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We consider a multi-objective optimization problem with objective functions that are expensive to evaluate. The decision maker (DM) has unknown preferences, and so the standard approach is to generate an approximation of the Pareto front and let the DM choose from the generated non-dominated designs. However, especially for expensive to evaluate problems where the number of designs that can be evaluated is very limited, the true best solution according to the DM's unknown preferences is unlikely to be among the small set of non-dominated solutions found, even if these solutions are truly Pareto optimal. We address this issue by using a multi-objective Bayesian optimization (BO) algorithm (see, e.g., [5]) and allowing the DM to select a preferred solution from a predicted continuous Pareto front just once before the end of the algorithm rather than selecting a solution after the end. This allows the algorithm to understand the DM's preferences and make a final attempt to identify a more preferred solution. We demonstrate the idea using ParEGO [2], and show empirically that the found solutions are significantly better in terms of true DM preferences than if the DM would simply pick a solution at the end.

This paper presents the multi-objective optimisation of the MH114 high-lift airfoil. We seek the set of Pareto optimal solutions that maximise the airfoil lift and minimise the drag. The lift and drag forces are considered uncertain due to geometrical uncertainties. The uncer- tainty quantification of the probabilistic aerodynamic force values re- quires a large number of samples. However, the prediction of the aero- dynamic forces is expensive due to the numerical solution of the Navier- Stokes equations. Therefore, a multi-fidelity surrogate assisted approach is employed to combine expensive RANS simulations with cheap poten- tial flow calculations. The multi-fidelity surrogate-based approach allows us to economically optimise the aerodynamic design of the airfoil under uncertainty.

Response Surface Methods (RSMs) are statistical and numerical models that approximate the relationship between multiple input variables and an output variable. This chapter introduces the methodology and its importance for engineer- ing design optimisation. The basic steps to build RSMs and validate the model accuracy are explained. An overview of three classical methods (Least Squares, Radial Basis Functions, and Kriging) is provided. A simple wing structure design optimisation problem is used to illustrate the different phases of the response surface methodology and its application to design optimisation. This example also includes the case of noisy data.

During a composite application’s initial design stages, the main objective is to have the optimal performance of the final structure. There is a vast demand for lightweight structures with minimum cost and enhanced safety features in all heavy-duty and performance-based industries such as aerospace, automobile, and sports. In order to make prudent decisions and establish the reliability of industrial decision-makers, it is paramount to consider the impacts of uncertainties on the strength and cost of the structure. For that reason, every source of uncertainty should be included when designing an optimal engineering device. This work focuses on applying multidisciplinary optimization tools for the optimal design of fiber-reinforced composites under uncertainties arising from different scales. For demonstration, we consider a composite leafspring for optimization under uncertainties. Material microstructure accounts for microscale uncertainties while composite layers stacking sequence and structural loading account for meso and macroscale uncertainties, respectively. Using a Sparse Polynomial Chaos Expansion (SPCE) method, a data-driven model that establishes a relationship between input parameters and system objectives is constructed by analyzing data. Results are provided with respect to both variations and probability distributions. The stiffness and the cost of the leafspring are the design objectives. Finally, the robust optimal designs are discussed using the Pareto front .

In this work, a design optimisation strategy is presented for expensive and uncertain single- and multi-objective problems. Computationally expensive design fitness evaluations prohibit the application of standard optimisation techniques and the direct calculation of risk measures. Therefore, a surrogate-assisted optimisation framework is presented. The computational budget limits the number of high-fidelity simulations which makes impossible to accurately approximate the landscape. This motivates the use of cheap low-fidelity simulations to obtain more information about the unexplored locations of the design space. The information stemming from numerical experiments of various fidelities can be fused together with multi-fidelity Gaussian process regression to build an accurate surrogate model despite the low number of high-fidelity simulations. We propose a new strategy for automatically selecting the fidelity level of the surrogate model update. The proposed method is extended to multi-objective applications. Although, Gaussian processes can inherently model uncertain processes, here the deterministic input and uncertain parameters are treated separately and only the design space is modelled with a Gaussian process. The probabilistic space is modelled with a polynomial chaos expansion to allow also uncertainties of non-Gaussian type. The combination of the above techniques allows us to efficiently carry out a (multi-objective) design optimisation under uncertainty which otherwise would be impractical.

