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.

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In this work a novel adaptive strategy for stochastic problems, inspired to the classical Harten's framework, is presented. The proposed algorithm allows building, in a very general manner, stochastic numerical schemes starting from a whatever type of deterministic schemes and handling a large class of problems, from unsteady to discontinuous solutions. Its formulations permits to recover the same results concerning the interpolation theory of the classical multiresolution approach, but with an extension to uncertainty quantification problems. The interest of the present strategy is demonstrated by performing several numerical problems where different forms of uncertainty distributions are taken into account, such as discontinuous and unsteady custom-defined probability density functions. In addition to algebraic and ordinary differential equations, numerical results for the challenging 1D Kraichnan-Orszag are reported in terms of accuracy and convergence. Finally, a two degree-of-freedom aeroelastic model for a subsonic case is presented. Though quite simple, the model allows recovering some physical key aspect, on the fluid/structure interaction, thanks to the quasi-steady aerodynamic approximation employed. The injection of an uncertainty is chosen in order to obtain a complete parameterization of the mass matrix. All the numerical results are compared with respect to classical Monte Carlo solution and with a non-intrusive Polynomial Chaos method.

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 derive two low-fidelity models (zero-and one-dimensional) that we use to obtain different estimators. Our goal is to investigate and compare the efficiency of estimators built from 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 workflow make uncertainty quantification feasible for constrained computational budgets. range of diseases and anatomies. Coronary artery disease is the most prevalent cause of death in the United States [4], and has thus been the subject of many modeling studies. These include computing fractional flow reserve for patients with coronary stenoses [5] and assessing differences in the hemodynamic and mechanical response of arterial and venous grafts towards determining cause of vein graft failure [6]. Computational models have also been applied to congenital heart disease for prognosis and treatment planning, from testing designs for surgical interventions of single ventricle congenital heart disease [7-9] to thrombotic risk assessment for Kawasaki disease [10][11][12], and assessment of disease progression and mechanical stimuli in pulmonary hypertension [13,14]. Hemodynamic studies are also used for additional disease cases, such as analyzing bloo...

This paper presents a bi-fidelity simulation approach to quantify the effect of uncertainty in the thermal boundary condition on the heat transfer in a ribbed channel. A numerical test case is designed where a random heat flux at the wall of a rectangular channel is applied to mimic the unknown temperature distribution in a realistic application. To predict the temperature distribution and the associated uncertainty over the channel wall, the fluid flow is simulated using 2D periodic steady Reynolds-Averaged Navier-Stokes (RANS) equations. The goal of this study is then to illustrate that the cost of propagating the heat flux uncertainty may be significantly reduced when two RANS models with different levels of fidelity, one low (cheap to simulate) and one high (expensive to evaluate), are used. The low-fidelity model is employed to learn a reduced basis and an interpolation rule that can be used, along with a small number of high-fidelity model evaluations, to approximate the high-fidelity solution at arbitrary samples of heat flux. Here, the low- and high-fidelity models are, respectively, the one-equation Spalart-Allmaras and the two-equation shear stress transport k–ω models. To further reduce the computational cost, the Spalart-Allmaras model is simulated on a coarser spatial grid and the non-linear solver is terminated prior to the solution convergence. It is illustrated that the proposed bi-fidelity strategy accurately approximates the target high-fidelity solution at randomly selected samples of the uncertain heat flux.