Publications

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A new method for fast evaluation of high dimensional integrals arising in quantum mechanics is proposed. Here, the method is based on sparse approximation of a high dimensional function followed by a low-rank compression. In the first step, we interpret the high dimensional integrand as a tensor in a suitable tensor product space and determine its entries by a compressed sensing based algorithm using only a few function evaluations. Secondly, we implement a rank reduction strategy to compress this tensor in a suitable low-rank tensor format using standard tensor compression tools. This allows representing a high dimensional integrand function as a small sum of products of low dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate this low-rank representation, thus alleviating the curse of dimensionality. Finally, numerical tests on synthetic functions, as well as on energy correction integrals for water and formaldehyde molecules demonstrate the efficiency of this method using very few function evaluations as compared to other integration strategies.

The development of scramjet engines is an important research area for advancing hypersonic and orbital flights. Progress toward optimal engine designs requires accurate flow simulations together with uncertainty quantification. However, performing uncertainty quantification for scramjet simulations is challenging due to the large number of uncertain parameters involved and the high computational cost of flow simulations. These difficulties are addressed in this paper by developing practical uncertainty quantification algorithms and computational methods, and deploying them in the current study to large-eddy simulations of a jet in crossflow inside a simplified HIFiRE Direct Connect Rig scramjet combustor. First, global sensitivity analysis is conducted to identify influential uncertain input parameters, which can help reduce the system’s stochastic dimension. Second, because models of different fidelity are used in the overall uncertainty quantification assessment, a framework for quantifying and propagating the uncertainty due to model error is presented. In conclusion, these methods are demonstrated on a nonreacting jet-in-crossflow test problem in a simplified scramjet geometry, with parameter space up to 24 dimensions, using static and dynamic treatments of the turbulence subgrid model, and with two-dimensional and three-dimensional geometries.

We conduct a global sensitivity analysis (GSA) of the Energy Exascale Earth System Model (E3SM), land model (ELM) to calculate the sensitivity of five key carbon cycle outputs to 68 model parameters. This GSA is conducted by first constructing a Polynomial Chaos (PC) surrogate via new Weighted Iterative Bayesian Compressive Sensing (WIBCS) algorithm for adaptive basis growth leading to a sparse, high-dimensional PC surrogate with 3,000 model evaluations. The PC surrogate allows efficient extraction of GSA information leading to further dimensionality reduction. The GSA is performed at 96 FLUXNET sites covering multiple plant functional types (PFTs) and climate conditions. About 20 of the model parameters are identified as sensitive with the rest being relatively insensitive across all outputs and PFTs. These sensitivities are dependent on PFT, and are relatively consistent among sites within the same PFT. The five model outputs have a majority of their highly sensitive parameters in common. A common subset of sensitive parameters is also shared among PFTs, but some parameters are specific to certain types (e.g., deciduous phenology). The relative importance of these parameters shifts significantly among PFTs and with climatic variables such as mean annual temperature.

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The development of scramjet engines is an important research area for advancing hypersonic and orbital flights. Progress toward optimal engine designs requires accurate flow simulations together with uncertainty quantification. However, performing uncertainty quantification for scramjet simulations is challenging due to the large number of uncertainparameters involvedandthe high computational costofflow simulations. These difficulties are addressedin this paper by developing practical uncertainty quantification algorithms and computational methods, and deploying themin the current studyto large-eddy simulations ofajet incrossflow inside a simplified HIFiRE Direct Connect Rig scramjet combustor. First, global sensitivity analysis is conducted to identify influential uncertain input parameters, which can help reduce the system's stochastic dimension. Second, because models of different fidelity are used in the overall uncertainty quantification assessment, a framework for quantifying and propagating the uncertainty due to model error is presented. These methods are demonstrated on a nonreacting jet-in-crossflow test problem in a simplified scramjet geometry, with parameter space up to 24 dimensions, using static and dynamic treatments of the turbulence subgrid model, and with two-dimensional and three-dimensional geometries.

