This report documents the results of an FY22 ASC V&V level 2 milestone demonstrating new algorithms for multifidelity uncertainty quantification. Part I of the report describes the algorithms, studies their performance on a simple model problem, and then deploys the methods to a thermal battery example from the open literature. Part II (restricted distribution) applies the multifidelity UQ methods to specific thermal batteries of interest to the NNSA/ASC program.
This project created and demonstrated a framework for the efficient and accurate prediction of complex systems with only a limited amount of highly trusted data. These next generation computational multi-fidelity tools fuse multiple information sources of varying cost and accuracy to reduce the computational and experimental resources needed for designing and assessing complex multi-physics/scale/component systems. These tools have already been used to substantially improve the computational efficiency of simulation aided modeling activities from assessing thermal battery performance to predicting material deformation. This report summarizes the work carried out during a two year LDRD project. Specifically we present our technical accomplishments; project outputs such as publications, presentations and professional leadership activities; and the project’s legacy.
We present an adaptive algorithm for constructing surrogate models of multi-disciplinary systems composed of a set of coupled components. With this goal we introduce “coupling” variables with a priori unknown distributions that allow surrogates of each component to be built independently. Once built, the surrogates of the components are combined to form an integrated-surrogate that can be used to predict system-level quantities of interest at a fraction of the cost of the original model. The error in the integrated-surrogate is greedily minimized using an experimental design procedure that allocates the amount of training data, used to construct each component-surrogate, based on the contribution of those surrogates to the error of the integrated-surrogate. The multi-fidelity procedure presented is a generalization of multi-index stochastic collocation that can leverage ensembles of models of varying cost and accuracy, for one or more components, to reduce the computational cost of constructing the integrated-surrogate. Extensive numerical results demonstrate that, for a fixed computational budget, our algorithm is able to produce surrogates that are orders of magnitude more accurate than methods that treat the integrated system as a black-box.
Bayesian optimization (BO) is an efficient and flexible global optimization framework that is applicable to a very wide range of engineering applications. To leverage the capability of the classical BO, many extensions, including multi-objective, multi-fidelity, parallelization, and latent-variable modeling, have been proposed to address the limitations of the classical BO framework. In this work, we propose a novel multi-objective BO formalism, called srMO-BO-3GP, to solve multi-objective optimization problems in a sequential setting. Three different Gaussian processes (GPs) are stacked together, where each of the GPs is assigned with a different task. The first GP is used to approximate a single-objective computed from the multi-objective definition, the second GP is used to learn the unknown constraints, and the third one is used to learn the uncertain Pareto frontier. At each iteration, a multi-objective augmented Tchebycheff function is adopted to convert multi-objective to single-objective, where the regularization with a regularized ridge term is also introduced to smooth the single-objective function. Finally, we couple the third GP along with the classical BO framework to explore the convergence and diversity of the Pareto frontier by the acquisition function for exploitation and exploration. The proposed framework is demonstrated using several numerical benchmark functions, as well as a thermomechanical finite element model for flip-chip package design optimization.
We present an adaptive algorithm for constructing surrogate models for integrated systems composed of a set of coupled components. With this goal we introduce ‘coupling’ variables with a priori unknown distributions that allow approximations of each component to be built independently. Once built, the surrogates of the components are combined and used to predict system-level quantities of interest (QoI) at a fraction of the cost of interrogating the full system model. We use a greedy experimental design procedure, based upon a modification of Multi-Index Stochastic Collocation (MISC), to minimize the error of the combined surrogate. This is achieved by refining each component surrogate in accordance with its relative contribution to error in the approximation of the system-level QoI. Our adaptation of MISC is a multi-fidelity procedure that can leverage ensembles of models of varying cost and accuracy, for one or more components, to produce estimates of system-level QoI. Several numerical examples demonstrate the efficacy of the proposed approach on systems involving feed-forward and feedback coupling. For a fixed computational budget, the proposed algorithm is able to produce approximations that are orders of magnitude more accurate than approximations that treat the integrated system as a black-box.
We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelerates the estimation of statistics of computationally expensive simulation models using an ensemble of models with lower cost. These lower cost models — which are typically lower fidelity with unknown statistics — are used to reduce the variance in statistical estimators relative to a MC estimator with equivalent cost. We derive the conditions under which our proposed approximate control variate framework recovers existing multifidelity variance reduction schemes as special cases. We demonstrate that existing recursive/nested strategies are suboptimal because they use the additional low-fidelity models only to efficiently estimate the unknown mean of the first low-fidelity model. As a result, they cannot achieve variance reduction beyond that of a control variate estimator that uses a single low-fidelity model with known mean. However, there often exists about an order-of-magnitude gap between the maximum achievable variance reduction using all low-fidelity models and that achieved by a single low-fidelity model with known mean. We show that our proposed approach can exploit this gap to achieve greater variance reduction by using non-recursive sampling schemes. The proposed strategy reduces the total cost of accurately estimating statistics, especially in cases where only low-fidelity simulation models are accessible for additional evaluations. Several analytic examples and an example with a hyperbolic PDE describing elastic wave propagation in heterogeneous media are used to illustrate the main features of the methodology.
