Paul G. Constantine

Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos

Polynomial chaos expansions (PCE) are an attractive technique for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability. When tailoring the orthogonal polynomial bases to match the forms of the input uncertainties in a Wiener-Askey scheme, excellent convergence properties can be achieved for general probabilistic analysis problems. Non-intrusive PCE methods allow the use of simulations as black boxes within UQ studies, and involve the calculation of chaos expansion coefficients based on a set of response function evaluations. These methods may be characterized as being either Galerkin projection methods, using sampling or numerical integration, or regression approaches (also known as point collocation or stochastic response surfaces), using linear least squares. Numerical integration methods may be further categorized as either tensor product quadrature or sparse grid Smolyak cubature and as either isotropic or anisotropic. Experience with these approaches is presented for algebraic and PDE-based benchmark test problems, demonstrating the need for accurate, efficient coefficient estimation approaches that sca le for problems with significant numbers of random variables.

Exploiting Active Subspaces to Quantify Uncertainty in the Numerical Simulation of the HyShot II Scramjet

Abstract We present a computational analysis of the reactive flow in a hypersonic scramjet engine with focus on effects of uncertainties in the operating conditions. We employ a novel methodology based on active subspaces to characterize the effects of the input uncertainty on the scramjet performance. The active subspace identifies one-dimensional structure in the map from simulation inputs to quantity of interest that allows us to reparameterize the operating conditions; instead of seven physical parameters, we can use a single derived active variable. This dimension reduction enables otherwise infeasible uncertainty quantification, considering the simulation cost of roughly 9500 CPU-hours per run. For two values of the fuel injection rate, we use a total of 68 simulations to (i) identify the parameters that contribute the most to the variation in the output quantity of interest, (ii) estimate upper and lower bounds on the quantity of interest, (iii) classify sets of operating conditions as safe or unsafe corresponding to a threshold on the output quantity of interest, and (iv) estimate a cumulative distribution function for the quantity of interest.

Reprint of

Integrated hydrologic models coupled to land surface models require several input parameters to characterize the land surface and to estimate energy fluxes. Uncertainty of input parameter values is inherent in any model and the sensitivity of output to these uncertain parameters becomes an important consideration. To better understand these connections in the context of hydrologic models, we use the ParFlow-Common Land Model (PF-CLM) to estimate energy fluxes given variations in 19 vegetation and land surface parameters over a 144-hour period of time. Latent, sensible and ground heat fluxes from bare soil and grass vegetation were estimated using single column and tilted-v domains. Energy flux outputs, along with the corresponding input parameters, from each of the four scenario simulations were evaluated using active subspaces. The active subspace method considers parameter sensitivity by quantifying a weight for each parameter. The method also evaluates the potential for dimension reduction by identifying the input-output relationship through the active variable - a linear combination of input parameters. The aerodynamic roughness length was the most important parameter for bare soil energy fluxes. Multiple parameters were important for energy fluxes from vegetated surfaces and depended on the type of energy flux. Relationships between land surface inputs and output fluxes varied between latent, sensible and ground heat, but were consistent between domain setup (i.e., with or without lateral flow) and vegetation type. A quadratic polynomial was used to describe the input-output relationship for these energy fluxes. The reduced-dimension model of land surface dynamics can be compared to observations or used to solve the inverse problem. Considering this work as a proof-of-concept, the active subspace method can be applied and extended to a range of domain setups, land cover types and time periods to obtain a reduced-form representation of any output of interest, provided that an active subspace exists. Active subspaces identify important input parameters and how they relate to output.Proof-of-concept domains show potential for dimension reduction of land surface.Important land surface parameters depend on land cover and flux type.Land surface inputs and energy flux outputs can be related by a quadratic polynomial.Lateral flow has negligible effect on the land surface parameter-flux relationship.

Active subspaces for sensitivity analysis and dimension reduction of an integrated hydrologic model

Integrated hydrologic models coupled to land surface models require several input parameters to characterize the land surface and to estimate energy fluxes. Uncertainty of input parameter values is inherent in any model and the sensitivity of output to these uncertain parameters becomes an important consideration. To better understand these connections in the context of hydrologic models, we use the ParFlow-Common Land Model (PF-CLM) to estimate energy fluxes given variations in 19 vegetation and land surface parameters over a 144-hour period of time. Latent, sensible and ground heat fluxes from bare soil and grass vegetation were estimated using single column and tilted-v domains. Energy flux outputs, along with the corresponding input parameters, from each of the four scenario simulations were evaluated using active subspaces. The active subspace method considers parameter sensitivity by quantifying a weight for each parameter. The method also evaluates the potential for dimension reduction by identifying the input-output relationship through the active variable - a linear combination of input parameters. The aerodynamic roughness length was the most important parameter for bare soil energy fluxes. Multiple parameters were important for energy fluxes from vegetated surfaces and depended on the type of energy flux. Relationships between land surface inputs and output fluxes varied between latent, sensible and ground heat, but were consistent between domain setup (i.e., with or without lateral flow) and vegetation type. A quadratic polynomial was used to describe the input-output relationship for these energy fluxes. The reduced-dimension model of land surface dynamics can be compared to observations or used to solve the inverse problem. Considering this work as a proof-of-concept, the active subspace method can be applied and extended to a range of domain setups, land cover types and time periods to obtain a reduced-form representation of any output of interest, provided that an active subspace exists. Active subspaces identify important input parameters and how they relate to output.Proof-of-concept domains show potential for dimension reduction of land surface.Important land surface parameters depend on land cover and flux type.Land surface inputs and energy flux outputs can be related by a quadratic polynomial.Lateral flow has negligible effect on the land surface parameter-flux relationship.