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The development of scramjet engines is an important research area for advancing hypersonic and orbital flights. Progress towards optimal engine designs requires accurate and computationally affordable flow simulations, as well as uncertainty quantification (UQ). While traditional UQ techniques can become prohibitive under expensive simulations and high-dimensional parameter spaces, polynomial chaos (PC) surrogate modeling is a useful tool for alleviating some of the computational burden. However, non-intrusive quadrature-based constructions of PC expansions relying on a single high-fidelity model can still be quite expensive. We thus introduce a two-stage numerical procedure for constructing PC surrogates while making use of multiple models of different fidelity. The first stage involves an initial dimension reduction through global sensitivity analysis using compressive sensing. The second stage utilizes adaptive sparse quadrature on a multifidelity expansion to compute PC surrogate coefficients in the reduced parameter space where quadrature methods can be more effective. The overall method is used to produce accurate surrogates and to propagate uncertainty induced by uncertain boundary conditions and turbulence model parameters, for performance quantities of interest from large eddy simulations of supersonic reactive flows inside a scramjet engine.

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Compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quantification analysis of expensive and high-dimensional physical models. We perform numerical investigations employing several compressive sensing solvers that target the unconstrained LASSO formulation, with a focus on linear systems that arise in the construction of polynomial chaos expansions. With core solvers l1_ls, SpaRSA, CGIST, FPC_AS, and ADMM, we develop techniques to mitigate overfitting through an automated selection of regularization constant based on cross-validation, and a heuristic strategy to guide the stop-sampling decision. Practical recommendations on parameter settings for these techniques are provided and discussed. The overall method is applied to a series of numerical examples of increasing complexity, including large eddy simulations of supersonic turbulent jet-in-crossflow involving a 24-dimensional input. Through empirical phase-transition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy, and computational trade-offs between polynomial bases of different degrees, and practicability of conducting compressive sensing for a realistic, high-dimensional physical application. Across test cases studied in this paper, we find ADMM to have demonstrated empirical advantages through consistent lower errors and faster computational times.

Data movement is considered the main performance concern for exascale, including both on-node memory and off-node network communication. Indeed, many application traces show significant time spent in MPI calls, potentially indicating that faster networks must be provisioned for scalability. However, equating MPI times with network communication delays ignores synchronization delays and software overheads independent of network hardware. Using point-to-point protocol details, we explore the decomposition of MPI time into communication, synchronization and software stack components using architecture simulation. Detailed validation using Bayesian inference is used to identify the sensitivity of performance to specific latency/bandwidth parameters for different network protocols and to quantify associated uncertainties. The inference combined with trace replay shows that synchronization and MPI software stack overhead are at least as important as the network itself in determining time spent in communication routines.

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We are conducting a large-scale, long-term climate change response experiment in an ombrotrophic peat bog in Minnesota to evaluate the effects of warming and elevated CO2 on ecosystem processes using empirical and modeling approaches. To better frame future assessments of peatland responses to climate change, we characterized and compared spatial vs. temporal variation in measured C cycle processes and their environmental drivers. We also conducted a sensitivity analysis of a peatland C model to identify how variation in ecosystem parameters contributes to model prediction uncertainty. High spatial variability in C cycle processes resulted in the inability to determine if the bog was a C source or sink, as the 95% confidence interval ranged from a source of 50 g C m-2 yr-1 to a sink of 67 g C m-2 yr-1. Model sensitivity analysis also identified that spatial variation in tree and shrub photosynthesis, allocation characteristics, and maintenance respiration all contributed to large variations in the pretreatment estimates of net C balance. Variation in ecosystem processes can be more thoroughly characterized if more measurements are collected for parameters that are highly variable over space and time, and especially if those measurements encompass environmental gradients that may be driving the spatial and temporal variation (e.g., hummock vs. hollow microtopographies, and wet vs. dry years). Together, the coupled modeling and empirical approaches indicate that variability in C cycle processes and their drivers must be taken into account when interpreting the significance of experimental warming and elevated CO2 treatments.

The UQ Toolkit (UQTk) is a collection of libraries and tools for the quantification of uncertainty in numerical model predictions. Version 3.0.4 offers intrusive and non-intrusive methods for propagating input uncertainties through computational models, tools for sensitivity analysis, methods for sparse surrogate construction, and Bayesian inference tools for inferring parameters from experimental data. This manual discusses the download and installation process for UQTk, provides pointers to the UQ methods used in the toolkit, and describes some of the examples provided with the toolkit.

A new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm−1 or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.

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