A robust optimization approach based on multi-fidelity metamodel

• Differentiable multi-fidelity inference: BMFIA enables gradient-based Bayesian inference for high-dimensional, non-differentiable multi-physics problems using low-fidelity model gradients. • Efficiency with minimal HF data: Accurate posterior estimates are achieved from only 100–300 HF–LF simulations with orders-of-magnitude speed-ups over HF-only inference. • Accuracy despite crude LF models: Robust posterior reconstruction is maintained even with weakly correlated or simplified low-fidelity models. High-dimensional Bayesian inverse analysis ( dim ≫ 100 ) is mostly unfeasible for computationally demanding, nonlinear physics-based high-fidelity (HF) models. Usually, the use of more efficient gradient-based inference schemes is impeded if the multi-physics models are provided by complex legacy codes. Adjoint-based derivatives are either exceedingly cumbersome to derive or nonexistent for practically relevant large-scale nonlinear and coupled multi-physics problems. Similarly, holistic automated differentiation w. r. t. primary variables of multi-physics codes is usually not yet an option and requires extensive code restructuring if not considered from the outset in the software design. This absence of differentiability further exacerbates the already present computational challenges. To overcome the existing limitations, we propose a novel inference approach called Bayesian multi-fidelity inverse analysis (BMFIA) , which leverages simpler and computationally cheaper lower-fidelity (LF) models that are designed to provide model derivatives. BMFIA learns a simple, probabilistic dependence of the LF and HF models, which is then employed in an altered likelihood formulation to statistically correct the inaccurate LF response. From a Bayesian viewpoint, this dependence represents a multi-fidelity (MF) conditional density (discriminative model). We demonstrate how this MF conditional density can be learned robustly in the small data regime from only a few HF and LF simulations (50 to 300), which would not be sufficient for naive surrogate approaches. The formulation is fully differentiable and allows the flexible design of a wide range of LF models. We demonstrate that BMFIA solves Bayesian inverse problems for scenarios that used to be prohibitive, such as finely-resolved and hence high-dimensional spatial reconstruction problems in two-dimensional Euclidean domains with static posteriors, given nonlinear and transient coupled poro-elastic media physics. We show that the resulting static MF posteriors are in excellent agreement with the (usually inaccessible) HF posteriors or ground-truth data and note that extending the framework to arbitrary three-dimensional domains is a natural and important direction for future work.

A novel robust optimization approach for an integrated municipal water distribution system design under uncertainty: A case study of Mashhad

Many researchers have studied Multi-fidelity (MF) models to obtain the optimum solution efficiently when the time of analysis and evaluation is long. The MF model is a meta-model that combines High fidelity (HF) data, which requires large computational cost for evaluation; and Low fidelity (LF) data, which needs small computation cost but has low accuracy. Therefore, integrating HF and LF data with a scale factor and correction model is highly important in MF modeling. This study investigates the performance of the MF model in terms of definition and estimation. First, three different MF models are built: An LF model multiplied by the scale factor, an LF model combined with a correction model, and an LF model with both scale factor and correction model. Second, the effects of scale factor on the MF model are analyzed in terms of constant vs. linear and maximum likelihood estimation vs. least squares estimation. Finally, a correction model is built using two methods, namely, interpolation and regression, to assess the influence of the correction model on the MF model. To evaluate the MF model performance, several test problems are applied, and the root mean square error of each model is estimated as the accuracy measure. In conclusion, the characteristics of different types of MF models are summarized and guidelines for the generation of MF models are proposed to approximate closely the precise models.

Coordinated operations of electricity and district heating networks offer a potential for mitigating inherent variability of renewable energy sources (RES) in the ongoing transition to smart grids. This paper proposes a two-stage distributionally robust optimization (DRO) approach to determine the optimal day-ahead unit commitment in coordinated electricity and district heating networks with variable RES power output. The proposed formulation is to minimize the worst-case expected total cost over an ambiguity set comprising a family of probability distributions with given support and moments of RES power output. As such, the proposed DRO approach can overcome the limitations of stochastic programming in its inherent dependence of exact probability distributions along with a huge computational burden, but also becomes less conservative than classical robust optimization. The pertinent DRO model is eventually reformulated as a tractable mixed-integer second-order cone (SOC) programming after employing linear decision rules and the SOC duality. Simplified affine policies are utilized to further improve computational tractability and performance. Finally, case studies are conducted based on Barry Island electricity and district heating networks. The numerical results demonstrate the decision-making superiority of the proposed method as compared with deterministic, stochastic programming, and robust optimization approaches. They also validate the computational improvement of the proposed approach by employing simplified affine policies.