Truly predictive numerical simulations can only be obtained by performing Uncertainty Quantification. However, many realistic engineering applications require extremely complex and computationally expensive high-fidelity numerical simulations for their accurate performance characterization. Very often the combination of complex physical models and extreme operative conditions can easily lead to hundreds of uncertain parameters that need to be propagated through high-fidelity codes. Under these circumstances, a single fidelity uncertainty quantification approach, i.e. a workflow that only uses high-fidelity simulations, is unfeasible due to its prohibitive overall computational cost. To overcome this difficulty, in recent years multifidelity strategies emerged and gained popularity. Their core idea is to combine simulations with varying levels of fidelity/accuracy in order to obtain estimators or surrogates that can yield the same accuracy of their single fidelity counterparts at a much lower computational cost. This goal is usually accomplished by defining a priori a sequence of discretization levels or physical modeling assumptions that can be used to decrease the complexity of a numerical model realization and thus its computational cost. Less attention has been dedicated to low-fidelity models that can be built directly from a small number of available high-fidelity simulations. In this work we focus our attention on reduced order models (ROMs). Our main goal in this work is to investigate the combination of multifidelity uncertainty quantification and ROMs in order to evaluate the possibility to obtain an efficient framework for propagating uncertainties through expensive numerical codes. We focus our attention on sampling-based multifidelity approaches, like the multifidelity control variate, and we consider several scenarios for a numerical test problem, namely the Kuramoto-Sivashinsky equation, for which the efficiency of the multifidelity-ROM estimator is compared to the standard (single-fidelity) Monte Carlo approach.
Bayesian optimization (BO) is an efficient and flexible global optimization framework that is applicable to a very wide range of engineering applications. To leverage the capability of the classical BO, many extensions, including multi-objective, multi-fidelity, parallelization, and latent-variable modeling, have been proposed to address the limitations of the classical BO framework. In this work, we propose a novel multi-objective (MO) extension, called srMOBO-3GP, to solve the MO optimization problems in a sequential setting. Three different Gaussian processes (GPs) are stacked together, where each of the GP is assigned with a different task: the first GP is used to approximate a single-objective computed from the MO definition, the second GP is used to learn the unknown constraints, and the third GP is used to learn the uncertain Pareto frontier. At each iteration, a MO augmented Tchebycheff function converting MO to single-objective is adopted and extended with a regularized ridge term, where the regularization is introduced to smooth the single-objective function. Finally, we couple the third GP along with the classical BO framework to explore the richness and diversity of the Pareto frontier by the exploitation and exploration acquisition function. The proposed framework is demonstrated using several numerical benchmark functions, as well as a thermomechanical finite element model for flip-chip package design optimization.
Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018
SNOWPAC (Stochastic Nonlinear Optimization With Path-Augmented Constraints) is a method for stochastic nonlinear constrained derivative-free optimization. For such problems, it extends the path-augmented constraints framework introduced by the deterministic optimization method NOWPAC and uses a noise-adapted trust region approach and Gaussian processes for noise reduction. In recent developments, SNOWPAC is available in the DAKOTA framework which offers a highly flexible interface to couple the optimizer with different sampling strategies or surrogate models. In this paper we discuss details of SNOWPAC and demonstrate the coupling with DAKOTA. We showcase the approach by presenting design optimization results of a shape in a 2D supersonic duct. This simulation is supposed to imitate the behavior of the flow in a SCRAMJET simulation but at a much lower computational cost. Additionally different mesh or model fidelities can be tested. Thus, it serves as a convenient test case before moving to costly SCRAMJET computations. Here, we study deterministic results and results obtained by introducing uncertainty on inflow parameters. As sampling strategies we compare classical Monte Carlo sampling with multilevel Monte Carlo approaches for which we developed new error estimators. All approaches show a reasonable optimization of the design over the objective while maintaining or seeking feasibility. Furthermore, we achieve significant reductions in computational cost by using multilevel approaches that combine solutions from different grid resolutions.
Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018
Wind energy is stochastic in nature; the prediction of aerodynamic quantities and loads relevant to wind energy applications involves modeling the interaction of a range of physics over many scales for many different cases. These predictions require a range of model fidelity, as predictive models that include the interaction of atmospheric and wind turbine wake physics can take weeks to solve on institutional high performance computing systems. In order to quantify the uncertainty in predictions of wind energy quantities with multiple models, researchers at Sandia National Laboratories have applied Multilevel-Multifidelity methods. A demonstration study was completed using simulations of a NREL 5MW rotor in an atmospheric boundary layer with wake interaction. The flow was simulated with two models of disparate fidelity; an actuator line wind plant large-eddy scale model, Nalu, using several mesh resolutions in combination with a lower fidelity model, OpenFAST. Uncertainties in the flow conditions and actuator forces were propagated through the model using Monte Carlo sampling to estimate the velocity defect in the wake and forces on the rotor. Coarse-mesh simulations were leveraged along with the lower-fidelity flow model to reduce the variance of the estimator, and the resulting Multilevel-Multifidelity strategy demonstrated a substantial improvement in estimator efficiency compared to the standard Monte Carlo method.
This paper presents a Bayesian multifidelity uncertainty quantification framework, called MFNets, which can be used to overcome three of the major challenges that arise when data from different sources are used to enhance statistical estimation and prediction with quantified uncertainty. Specifically, we demonstrate that MFNets can (1) fuse heterogeneous data sources arising from simulations with different parameterizations, e.g., simulation models with different uncertain parameters or data sets collected under different environmental conditions; (2) encode known relationships among data sources to reduce data requirements; and (3) improve the robustness of existing multifidelity approaches to corrupted data. In this paper we use MFNets to construct linear-subspace surrogates and estimate statistics using Monte Carlo sampling. In addition to numerical examples highlighting the efficacy of MFNets we also provide a number of theoretical results. Firstly we provide a mechanism to assess the quality of the posterior mean of a MFNets Monte Carlo estimator as a frequentist estimator. We then use this result to compare MFNets estimators to existing single fidelity, multilevel, and control variate Monte Carlo estimators. In this context, we show that the Monte Carlo-based control variate estimator can be derived entirely from the use of Bayes rule and linear-Gaussian models—to our knowledge the first such derivation. Finally, we demonstrate the ability to work with different uncertain parameters across different models.