decomposition-based approaches used in multidisciplinary analysis and optimization. Specifically, we perform uncertainty analysis and global sensitivity analysis of our multicomponent aviation system in a decomposition-based manner. In this work, we demonstrate how to handle a high-dimensional multicomponent interface using sensitivity-based dimension reduction and a novel importance sampling method. Our results demonstrate that the decomposition-based uncertainty quantification approach can effectively quantify the uncertainty of a feed-forward multicomponent system for which the component models are housed in different locations and owned by different groups. Keywords: Aviation Environmental Impact; Decomposition; Global Sensitivity Analysis; Uncertainty Quantification

A self-aware aerospace vehicle can dynamically adapt the way it performs missions by gathering information about itself and its surroundings and responding intelligently. Achieving this DDDAS paradigm enables a revolutionary new generation of self-aware aerospace vehicles that can perform missions that are impossible using current design, flight, and mission planning paradigms. To make self-aware aerospace vehicles a reality, fundamentally new algorithms are needed that drive decision-making through dynamic response to uncertain data, while incorporating information from multiple modeling sources and multiple sensor fidelities.In this work, the specific challenge of a vehicle that can dynamically and autonomously sense, plan, and act is considered. The challenge is to achieve each of these tasks in real time executing online models and exploiting dynamic data streams–while also accounting for uncertainty. We employ a multifidelity approach to inference, prediction and planning an approach that incorporates information from multiple modeling sources, multiple sensor data sources, and multiple fidelities.

Numerical simulation models to support decision-making and policy-making processes are often complex, involving many disciplines, many inputs, and long computation times. Inputs to such models are inherently uncertain, leading to uncertainty in model outputs. Characterizing, propagating, and analyzing this uncertainty is critical both to model development and to the effective application of model results in a decision-making setting; however, the many thousands of model evaluations required to sample the uncertainty space (e.g., via Monte Carlo sampling) present an intractable computational burden. This paper presents a novel surrogate modeling methodology designed specifically for propagating uncertainty from model inputs to model outputs and for performing a global sensitivity analysis, which characterizes the contributions of uncertainties in model inputs to output variance, while maintaining the quantitative rigor of the analysis by providing confidence intervals on surrogate predictions. The approach is developed for a general class of models and is demonstrated on an aircraft emissions prediction model that is being developed and applied to support aviation environmental policy-making. The results demonstrate how the confidence intervals on surrogate predictions can be used to balance the tradeoff between computation time and uncertainty in the estimation of the statistical outputs of interest., United States. Federal Aviation Administration (contract no. DTFAWA-05-D-00012, Task Order 0002)

This work presents a nonintrusive projection-based model reduction approach for full models based on time-dependent partial differential equations. Projection-based model reduction constructs the operators of a reduced model by projecting the equations of the full model onto a reduced space. Traditionally, this projection is intrusive, which means that the full-model operators are required either explicitly in an assembled form or implicitly through a routine that returns the action of the operators on a given vector; however, in many situations the full model is given as a black box that computes trajectories of the full-model states and outputs for given initial conditions and inputs, but does not provide the full-model operators. Our nonintrusive operator inference approach infers approximations of the reduced operators from the initial conditions, inputs, trajectories of the states, and outputs of the full model, without requiring the full-model operators. Our operator inference is applicable to full models that are linear in the state or have a low-order polynomial nonlinear term. The inferred operators are the solution of a least-squares problem and converge, with sufficient state trajectory data, in the Frobenius norm to the reduced operators that would be obtained via an intrusive projection of the full-model operators. Our numerical results demonstrate operator inference on a linear climate model and on a tubular reactor model with a polynomial nonlinear term of third order. Keywords: Nonintrusive model reduction; Data-driven model reduction; Black-box full model; Inference

Fixed point iteration is a common strategy to handle interdisciplinary coupling within a coupled multidisciplinary analysis. For each coupled analysis, this requires a large number of disciplinary high-fidelity simulations to resolve the interactions between different disciplines. When embedded within an uncertainty analysis loop (e.g., with Monte Carlo sampling over uncertain parameters) the number of high-fidelity disciplinary simulations quickly becomes prohibitive, since each sample requires a fixed point iteration and the uncertainty analysis typically involves thousands or even millions of samples. This paper develops a method for uncertainty analysis in feedback-coupled black-box systems that leverages adaptive surrogates to reduce the number of cases for which fixed point iteration is needed. The multifidelity coupled uncertainty propagation method is an iterative process that uses surrogates for approximating the coupling variables and adaptive sampling strategies to refine the surrogates. The adaptive sampling strategies explored in this work are residual error, information gain, and weighted information gain. The surrogate models are adapted in a way that does not compromise accuracy of the uncertainty analysis relative to the original coupled high-fidelity problem., United States. Air Force Office of Scientific Research. Multidisciplinary University Research Initiative (Award FA9550-15-1-0038)

Designing and optimizing complex systems often requires numerous evaluations of a quantity of interest. This is typically achieved by querying potentially expensive numerical models in an optimization process. To alleviate the cost of optimization, surrogate models can be used in lieu of the original model, as they are cheaper to evaluate. In addition, different information sources with varying fidelity, such as numerical models, experimental results or historical data may be available to estimate the quantity of interest. This work proposes a strategy to adaptively construct and exploit a multifidelity surrogate when multiple information sources of varying fidelity are available. One of the distinguishing features of the proposed approach is the relaxation of the common assumption of hierarchical relationships among information sources. This is achieved by endowing the surrogate representation with uncertainty functions that can vary across the design space; this uncertainty quantifies the fidelity of the underlying information source. The resulting multifidelity surrogate is used in an optimization setting to identify the next design to evaluate, as well as to select the information sources with which to perform the evaluation, based on information source evaluation cost and fidelity. For an aerodynamic design example, the proposed strategy leverages multifidelity information to reduce the number of evaluations of the expensive information source needed during the optimization., United States. Air Force. Office of Scientific Research. Multidisciplinary University Research Initiative (Grant FA9550- 09-0613), Singapore-MIT Alliance Computational Engineering Programme

We present a local density estimator based on first-order statistics. To estimate the density at a point, x, the original sample is divided into subsets and the average minimum sample distance to x over all such subsets is used to define the density estimate at x. The tuning parameter is thus the number of subsets instead of the typical bandwidth of kernel or histogram-based density estimators. The proposed method is similar to nearest-neighbor density estimators but it provides smoother estimates. We derive the asymptotic distribution of this minimum sample distance statistic to study globally optimal values for the number and size of the subsets. Simulations are used to illustrate and compare the convergence properties of the estimator. The results show that the method provides good estimates of a wide variety of densities without changes of the tuning parameter, and that it offers competitive convergence performance., United States. Department of Energy. Applied Mathematical Sciences Program (Award DE-FG02-08ER2585), United States. Department of Energy. Applied Mathematical Sciences Program (Award de-sc0009297)