Uncertainty-based optimisation techniques provide optimal airfoil de- signs that are less vulnerable to the presence of uncertainty in the operational conditions (i.e., Mach number, angle-of-attack, etc.) at which an airfoil is func- tioning. These uncertainty-based techniques typically require numerous function evaluations to accurately calculate the statistical measure of the quantity of inter- est. To render the computational burden down, the design optimisation of the air- foil is performed by a multi-fidelity surrogate-based technique. The high-fidelity aerodynamic performance is calculated with a compressible RANS solver using a fine grid. At the low-fidelity level a coarser grid is used. To obtain accurate drag predictions despite the lower grid resolution the so-called far-field drag approxi- mation is employed.

Benchmarks are a useful tool for empirical performance comparisons. However, one of the main shortcomings of existing benchmarks is that it remains largely unclear how they relate to real-world problems. What does an algorithm's performance on a benchmark say about its potential on a specific real-world problem? This work aims to identify properties of real-world problems through a questionnaire on real-world single-, multi-, and many-objective optimization problems. Based on initial responses, a few challenges that have to be considered in the design of realistic benchmarks can already be identified. A key point for future work is to gather more responses to the questionnaire to allow an analysis of common combinations of properties. In turn, such common combinations can then be included in improved benchmark suites. To gather more data, the reader is invited to participate in the questionnaire at: https://tinyurl.com/opt-survey, 2 pages, GECCO2020 Poster Paper

It is known that seismic isolation is able to protect structures from damage by reducing the earthquake effects on the superstructure rather than increasing the structural resistance. Base-isolated buildings are becoming more numerous all around the world, especially in areas subject to a high seismic hazard or where high safety levels are required. The cost of the isolation devices for ordinary buildings hinders a widespread adoption of the new technology. However, a well-designed base isolation system can largely reduce seismic loadings transferred to the superstructure and it not only enables to immediately reduce the superstructure building cost, but also to reduce the maintenance costs incurred after every earthquake during the building lifetime. To better understand these factors, this paper presents an efficient numerical optimization technique for comparing the responses of a base-isolated and a traditional fixed-base reinforced concrete ordinary building under the same type of solicitations and seismic spectra, as appropriate for each case. We start from a multiobjective optimization. The superstructure and the isolation system are generally designed separately in a building. In this work, we consider elastomeric isolators and we optimize at the same time the structural elements of the building (superstructure column and beam sections and reinforcements) and the isolator parameters (rubber type, maximum allowed displacement and elastomer size). We consider three objectives: minimization of the superstructure material cost, minimization of the top-floor acceleration and minimization of the top-floor displacement. This multiobjective optimization yields a set of trade-off optimal solutions (the so-called Pareto optimal designs) that can be post-processed with tools such as hierarchical clustering or decision-making algorithms and further analyzed. The purpose of this analysis is to identify similarities within data sets or choose a final optimal solution based on pre-defined priority criteria applied to the optimization objectives. We compare the base-isolated structure results with the solutions found for the same building with a traditional foundation fixed to the ground.

Engineering design optimization aims at designing low-cost systems and structures that fulfill certain performance objectives. In civil engineering Reliability-Based Design Optimization (RBDO) is extremely important because it enables to find optimal structure configurations while considering the effects of uncertainties, such as loading, geometry, structural parameters, modeling assumptions etc., on the building structural performance and reliability. The impact of uncertainties on system response and safety levels may escape the deterministic optimization approach. In this work we perform a multiobjective RBDO of reinforced concrete structures with elastomeric base isolators. Seismic isolation makes a superstructure less susceptible to ground motion caused by earthquakes and reduces the damages on the building. We perform both a sizing optimization of the superstructure (with the beam and column sections and reinforcements as input decision variables) and an optimization of the isolator catalog variables (elastomer rubber type, allowed displacement and dimensions). Uncertainty sources are the vertical loadings and the isolator damping coefficient. We aim at minimizing the superstructure cost as well as minimizing the top floor acceleration and displacement to reduce maintenance costs, while considering probabilistic constraints (and objectives if appropriate). The probabilistic responses are given in terms of percentiles computed with the Polynomial Chaos Expansion (PCE) method [1]. PCE statistical responses can be very accurate but its computational cost is prohibitive for a large number of uncertainties or a high polynomial order. We try to mitigate this effect with an Adaptive Sparse PCE (ASPCE) approach [2] which builds a high-order sparse polynomial with limited sample evaluations. ASPCE reduces the overall computational cost while preserving accuracy. We compare the deterministic optimization and the RBDO results for a base-isolated and a fixed-base structure subject to the same type of actions.

