A Generalized Approximate Control Variate Framework for Multifidelity Uncertainty Quantification

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 multi-model variance reduction schemes as special cases. We demonstrate that these existing strategies use recursive sampling strategies, and as a result, their maximum possible variance reduction is limited to that of a control variate algorithm that uses only a single low-fidelity model with known mean. This theoretical result holds regardless of the number of low-fidelity models and/or samples used to build the estimator. We then derive new sampling strategies within our framework that circumvent this limitation to make efficient use of all available information sources. In particular, we demonstrate that a significant gap can exist, of orders of magnitude in some cases, between the variance reduction achievable by using a single low-fidelity model and our non-recursive approach. We also present initial sample allocation approaches for exploiting this gap. They yield the greatest benefit when augmenting the high-fidelity model evaluations is impractical because, for instance, they arise from a legacy database. 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

On the Optimization of Approximate Control Variates with Parametrically Defined Estimators

Multi-model Monte Carlo methods, such as multi-level Monte Carlo (MLMC) and multifidelity Monte Carlo (MFMC), allow for efficient estimation of the expectation of a quantity of interest given a set of models of varying fidelities. Recently, it was shown that the MLMC and MFMC estimators are both instances of the approximate control variates (ACV) framework [Gorodetsky et al. 2020]. In that same work, it was also shown that hand-tailored ACV estimators could outperform MLMC and MFMC for a variety of model scenarios. Because there is no reason to believe that these hand-tailored estimators are the best among a myriad of possible ACV estimators, a more general approach to estimator construction is pursued in this work. First, a general form of the ACV estimator variance is formulated. Then, the formulation is utilized to generate parametrically-defined estimators. These parametrically-defined estimators allow for an optimization to be pursued over a larger domain of possible ACV estimators. The parametrically-defined estimators are tested on a large set of model scenarios, and it is found that the broader search domain enabled by parametrically-defined estimators leads to greater variance reduction

Survey of multifidelity methods in uncertainty propagation, inference, and optimization

In many situations across computational science and engineering, multiple computational models are available that describe a system of interest. These different models have varying evaluation costs and varying fidelities. Typically, a computationally expensive high-fidelity model describes the system with the accuracy required by the current application at hand, while lower-fidelity models are less accurate but computationally cheaper than the high-fidelity model. Outer-loop applications, such as optimization, inference, and uncertainty quantification, require multiple model evaluations at many different inputs, which often leads to computational demands that exceed available resources if only the high-fidelity model is used. This work surveys multifidelity methods that accelerate the solution of outer-loop applications by combining high-fidelity and low-fidelity model evaluations, where the low-fidelity evaluations arise from an explicit low-fidelity model (e.g., a simplified physics approximation, a reduced model, a data-fit surrogate, etc.) that approximates the same output quantity as the high-fidelity model. The overall premise of these multifidelity methods is that low-fidelity models are leveraged for speedup while the high-fidelity model is kept in the loop to establish accuracy and/or convergence guarantees. We categorize multifidelity methods according to three classes of strategies: adaptation, fusion, and filtering. The paper reviews multifidelity methods in the outer-loop contexts of uncertainty propagation, inference, and optimization.Comment: will appear in SIAM Revie

Global Sensitivity Analysis and Estimation of Model Error, Toward Uncertainty Quantification in Scramjet Computations

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 systems 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.Comment: Preprint 29 pages, 10 figures (26 small figures); v1 submitted to the AIAA Journal on May 3, 2017; v2 submitted on September 17, 2017. v2 changes: (a) addition of flowcharts in Figures 4 and 5 to summarize the tools used; (b) edits to clarify and reorganize certain parts; v3 submitted on February 7, 2018. v3 changes: (a) title; (b) minor edit

Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices

We introduce a multifidelity estimator of covariance matrices formulated as the solution to a regression problem on the manifold of symmetric positive definite matrices. The estimator is positive definite by construction, and the Mahalanobis distance minimized to obtain it possesses properties which enable practical computation. We show that our manifold regression multifidelity (MRMF) covariance estimator is a maximum likelihood estimator under a certain error model on manifold tangent space. More broadly, we show that our Riemannian regression framework encompasses existing multifidelity covariance estimators constructed from control variates. We demonstrate via numerical examples that our estimator can provide significant decreases, up to one order of magnitude, in squared estimation error relative to both single-fidelity and other multifidelity covariance estimators. Furthermore, preservation of positive definiteness ensures that our estimator is compatible with downstream tasks, such as data assimilation and metric learning, in which this property is essential.Comment: 30 pages + 15-page supplemen

Bi-fidelity Stochastic Gradient Descent for Structural Optimization under Uncertainty

The presence of uncertainty in material properties and geometry of a structure is ubiquitous. The design of robust engineering structures, therefore, needs to incorporate uncertainty in the optimization process. Stochastic gradient descent (SGD) method can alleviate the cost of optimization under uncertainty, which includes statistical moments of quantities of interest in the objective and constraints. However, the design may change considerably during the initial iterations of the optimization process which impedes the convergence of the traditional SGD method and its variants. In this paper, we present two SGD based algorithms, where the computational cost is reduced by employing a low-fidelity model in the optimization process. In the first algorithm, most of the stochastic gradient calculations are performed on the low-fidelity model and only a handful of gradients from the high-fidelity model are used per iteration, resulting in an improved convergence. In the second algorithm, we use gradients from low-fidelity models to be used as control variate, a variance reduction technique, to reduce the variance in the search direction. These two bi-fidelity algorithms are illustrated first with a conceptual example. Then, the convergence of the proposed bi-fidelity algorithms is studied with two numerical examples of shape and topology optimization and compared to popular variants of the SGD method that do not use low-fidelity models. The results show that the proposed use of a bi-fidelity approach for the SGD method can improve the convergence. Two analytical proofs are also provided that show the linear convergence of these two algorithms under appropriate assumptions.Comment: 38 pages, 24 figure

Active Learning with Multifidelity Modeling for Efficient Rare Event Simulation

While multifidelity modeling provides a cost-effective way to conduct uncertainty quantification with computationally expensive models, much greater efficiency can be achieved by adaptively deciding the number of required high-fidelity (HF) simulations, depending on the type and complexity of the problem and the desired accuracy in the results. We propose a framework for active learning with multifidelity modeling emphasizing the efficient estimation of rare events. Our framework works by fusing a low-fidelity (LF) prediction with an HF-inferred correction, filtering the corrected LF prediction to decide whether to call the high-fidelity model, and for enhanced subsequent accuracy, adapting the correction for the LF prediction after every HF model call. The framework does not make any assumptions as to the LF model type or its correlations with the HF model. In addition, for improved robustness when estimating smaller failure probabilities, we propose using dynamic active learning functions that decide when to call the HF model. We demonstrate our framework using several academic case studies and two finite element (FE) model case studies: estimating Navier-Stokes velocities using the Stokes approximation and estimating stresses in a transversely isotropic model subjected to displacements via a coarsely meshed isotropic model. Across these case studies, not only did the proposed framework estimate the failure probabilities accurately, but compared with either Monte Carlo or a standard variance reduction method, it also required only a small fraction of the calls to the HF model

Multi-scale variance reduction methods based on multiple control variates for kinetic equations with uncertainties

The development of efficient numerical methods for kinetic equations with stochastic parameters is a challenge due to the high dimensionality of the problem. Recently we introduced a multiscale control variate strategy which is capable to accelerate considerably the slow convergence of standard Monte Carlo methods for uncertainty quantification. Here we generalize this class of methods to the case of multiple control variates. We show that the additional degrees of freedom can be used to improve further the variance reduction properties of multiscale control variate methods.Comment: arXiv admin note: text overlap with arXiv:1810.1084

