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The equations are justified in the case of all positive variables by observing that the first two springs are elongated by x; and x2 — x1, respectively. The last spring is compressed by x2, which accounts for the minus sign. Eq. (18) can be written as a second-order vector-matrix system

Figure 6 The equations are justified in the case of all positive variables by observing that the first two springs are elongated by x; and x2 — x1, respectively. The last spring is compressed by x2, which accounts for the minus sign. Eq. (18) can be written as a second-order vector-matrix system

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Table 1: Analytical benchmarks main features een SE The multifidelity benchmark problems are selected to capture fundamental mathematical char- teristics and properties which also mimic real-world engineering problems. The distinguishing athematical features of the selected benchmark problems can be listed as: behaviors, scalability, screpancy type, and noise. Function behaviors include multi-modality, discontinuities, and atypical cal behaviors, that cannot be neglected a priori, especially for real world problems where it could be 1portant to represent the whole variable domain. In particular, multi-modality and discontinuities e challenging from both a modelling and optimization viewpoint. Scalabilty takes into consideration th the function parameterization and the fidelity spectrum. The former is one of the important iteria for the comprehensive assessment of the performance of the multifidelity methods which lables to represent the same parametric function with different input dimensions, whereas the latter useful to demonstrate how the modelling process can be improved depending on the fidelity level railable. This last point is particularly relevant, because its relation with the discrepancy type, that scribes the relation among fidelities. In general, a linear discrepancy is simpler to model than a mnlinear one. For this reason the discrepancy type allows for a deeper assessment of the multifidelity ethods, since the number of fidelities available have to be correlated to the associated discrepancy pe. Finally, in real-world engineering problems, noise may exist, it is undesired but inescapable in e overall response of a system which may exhibit abrupt changes within the solution domain. It is fficult or even impossible to distinguish the individual impact of the noise in the overall response id eliminate it. Thus, the user has to deal with a function with some embedded noise, and is 1portant to asses the ability of multifidelity methods to model a noisy function. Considering all ese mathematical characteristics over the benchmark problems will allow to evaluate the perfor-
Table 1: Analytical benchmarks main features een SE The multifidelity benchmark problems are selected to capture fundamental mathematical char- teristics and properties which also mimic real-world engineering problems. The distinguishing athematical features of the selected benchmark problems can be listed as: behaviors, scalability, screpancy type, and noise. Function behaviors include multi-modality, discontinuities, and atypical cal behaviors, that cannot be neglected a priori, especially for real world problems where it could be 1portant to represent the whole variable domain. In particular, multi-modality and discontinuities e challenging from both a modelling and optimization viewpoint. Scalabilty takes into consideration th the function parameterization and the fidelity spectrum. The former is one of the important iteria for the comprehensive assessment of the performance of the multifidelity methods which lables to represent the same parametric function with different input dimensions, whereas the latter useful to demonstrate how the modelling process can be improved depending on the fidelity level railable. This last point is particularly relevant, because its relation with the discrepancy type, that scribes the relation among fidelities. In general, a linear discrepancy is simpler to model than a mnlinear one. For this reason the discrepancy type allows for a deeper assessment of the multifidelity ethods, since the number of fidelities available have to be correlated to the associated discrepancy pe. Finally, in real-world engineering problems, noise may exist, it is undesired but inescapable in e overall response of a system which may exhibit abrupt changes within the solution domain. It is fficult or even impossible to distinguish the individual impact of the noise in the overall response id eliminate it. Thus, the user has to deal with a function with some embedded noise, and is 1portant to asses the ability of multifidelity methods to model a noisy function. Considering all ese mathematical characteristics over the benchmark problems will allow to evaluate the perfor-
Figure 2: Rosenbrock Function: from left to right, f; (highest-fidelity), f2 , and f3 (lowest-fidelity) The three-fidelity levels are shown for two dimensions in Figure 2. 2.3 Shifted-rotated Rastrigin function To address real-world optimization problems, where the landscape of the objective function is usually multi-modal, the Rastrigin function is selected as benchmark. The function is shifted to change the position of the minimum and rotated to change the properties of the function itself within the variable space. The equation of the shifted-rotated Rastrigin function reads as follows
Figure 2: Rosenbrock Function: from left to right, f; (highest-fidelity), f2 , and f3 (lowest-fidelity) The three-fidelity levels are shown for two dimensions in Figure 2. 2.3 Shifted-rotated Rastrigin function To address real-world optimization problems, where the landscape of the objective function is usually multi-modal, the Rastrigin function is selected as benchmark. The function is shifted to change the position of the minimum and rotated to change the properties of the function itself within the variable space. The equation of the shifted-rotated Rastrigin function reads as follows
2.4 Heterogeneous function This problem employs heterogeneous non-polynomial analytic functions defined on unit hypercubes (0 < x; < 1 fori = 1,...D) in one, two (taken from [7]), and three dimensions, with low-fidelity functions obtained by using linear additive and multiplicative bridge functions. The benchmark reads
2.4 Heterogeneous function This problem employs heterogeneous non-polynomial analytic functions defined on unit hypercubes (0 < x; < 1 fori = 1,...D) in one, two (taken from [7]), and three dimensions, with low-fidelity functions obtained by using linear additive and multiplicative bridge functions. The benchmark reads
The one- and two-dimensional functions are displayed together with their low-fidelity approxima-
The one- and two-dimensional functions are displayed together with their low-fidelity approxima-
2.5 Coupled Spring-Mass system The proposed problem represents a general coupled spring-mass system. Consider two point masses m, and m2 concentrated at their center of gravity and attached to each other by three springs, that operate according to Hooke’s law with constant k;, kz, and k3. The mass of each spring is negligible and they restore after compression and extension. The masses can slide along a frictionless horizontal surface, while the first and last springs (k, and k3) are attached to fixed walls, as shown in Figure 6.
2.5 Coupled Spring-Mass system The proposed problem represents a general coupled spring-mass system. Consider two point masses m, and m2 concentrated at their center of gravity and attached to each other by three springs, that operate according to Hooke’s law with constant k;, kz, and k3. The mass of each spring is negligible and they restore after compression and extension. The masses can slide along a frictionless horizontal surface, while the first and last springs (k, and k3) are attached to fixed walls, as shown in Figure 6.
A is a parameter that models the error among the fidelities and can vary between 0 and 1, when A = 0 the low- and high-fidelity equations are the same, while the change between the low-fidelity and high-fidelity models increases as the value of A increases. In this study it is set as A = 0.5, while a; is the coefficient that defines the noise level. The benchmark is defined considering the variable ranges as 0.3 < x; < 1.0 for Vi. For the high-fidelity case a noise level is added to the equation
A is a parameter that models the error among the fidelities and can vary between 0 and 1, when A = 0 the low- and high-fidelity equations are the same, while the change between the low-fidelity and high-fidelity models increases as the value of A increases. In this study it is set as A = 0.5, while a; is the coefficient that defines the noise level. The benchmark is defined considering the variable ranges as 0.3 < x; < 1.0 for Vi. For the high-fidelity case a noise level is added to the equation
Table 2: Experiments setup summary
Table 2: Experiments setup summary
Table 3: Summary of multifidelity benchmark values for the evaluation of the goal-sensitive metrics > Concluding Remarks
Table 3: Summary of multifidelity benchmark values for the evaluation of the goal-sensitive metrics > Concluding Remarks
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