Publications

Gorodetsky, Alex A.; Tsuji, Kazuya T.; Jakeman, John D.; Geraci, Gianluca G.; Eldred, Michael S.

Abstract not provided.

Geraci, Gianluca G.; Eldred, Michael S.; Gorodetsky, Alex A.; Jakeman, John D.

Abstract not provided.

Computer Methods in Applied Mechanics and Engineering

Fleeter, Casey M.; Geraci, Gianluca G.; Schiavazzi, Daniele E.; Kahn, Andrew M.; Marsden, Alison L.

Standard approaches for uncertainty quantification in cardiovascular modeling pose challenges due to the large number of uncertain inputs and the significant computational cost of realistic three-dimensional simulations. We propose an efficient uncertainty quantification framework utilizing a multilevel multifidelity Monte Carlo (MLMF) estimator to improve the accuracy of hemodynamic quantities of interest while maintaining reasonable computational cost. This is achieved by leveraging three cardiovascular model fidelities, each with varying spatial resolution to rigorously quantify the variability in hemodynamic outputs. We employ two low-fidelity models (zero- and one-dimensional) to construct several different estimators. Our goal is to investigate and compare the efficiency of estimators built from combinations of these two low-fidelity model alternatives and our high-fidelity three-dimensional models. We demonstrate this framework on healthy and diseased models of aortic and coronary anatomy, including uncertainties in material property and boundary condition parameters. Our goal is to demonstrate that for this application it is possible to accelerate the convergence of the estimators by utilizing a MLMF paradigm. Therefore, we compare our approach to single fidelity Monte Carlo estimators and to a multilevel Monte Carlo approach based only on three-dimensional simulations, but leveraging multiple spatial resolutions. We demonstrate significant, on the order of 10 to 100 times, reduction in total computational cost with the MLMF estimators. We also examine the differing properties of the MLMF estimators in healthy versus diseased models, as well as global versus local quantities of interest. As expected, global quantities such as outlet pressure and flow show larger reductions than local quantities, such as those relating to wall shear stress, as the latter rely more heavily on the highest fidelity model evaluations. Similarly, healthy models show larger reductions than diseased models. In all cases, our workflow coupling Dakota's MLMF estimators with the SimVascular cardiovascular modeling framework makes uncertainty quantification feasible for constrained computational budgets.

Journal of Computational Physics

Gorodetsky, Alex A.; Geraci, Gianluca G.; Eldred, Michael S.; Jakeman, John D.

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.

Maniaci, David C.; Hsieh, Alan H.; Geraci, Gianluca G.; Seidl, Daniel T.; Herges, Thomas G.; Eldred, Michael S.; Blaylo, Myra L.; Houchens, Brent C.

Abstract not provided.

AIAA Scitech 2020 Forum

Blonigan, Patrick J.; Geraci, Gianluca G.; Rizzi, Francesco N.; Eldred, Michael S.

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.

Blonigan, Patrick J.; Geraci, Gianluca G.; Rizzi, Francesco N.; Eldred, Michael S.

Abstract not provided.

Hsieh, Alan H.; Maniaci, David C.; Herges, Thomas H.; Geraci, Gianluca G.; Seidl, Daniel T.; Eldred, Michael S.; Blaylock, Myra L.; Houchens, Brent C.

Abstract not provided.

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

Schiavazzi, Daniele E.; Fleeter, Casey M.; Geraci, Gianluca G.; Marsden, Alison L.

Predictions from numerical hemodynamics are increasingly adopted and trusted in the diagnosis and treatment of cardiovascular disease. However, the predictive abilities of deterministic numerical models are limited due to the large number of possible sources of uncertainty including boundary conditions, vessel wall material properties, and patient specific model anatomy. Stochastic approaches have been proposed as a possible improvement, but are penalized by the large computational cost associated with repeated solutions of the underlying deterministic model. We propose a stochastic framework which leverages three cardiovascular model fidelities, i.e., three-, one- and zero-dimensional representations of cardiovascular blood flow. Specifically, we employ multilevel and multifidelity estimators from Sandia's open-source Dakota toolkit to reduce the variance in our estimated quantities of interest, while maintaining a reasonable computational cost. The performance of these estimators in terms of computational cost reductions is investigated for both global and local hemodynamic indicators.

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

Menhorn, Friedrich M.; Geraci, Gianluca G.; Eldred, Michael S.; Marzouk, Youssef M.

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

Maniaci, David C.; Frankel, Ari L.; Geraci, Gianluca G.; Blaylock, Myra L.; Eldred, Michael S.

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.

International Journal for Uncertainty Quantification

Gorodetsky, Alex A.; Jakeman, John D.; Geraci, Gianluca G.; Eldred, Michael S.

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.

Friedrich, Menhorn F.; Geraci, Gianluca G.; Seidl, Daniel T.; Eldred, Michael S.; Ryan, King R.; Hans-Joachim, Bungartz H.; Youssef, Marzouk Y.

Abstract not provided.

Geraci, Gianluca G.; Blonigan, Patrick J.; Rizzi, Francesco N.; Gorodetsky, Alex A.; carlberg, Kevin c.; Eldred, Michael S.

Abstract not provided.

Swiler, Laura P.; Geraci, Gianluca G.; Debusschere, Bert D.; Crussell, Jonathan C.

Abstract not provided.

Geraci, Gianluca G.; Eldred, Michael S.; Gorodetsky, Alex A.; Jakeman, John D.

Abstract not provided.

Eldred, Michael S.; Geraci, Gianluca G.; Seidl, Daniel T.; Menhorn, Friedrich M.; King, Ryan N.; Herges, Thomas H.; Hsieh, Alan H.; Maniaci, David C.

Abstract not provided.

Blonigan, Patrick J.; Geraci, Gianluca G.; Rizzi, Francesco N.; Eldred, Michael S.; carlberg, Kevin c.

Abstract not provided.

Eldred, Michael S.; Geraci, Gianluca G.; Gorodetsky, Alex A.; Jakeman, John D.

Abstract not provided.

Najm, H.N.; Safta, Cosmin S.; Huan, Xun H.; Casey, Tiernan A.; Sargsyan, Khachik S.; Oefelein, Joseph C.; Lacaze, Guilhem L.; Vane, Zachary P.; Eldred, Michael S.; Geraci, Gianluca G.

Abstract not provided.

Geraci, Gianluca G.; Eldred, Michael S.; Gorodetsky, Alex A.; Jakeman, John D.

Abstract not provided.

Stephens, John A.; Adams, Brian M.; Geraci, Gianluca G.

Abstract not provided.

Eldred, Michael S.; Geraci, Gianluca G.; Gorodetsky, Alex A.; Jakeman, John D.

Abstract not provided.

Casey, Tiernan A.; Debusschere, Bert D.; Eldred, Michael S.; Geraci, Gianluca G.; Ghanem, Roger G.; Jakeman, John D.; Marzouk, Youssef M.; Najm, H.N.; Safta, Cosmin S.; Sargsyan, Khachik S.

Abstract not provided.

Najm, H.N.; Safta, Cosmin S.; Huan, Xun H.; Casey, Tiernan A.; Sargsyan, Khachik S.; Oefelein, Joseph C.; Lacaze, Guilhem L.; Eldred, Michael S.; Geraci, Gianluca G.

Abstract not provided.