A density-matching approach for optimization under uncertainty

Abstract Modern computers enable methods for design optimization that account for uncertainty in the system—so-called optimization under uncertainty (OUU). We propose a metric for OUU that measures the distance between a designer-specified probability density function of the system response (the target ) and the system response’s density function at a given design. We study an OUU formulation that minimizes this distance metric over all designs. We discretize the objective function with numerical quadrature, and we approximate the response density function with a Gaussian kernel density estimate. We offer heuristics for addressing issues that arise in this formulation, and we apply the approach to a CFD-based airfoil shape optimization problem. We qualitatively compare the density-matching approach to a multi-objective robust design optimization to gain insight into the method.

Discovering an active subspace in a single-diode solar cell model

Predictions from science and engineering models depend on the values of the model’s input parameters. As the number of parameters increases, algorithmic parameter studies like optimization or uncertainty quantification require many more model evaluations. One way to combat this curse of dimensionality is to seek an alternative parameterization with fewer variables that produces comparable predictions. The active subspace is a low-dimensional linear subspace of the space of model inputs that captures the variability in the model’s predictions. We describe a method for checking if a model admits an exploitable active subspace, and we apply this method to a single-diode solar cell model. We find that the maximum power of the solar cell has a dominant one-dimensional active subspace in its space of five input parameters.

On Active Subspaces in Car Aerodynamics

The Active Subspace Method (ASM) is an emerging set of tools for dimensionality reduction in complex physical systems. It allows to discover low-dimensional trends in the quantity of interest by exploiting redundancies in the input variables and combining them linearly into so-called active variables. The purpose of this study is to assess the applicability and the benefit of the ASM in car aerodynamics. To that end, we apply the ASM to drag and lift computations of three different parameterized vehicle geometries of increasing complexity. We thereby assess the impact of adjoint-based gradient inaccuracies on the results of the ASM, devise and validate a methodology to apply the ASM in the absence of adjoint-based gradients, and exemplify the practical use of this methodology in car aerodynamics. For all investigated cases, the ASM reveals that a large portion of the overall variability of drag or lift is captured already by an active subspace of dimension one, thus providing physical insight into the main shape parameter dependencies. By projection into an active subspace of a suitably chosen dimension larger than one, it is demonstrated that the predictive accuracy of surrogate models for drag and lift can consistently be improved.

MODEL REDUCTION WITH MAPREDUCE-ENABLED TALL AND SKINNY SINGULAR VALUE DECOMPOSITION ∗

We present a method for computing reduced-order models of parameterized partial differential equation solutions. The key analytical tool is the singular value expansion of the parameterized solution, which we approximate with a singular value decomposition of a parameter snapshot matrix. To evaluate the reduced-order model at a new parameter, we interpolate a subset of the right singular vectors to generate the reduced-order models coefficients. We employ a novel method to select this subset that uses the parameter gradient of the right singular vectors to split the terms in the expansion, yielding a mean prediction and a prediction covariance---similar to a Gaussian process approximation. The covariance serves as a confidence measure for the reduced-order model. We demonstrate the efficacy of the reduced-order model using a parameter study of heat transfer in random media. The high-fidelity simulations produce more than 4TB of data; we compute the singular value decomposition and evaluate the reduced-or...

Time-dependent global sensitivity analysis with active subspaces for a lithium ion battery model

Renewable energy researchers use computer simulation to aid the design of lithium ion storage devices. The underlying models contain several physical input parameters that affect model predictions. Effective design and analysis must understand the sensitivity of model predictions to changes in model parameters, but global sensitivity analyses become increasingly challenging as the number of input parameters increases. Active subspaces are part of an emerging set of tools for discovering and exploiting low-dimensional structures in the map from high-dimensional inputs to model outputs. We extend linear and quadratic model-based heuristic for active sub- space discovery to time-dependent processes and apply the resulting technique to a lithium ion battery model. The results reveal low-dimensional structure and sensitivity metrics that a designer may exploit to study the relationship between parameters and predictions.

Scalable Environment for Quantification of Uncertainty and Optimization in Industrial Applications (SEQUOIA)

As part of the DARPA EQUiPS (Enabling Quantification of Uncertainty in Physical Systems) program, the SEQUOIA project provides an integrated plan for performing uncertainty quantification (UQ) and design under uncertainty (DUU) that aggressively pursues new frontiers in scale and complexity. In particular, the coordinated investments in this e↵ort will create advancements in scalable forward and inverse UQ algorithms and the rigorous quantification of model inadequacy, providing the primary foundation for the development of generalized stochastic design approaches that address the robustness and reliability of complex multi-disciplinary systems. This project will demonstrate new UQ methods on high-performance aircraft nozzle analysis and design problems that are simultaneously designed for aerodynamic performance, thermal and pressure loads, and fatigue, while subject to geometric constraints to be fully integrated with complex vehicle shapes. The ability to handle all kinds of uncertainty at very large scale will enable the design of future components and vehicles that can have a substantial impact on DARPAs mission. This paper describes the e↵orts of our team thus far, and the accomplishments we have completed in the pursuit of the overall goals of the project.