Multi-fidelity Algorithm Research Articles

An iterative multi-fidelity scheme for simulating multi-dimensional non-Gaussian random fields

This paper presents an efficient multi-fidelity scheme to simulate multi-dimensional non-Gaussian random fields that are specified by covariance functions and marginal probability distribution functions. To develop the multi-fidelity scheme, two numerical algorithms are proposed in turn. The first algorithm is used to simulate random samples of the random field. In this algorithm, initial random samples are first generated to meet the marginal distribution and an iterative procedure is adopted to change the ranking of the random samples to match the target covariance function. By using the above algorithm, random samples satisfying the target covariance function and the target marginal distribution are obtained, but it is computationally intensive and is not suitable to simulate large-scale random fields. By taking advantage of Karhunen–Loève expansion, a multi-fidelity algorithm is then proposed to reduce the computational effort. The random variables in Karhunen–Loève expansion are calculated via performing the first algorithm on the low-fidelity model and the deterministic functions in Karhunen–Loève expansion are solved on the high-fidelity model. In this way, the proposed method has low computational effort and a high fidelity simultaneously. Three numerical examples, including two- and three-dimensional non-Gaussian random fields, are used to verify the good performance of the proposed algorithms.

Multifidelity Algorithm for the Sensitivity Analysis of Multidisciplinary Problems

AbstractThe present article proposes an algorithm for the sensitivity analysis of a multidisciplinary problem, in which the derivative-based global sensitivity indices are computed with multifidelity Gaussian process models. Two levels of fidelity are used to estimate the indices, where the low-fidelity samples are obtained by stopping the multidisciplinary analysis solver before convergence. A dedicated refinement strategy for the multifidelity Gaussian process is proposed to ensure the accuracy of the sensitivity index estimation. This algorithm is tested on three multidisciplinary problems of increasing complexity (one analytical and two representative engineering design problems), and proved to be both reliable in detecting the noninfluential variables and computationally efficient, compared to classical Monte Carlo integration and to three other candidate algorithms.

Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations

The simultaneous optimization of aircraft shape and internal structural size for transonic flight is excessively costly. The analysis of the governing physics is expensive, in particular for highly flexible aircraft, and the search for optima using analysis samples can scale poorly with design space size. This paper has a two-fold purpose targeting the scalable reduction of analysis sampling. First, a new algorithm is explored for computing design derivatives by analytically linking objective definition, geometry differentiation, mesh construction, and analysis. The analytic computation of design derivatives enables the accurate use of more efficient gradient-based optimization methods. Second, the scalability of a multi-fidelity algorithm is assessed for optimization in high dimensions. This method leverages a multi-fidelity model during the optimization line search for further reduction of sampling costs. The multi-fidelity optimization is demonstrated for cases of aerodynamic and aeroelastic design considering both shape and structural sizing separately and in combination with design spaces ranging from 17 to 321 variables, which would be infeasible using typical, surrogate-based methods. The multi-fidelity optimization consistently led to a reduction in high-fidelity evaluations compared to single-fidelity optimization for the aerodynamic shape problems, but frequently resulted in a cost penalty for cases involving structural sizing. While the multi-fidelity optimizer was successfully applied to problems with hundreds of variables, the results underscore the importance of accurately computing gradients and motivate the extension of the approach to constrained optimization methods.

A Derivative-Free Line-Search Algorithm for Simulation-Driven Design Optimization Using Multi-Fidelity Computations

The paper presents a multi-fidelity extension of a local line-search-based derivative-free algorithm for nonsmooth constrained optimization (MF-CS-DFN). The method is intended for use in the simulation-driven design optimization (SDDO) context, where multi-fidelity computations are used to evaluate the objective function. The proposed algorithm starts using low-fidelity evaluations and automatically switches to higher-fidelity evaluations based on the line-search step length. The multi-fidelity algorithm is driven by a suitably defined threshold and initialization values for the step length, which are associated to each fidelity level. These are selected to increase the accuracy of the objective evaluations while progressing to the optimal solution. The method is demonstrated for a multi-fidelity SDDO benchmark, namely pertaining to the hull-form optimization of a destroyer-type vessel, aiming at resistance minimization in calm water at fixed speed. Numerical simulations are based on a linear potential flow solver, where seven fidelity levels are used selecting systematically refined computational grids for the hull and the free surface. The method performance is assessed varying the steplength threshold and initialization approach. Specifically, four MF-CS-DFN setups are tested, and the optimization results are compared to its single-fidelity (high-fidelity-based) counterpart (CS-DFN). The MF-CS-DFN results are promising, achieving a resistance reduction of about 12% and showing a faster convergence than CS-DFN. Specifically, the MF extension is between one and two orders of magnitude faster than the original single-fidelity algorithm. For low computational budgets, MF-CS-DFN optimized designs exhibit a resistance that is about 6% lower than that achieved by CS-DFN.