Ducted propellers are promising candidates for electrical aircrafts where the propulsion efficiency must be highly optimised. The aerodynamic performance and efficiency of ducted propellers can be obtained through models and experiments of various fidelity, ranging from cheap analytic formulas to expensive Computational Fluid Dynamics simulations. The ducted propeller design problem also involves several uncertainties arising from environmental conditions or manufacturing imperfections. To alleviate the computational burden of the uncertainty-based design optimisation of ducted propellers, we propose a new approach that combines multi-fidelity information fusion and dimensionality reduction by using different surrogate models for the design and probabilistic spaces. The method is validated on a simplified propeller model.

This paper introduces the concept of Resilience Engineering in the context of space systems design and a model of Global System Reliability and Robustness that accounts for epistemic uncertainty and imprecision. In particular, Dempster-Shafer Theory of evidence is used to model uncertainty in both system and environmental parameters. A resilience model is developed to account for the transition from functional to degraded states, and back, during the operational life and the dependency of these transitions on system level design choices and uncertainties. The resilience model is embedded in a network representation of a complex space system. This network representation, called Evidence Network Model (ENM), allows for a fast quantification of the global robustness and reliability of system. A computational optimisation algorithm is then proposed to derive design solutions that provide an optimal compromise between resilience and performance. The result is a set of design solutions that maximise the probability of a system to recover functionalities in the case of a complete or partial failure and at the same time maximises the belief in the desired target value of the performance index.

We present a simple strategy for multiobjective target-driven optimization and apply it to the sizing optimization of a steel girder bridge. Users or decision makers are asked to express their preferences (based on their previous experience) in terms of desired target objective values to drive the optimization towards the most preferred regions of the Pareto front. This can lead to a more efficient exploration of specific regions of the objective space and reduce the computational cost of finding desirable solutions. This strategy combines a-priori with interactive preference-handling approaches. These methods have recently received more attention in the evolutionary multiobjective optimization community. The proposed algorithm is described in detail and compared with existing methods. Benchmarks on standard mathematical test functions as well as on a realistic structural engineering sizing optimization problem are provided.

We revise a method recently introduced by the authors for the estimation of robustness and reliability in design optimization problems with uncertainties in the input variable space. Percentile values of system output properties are estimated by means of polynomial chaos expansions used as stochastic response surfaces. The percentiles can be used as objectives or constraints in multiobjective optimization problems. We clarify the theoretical background and motivations of our approach, and we show benchmark results, as well as applications of multiobjective optimization problems solved with evolutionary algorithms. The advantages of the method are also presented.

We discuss the role of the U(1) axial symmetry for the phase structure of QCD at finite temperature. We expect that, above a certain critical temperature, also the U(1) axial symmetry will be (effectively) restored. We will try to see if this transition has (or has not) anything to do with the usual chiral transition: various possible scenarios are discussed. In particular, supported by recent lattice results, we analyse a scenario in which a U(1)-breaking condensate survives across the chiral transition. This scenario can be consistently reproduced using an effective Lagrangian model. The effects of the U(1) chiral condensate on the slope of the topological susceptibility in the full theory with quarks are studied: we find that this quantity (in the chiral limit of zero quark masses) acts as an order parameter for the U(1) axial symmetry above the chiral transition. Further information on the new U(1) chiral order parameter is derived from the study (at zero temperature) of the radiative decays of the pseudoscalar mesons in two photons: a comparison of our results with the experimental data is performed., 27 pages, LaTeX file. Completely revised version including new references and new comments on the results

One of the major tasks of structural engineering design optimization is the handling of uncertainties (such as variations in material properties, loading conditions, unknown environmental conditions or even uncertainties in modeling assumptions), which affect system performance in terms of robustness and reliability (or, in other words, the ability to respond to input variations with minimal alteration, loss of functionality or damage). This task is usually tackled with Optimization Under Uncertainty (OUU) methods[1], like robust design optimization and reliability-based design optimization. In most cases, the optimization has to deal with multi-objective problems (such as maximizing the performance while minimizing costs, system response variations, etc). These problems do not have a unique solution, but a set of tradeoff optimal solutions (the so-called Pareto front). The action of a decision maker (DM) is necessary for choosing the final optimal design according to some (pre-defined) preferences or criteria. Multi-Criteria Decision Making (MCDM) techniques[2] have been developed over the past years to try to make these choices objective and rational. In most MCDM methods, the preferences are usually taken into account during some a-posteriori analyses of the optimization outcomes. Here we address both OUU and MCDM problems with an approach that integrates directly the action of the DM with the optimization process. The DM is asked to express their preferences (based on their previous experience) to drive the optimization towards the most preferred regions of the Pareto front. This can lead to a more efficient exploration of specific regions of the Pareto front and reduce the computational cost of finding desirable solutions. Interactive MCDM approaches have been recently given more attention in the multi-objective optimization community [3, 4, 5]. A validation of this approach on simple test-cases is shown as well as its application to the design of a simple building structure under uncertainties with seismic hazard and snow loads.