Scaling Properties of Multi-Fidelity Shape Optimization Algorithms

Intelligent sensitivity analysis framework based on machine learning for spacecraft thermal design

Design approach for tilt propellers of UAM/eVTOLs for cruise and hover considering aerodynamic and aeroacoustic characteristics via a multi-fidelity model

This paper presents a methodology to move toward reliable real-time structural health monitoring (SHM). The proposed procedure relies upon surrogate modeling based on a multi-fidelity (MF) deep neural network (DNN), conceived to map damage and operational parameters onto sensor recordings. Within a stochastic framework, the MF-DNN is adopted by a Markov chain Monte Carlo sampling procedure to update the probability distribution of the structural state, conditioned on noisy observations. The MF-DNN enables to locate and possibly quantify the presence of damage, and its multi-fidelity configuration effectively blends datasets featuring different fidelities without any prior assumption. The training datasets are generated with physics-based models of the monitored structure: high fidelity (HF) and low fidelity (LF) models are considered to simulate the structural response under varying operational conditions, respectively in the presence or absence of a structural damage. The MF-DNN is a composition of a fully-connected LF-DNN, which mimics sensor recordings in the undamaged condition, and of a long short-term memory HF-DNN, which is exploited to enrich the LF approximation for the considered damaged scenarios. By framing the model updating strategy as an incremental or residual modeling problem, the MF-DNN is reported to provide numerous advantages over single-fidelity based models for SHM purposes.KeywordsBayesian model updatingDeep learningMarkov chain Monte CarloStructural health monitoringMulti-fidelity methodsReduced-order modelingReal-time damage identification

In this research, a self-scheduling problem for a power generation company (Genco), participating in a day-ahead power market is studied. A robust optimisation approach is followed to tackle uncertainty on the market prices. Due to the existing correlations among hourly market prices and in order to enhance the value of the objective function in an uncertain environment, a new robust optimisation approach is developed and presented to prevent over-conservative solutions. A couple of polyhedral uncertainty sets are applied to protect the optimal solution solely against any correlated perturbation. In addition two robust self-scheduling models are formulated under these uncertainty sets. The results of this study justify the performance of the proposed models compared to those of the existing robust self-scheduling model applied for conventional polyhedral uncertainty set.

PurposeEngineering system design and optimization problems are usually multi-objective and constrained and have uncertainties in the inputs. These uncertainties might significantly degrade the overall performance of engineering systems and change the feasibility of the obtained solutions. This paper aims to propose a multi-objective robust optimization approach based on Kriging metamodel (K-MORO) to obtain the robust Pareto set under the interval uncertainty.Design/methodology/approachIn K-MORO, the nested optimization structure is reduced into a single loop optimization structure to ease the computational burden. Considering the interpolation uncertainty from the Kriging metamodel may affect the robustness of the Pareto optima, an objective switching and sequential updating strategy is introduced in K-MORO to determine (1) whether the robust analysis or the Kriging metamodel should be used to evaluate the robustness of design alternatives, and (2) which design alternatives are selected to improve the prediction accuracy of the Kriging metamodel during the robust optimization process.FindingsFive numerical and engineering cases are used to demonstrate the applicability of the proposed approach. The results illustrate that K-MORO is able to obtain robust Pareto frontier, while significantly reducing computational cost.Practical implicationsThe proposed approach exhibits great capability for practical engineering design optimization problems that are multi-objective and constrained and have uncertainties.Originality/valueA K-MORO approach is proposed, which can obtain the robust Pareto set under the interval uncertainty and ease the computational burden of the robust optimization process.

A robust convex optimization approach to design a hierarchical organ transplant network: A case study

A reanalysis-based multi-fidelity (RBMF) surrogate framework for efficient structural optimization

A new approach for multi-objective robust design optimization has been proposed and applied to a real-world design problem with a large number of objective functions. The present approach is assisted by response surface approximation and visual data-mining, which results in two major gains regarding computational time and data interpretation. The Kriging model for response surface approximation can realize accurate predictions of robustness measures, and dramatically reduces the computational time for objective function evaluation. In addition, the use of self-organizing maps as a data-mining technique allows visualization of complicated design information between optimality and robustness of design in a comprehensible two-dimensional form. Therefore, the extraction and interpretation of trade-off relations between optimality and robustness of design, and also the location of sweet-spots in the design space, can be performed in a comprehensive manner.

In this work, the authors compare single point, multi-point, and non-deterministic optimization approaches applied to airfoil design. The objective is to minimize the drag coefficient using PABLO as the aerodynamics solver and Genetic Algorithm as the optimizer. First, the authors optimized the airfoil at fixed operating conditions for single point and multi-point optimization. Second, the airfoil was optimized at varying operating conditions. Comparison of drag coefficients from single and multi-point optimization reveal that single point optimization provides the lowest values at its operating conditions while multi-point optimization provides lower values at off-design conditions. Results from nondeterministic optimization shows that drag coefficient values are lower than values from multi-point optimization around cruise speed by 16% while providing acceptable values at lower speeds.

A generic class of multistage optimisation problems related to production/inventory management under Markovian uncertainty is introduced and investigated. For each instance in the class, it is shown how to construct state-space representable uncertainty sets at any probability level, thus leading to efficient resolution of both the stochastic and robust versions of the problem. Computational experiments aimed at comparing the optimal strategies corresponding to both versions in terms of risk are then reported and discussed; it is observed that the robust optimisation approach can significantly outperform the stochastic optimisation approach when targeting lower risk levels (typically less than 2%).