A Bilevel Multi-Fidelity Polynomial Chaos Approach for the Uncertainty Quantification of MWCNT Interconnect Networks With Variable Imperfect Contact Resistances

With on-chip copper interconnects reaching their performance limits at 22 nanometer technology nodes, multi-walled carbon nanotube (MWCNT) interconnects are projected to replace them below this point. A major aspect of MWCNT interconnect design is to perform uncertainty quantification (UQ) in an efficient yet accurate manner. In this paper, a polynomial chaos (PC) based approach is developed for the UQ of MWCNT interconnect networks under the condition that some shells of each conductor of the network are perfectly contacted while others are imperfectly contacted. The key feature of the proposed approach is the development of a bilevel multi-fidelity algorithm where two different low-fidelity models are combined together. The main outcome of using this bilevel approach is to further reduce the computational time cost of state-of-the-art single level multi-fidelity algorithms, especially in the presence of variable imperfect contact resistances where single level multi-fidelity algorithms fail to provide much speedup over conventional PC approaches. The proposed approach adopts a SPICE hybrid model that combines the features of the equivalent single conductor (ESC) model and the rigorous multiconductor circuit (MCC) model of the MWCNT conductors. Then the low-fidelity ESC model, the intermediate-fidelity hybrid model, and the high-fidelity MCC model are exploited in a bilevel multi-fidelity algorithm for the recovery of the PC metamodel of the interconnect network. This proposed bilevel multi-fidelity algorithm is demonstrably 3-5x more numerically efficient than state-of-the-art single level multi-fidelity algorithms while being even more accurate. Once recovered, the PC metamodel is used to derive all statistical information of the network transient responses.

Two-Level Multifidelity Algorithm With Dimension Reduction for Efficient Uncertainty Quantification of MWCNT Interconnects

In this article, a polynomial chaos (PC) approach to quantify the uncertainty in transient simulation results of multiwalled carbon nanotube interconnect networks is presented. The proposed algorithm offers two distinct levels of numerical efficiency. The first level of efficiency comes from the use of a multifidelity formulation where the numerical expediency of a crude low-fidelity model of the network is combined with the accuracy of a high-fidelity model. At the second level, the crude low-fidelity model is further leveraged to perform dimension reduction. Consequently, the proposed algorithm offers substantially more speedup in the training of PC metamodels than existing techniques. The advantages of the proposed approach are validated using multiple numerical examples.

Aerostructural Design Optimization Using a Multifidelity Quasi-Newton Method

The traditional aircraft design process relies upon low-fidelity models for expedience and resource savings. However, the reduced accuracy and reliability of low-fidelity tools often lead to the discovery of design defects or inadequacies late in the design process. These deficiencies result in either costly changes or the acceptance of a configuration that does not meet expectations. Multifidelity methods attempt to blend the increased accuracy and reliability of high-fidelity models with the reduced cost of low-fidelity models. A new multifidelity algorithm has been proposed, combining elements from typical trust region model management and classical quasi-Newton methods. In this paper, the algorithm is compared with a single-fidelity quasi-Newton method for complex aeroelastic simulations. The vehicle design problem includes variables for planform shape, structural sizing, and cruise condition with constraints on trim and structural stresses. Considering the objective function reduction versus computational expenditure, the multifidelity process performs better in three of four cases in early iterations compared with a single-fidelity approach. A contracting trust region is found to slow the progress of the multifidelity optimizer. However, by leveraging the approximate inverse Hessian, the optimization can be seamlessly continued using high-fidelity data alone. Ultimately, the proposed new algorithm produces better designs in the cases considered. Investigating the return on investment confirms that the multifidelity advantage is greatest in early iterations, and managing the transition to pure high-fidelity optimization is critical.

One-dimensional modeling of fractional flow reserve in coronary artery disease: Uncertainty quantification and Bayesian optimization

Non-invasive estimation of fractional flow reserve (FFR) values, the key index in the diagnosis of obstructive coronary artery disease, is a promising alternative to traditional way of performing invasive coronary angiography. With the advances in computational fluid dynamics (CFD), one can estimate FFR based on the solution obtained in a reconstructed coronary geometry from coronary computed tomography (CT) angiography. However, the computational cost to perform three-dimensional (3D) simulations has limited the use of CFD in most clinical settings. This could become more restrictive if one aims to quantify the uncertainty associated with FFR calculations due to the uncertainty in anatomic and physiologic properties as a significant number of 3D simulations is required to sample a relatively large parametric space. We have developed a predictive probabilistic model of FFR, which quantifies the uncertainty of the predicted values with significantly lower computational costs. Based on global sensitivity analysis, we first identify the important physiologic and anatomic parameters that impact the predictions of FFR. Our approach is to employ one-dimensional blood flow simulations of coronary trees that offer fast FFR predictions with uncertainty quantification in computing blood pressure and flow distributions within the coronaries. This is complemented with a multifidelity algorithm that is used to infer their optimal values using available patient-specific clinical measurements.

Multi-fidelity stochastic collocation method for computation of statistical moments

We present an efficient numerical algorithm to approximate the statistical moments of stochastic problems, in the presence of models with different fidelities. The method extends the multi-fidelity approximation method developed in [18,26]. By combining the efficiency of low-fidelity models and the accuracy of high-fidelity models, our method exhibits fast convergence with a limited number of high-fidelity simulations. We establish an error bound of the method and present several numerical examples to demonstrate the efficiency and applicability of the multi-fidelity algorithm.