We apply a variational wave function capable of describing qualitatively and quantitatively the so called "resonating valence bond" in realistic materials, by improving standard ab initio calculations by means of quantum Monte Carlo methods. In this framework we clearly identify the Kekul\'e and Dewar contributions to the chemical bond of the benzene molecule, and we establish the corresponding resonating valence bond energy of these well known structures ($\simeq 0.01$eV/atom). We apply this method to unveil the nature of the chemical bond in undoped graphene and show that this picture remains only within a small "resonance length" of few atomic units., Comment: 4 pages, 4 figures, submitted to Phys. Rev. Lett

We study the iron dimer by using an accurate ansatz for quantum chemical calculations based on a simple variational wave function, defined by a single geminal expanded in molecular orbitals and combined with a real space correlation factor. By means of this approach we predict that, contrary to previous expectations, the neutral ground state is $^7 \Delta$ while the ground state of the anion is $^8 \Sigma_g^-$, hence explaining in a simple way a long standing controversy in the interpretation of the experiments. Moreover, we characterize consistently the states seen in the photoemission spectroscopy by Leopold \emph{et al.}. It is shown that the non-dynamical correlations included in the geminal expansion are relevant to correctly reproduce the energy ordering of the low-lying spin states., Comment: 5 pages, submitted to the Chemical Physics Letters

We introduce a method for accurate quantum chemical calculations based on a simple variational wave function, defined by a single geminal that couples all the electrons into singlet pairs, combined with a real space correlation factor. The method uses a constrained variational optimization, based on an expansion of the geminal in terms of molecular orbitals. It is shown that the most relevant non-dynamical correlations are correctly reproduced once an appropriate number $n$ of molecular orbitals is considered. The value of $n$ is determined by requiring that, in the atomization limit, the atoms are described by Hartree-Fock Slater determinants with Jastrow correlations. The energetics, as well as other physical and chemical properties, are then given by an efficient variational approach based on standard quantum Monte Carlo techniques. We test this method on a set of homonuclear (Be2, B2, C2, N2, O2, and F2) and heteronuclear (LiF, and CN) dimers for which strong non-dynamical correlations and/or weak van der Waals interactions are present., 13 pages, 3 figures, final version accepted for publication

We find that the spin susceptibility of a two-dimensional electron system with valley degeneracy does not grow critically at low densities, at variance with experimental results [A. Shashkin et al., Phys. Rev. Lett. 96, 036403 (2006)]. We ascribe this apparent discrepancy to the weak disorder present in experimental samples. Our prediction is obtained from accurate correlation energies computed with state-of-the-art diffusion Monte Carlo simulations and fitted with an analytical expression which also provides a local spin density functional for the system under investigation.

We present component-resolved and total pair distribution functions for a 2DEG with two symmetric valleys. Our results are based on quantum Monte Carlo simulations performed at several densities and spin polarizations., 5 pages, 3 figure, accepted for publication in J. Phys. A: Math. Theor. as a special issue article for the SCCS2008 conference (Camerino, Italy)

This paper proposes to apply multi-fidelity learning for reliability-based design optimisation of a ducted propeller. Theoretically, the efficiency of a propeller can be increased by placing the propeller into a duct. The increased efficiency makes the ducted propeller an appealing option for electrical aviation where optimal electricity consumption is vital. The electricity consumption is mainly dictated by the required power to reach the required thrust force. Recent design optimisation techniques such as machine learning can help us to reach high thrust to power ratios. Due to the expensive computational fluid dynamics simulations a multi-fidelity learning algorithm is investigated here for the application of ducted propeller design. The limited number of high-fidelity numerical experiments cannot provide sufficient information about the landscape of the design field and probability field. Therefore, information from lower fidelity simulations is fused into the high-fidelity surrogate using the recently published recursive co-Kriging technique augmented with Gaussian-Markov Random Fields. At each level the uncertainty can be modelled via a polynomial chaos expansion which providesa variable-fidelity quantification technique of the uncertainty. This facilitates the calculation of risk measures, like conditional Value-at-Risk, for reliability-based design optimisation. The multi-fidelity surrogate model can be adaptively refined following a similar strategy to the Efficient Global Optimisation using the expected improvement measure. The proposed combination of techniques provides an efficient manner to conduct reliability-based optimisation on expensive realistic problems using a multi-fidelity learning technique.