Multidelity approaches for design under uncertainty

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2013.This electronic version was submitted and approved by the author's academic department as part of an electronic thesis pilot project. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from department-submitted PDF version of thesis.Includes bibliographical references (p. 113-117).Uncertainties are present in many engineering applications and it is important to account for their effects during engineering design to achieve robust and reliable systems. One approach is to represent uncertainties as random inputs to the numerical model of the system and investigate the probabilistic behaviour of the model outputs. However, performing optimization in this setting can be computationally expensive, requiring many evaluations of the numerical model to compute the statistics of the system metrics, such as the mean and the variance of the system performance. Fortunately, in many engineering applications, there are one or more lower fidelity models that approximate the original (high-fidelity) numerical model at lower computational costs. This thesis presents rigorous multifidelity approaches to leverage cheap low-fidelity models and other approximations of the expensive high-fidelity model to reduce the computational expense of optimization under uncertainty. Solving an optimization under uncertainty problem can require estimates of the statistics at many different design points, incurring a significant number of expensive high-fidelity model evaluations. The multifidelity estimator is developed based on the control variate method to reduce the computational cost of achieving a specified root mean square error in the statistic estimate by making use of the correlation between the outputs of the expensive high-fidelity model and the outputs of the cheap low-fidelity model. The method optimally relegates some of the computational load to the low-fidelity model based on the relative model evaluation cost and the strength of the correlation. It has demonstrated 85% computational savings in an acoustic horn robust optimization example. When the model is sufficiently smooth in the design space in the sense that a small change in the design variables produces a small change in the model outputs, it has an autocorrelation structure that can be exploited by the control variate method. The information reuse estimator is developed to reduce the computational cost of achieving a specified root mean square error in the statistic estimate by making use of the correlation between the high-fidelity model outputs at one design point and those at a previously visited design point. As the optimization progresses towards the optimum in the design space, the steps taken in the design space often become shorter, increasing the correlation and making the information reuse estimator more efficient. To further reduce the computational cost, the combined estimator is developed to incorporate the features of both the multifidelity estimator and the information reuse estimator. It has demonstrated 90% computational savings in the acoustic horn robust optimization example. The methods developed in this thesis are applied to two practical aerospace applications. In conceptual aircraft design, there are often uncertainties about the future developments of the underlying technologies. The information reuse estimator can be used to efficiently generate a Pareto front to study the trade off between the expected performance and the risk induced by the uncertainties in the different aircraft designs. In a large-scale wing robust optimization problem with uncertainties in material properties and flight conditions, the combined estimator demonstrated a reasonable solution turnaround time of 9.7 days on a 16-processor desktop machine, paving the way to a larger scale wing optimization problem with distributed uncertainties to account for degradation or damage.by Leo Wai-Tsun Ng.Ph.D

Fast Adjoint-assisted Multilevel Multi delity Method for Uncertainty Quanti cation of the Aleatoric Kind

PhDIn this thesis an adjoint-based multilevel multi delity Monte Carlo (MLMF) method is proposed, analysed, and demonstrated using test problems. Firstly, a multifi delity framework using the approximate function evaluation [1] based on the adjoint error correction of Giles et al. [2] is employed as a low fidelity model. This multifi delity framework is analysed using the method proposed by Ng and Wilcox [3]. The computational cost reduction and accuracy is demonstrated using the viscous Burgers' equation subject to uncertain boundary condition. The multi fidelity framework is extended to include multilevel meshes using the MLMF of Geraci [4] called the FastUQ. Some insights on parameters affecting computational cost are shown. The implementation of FastUQ in Dakota toolkit is outlined. As a demonstration, FastUQ is used to quantify uncertainties in aerodynamic parameters due to surface variations caused by manufacturing process. A synthetic model for surface variations due to manufacturing process is proposed based on Gaussian process. The LS89 turbine cascade subject to this synthetic disturbance model at two o -design conditions is used as a test problem. Extraction of independent random modes and truncation using a goal-based principal component analysis is shown. The analysis includes truncation for problems involving multiple QoIs and test conditions. The results from FastUQ are compared to the state-of-art SMLMC method and the approximate function evaluation using adjoint error correction called the inexpensive Monte Carlo method (IMC). About 70% reduction in computational cost compared to SMLMC is achieved without any loss of accuracy. The approximate model based on the IMC has high deviations for non-linear and sensitive QoI, namely the total-pressure loss. FastUQ control variate effectively balances the low fi delity model errors and additional high fidelity evaluations to yield accurate results comparable to the high fidelity model.This work has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 642959