text: presented. The right panel illustrates convergence of CMA-ES algorithm. The red points violate the performance criteria (failure points). The blue and green ones are in the safe domain but only the latter improves the current best design during optimization. c(d) 1 10 0.8 8 0.6 6 4 0.4 2 0.2 0 0 0 30 60 90 120 150 0 2 4 6 8 10 Iteration # d1 (a) Evolution of the quantile accuracy criterion (b) Convergence of the constrained (1+1)-CMA-ES algo- rithm Figure 7: Illustration of the second stage of enrichment for the two-dimensional problem Finally, Table 2 compares the results presented here with those reported in the literature for benchmark. All the selected methods provide a solution with a good accuracy. The diﬀerence rather lies in their costs. In these results, the single loop and the reliable design space (RDS) approaches require a relatively small number of functions evaluation despite they do not rely upon surrogate models. The two cases considering surrogate models (Meta-RBDO and Quantile- RBDO) are among the best in terms of model evaluations. In this example, the quantile-based approach we propose is on average the less expensive one. Since the initial design is random, the analysis is replicated 50 times. The number of calls to the true models varies between 11 and 23, all of them leading to good results. On average the number of calls is 14.6 and among them only three are above 20. 5.3 Bracket structure This mechanical example consists of the two-member bracket structure illustrated in Figure 8 (Chateauneuf and Aoues, 2008). The two-members are pin-joined at the point B and a vertical load P is applied on the right end of the member CD at a distance L of its hinge. The aim is to minimize its weight while considering two failure modes: 24 qη 2d ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: presented.

text: The

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text: algorithm.

text: The

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text: The

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text: c(d)

text: Iteration

text: (a)

text: Evolution

text: of

text: the

text: quantile

text: accuracy

text: criterion

text: (b)

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text: of

text: the

text: constrained

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text: second

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text: enrichment

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text: two-dimensional

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text: Table

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text: upon

text: surrogate

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text: The

text: two

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text: π2EI (cid:18) π2Etw3 (cid:19) g (d, z) = F − F with: F = = AB , 2 b AB b L2 12 (2L/3sinθ)2 AB (38) 1 3P 3ρgw tL F = + CD , AB cosθ 2 4 where w and L are respectively the width and length of AB and θ is its inclination AB AB angle. The RBDO application problem as in Chateauneuf and Aoues (2008) and Dubourg et al. (cid:32) (cid:33) (2011) consists in minimizing the weight of the structure, given the following objective function: √ 4 3 c (d) = ρtL w + w , (39) AB CD 9 where d = {w , w , t}T ∈ D = [5, 30]3 represents the set of design parameters. AB CD The target reliability index for this problem is set to β = β = 2, thus corresponding to a 1 2 component failure probability of 0.0227. The RBDO problem therefore reads:  (cid:16) (cid:16) (cid:17) (cid:17) (cid:16) (cid:17)  (cid:16) (cid:16) (cid:17) (cid:17)  d∗ = arg min c d(cid:126) subject to: P g1 X(cid:126) (d) , Z ≤ 0 ≤ P¯f1 (40) ∈D d P g X(cid:126) (d) , Z ≤ 0 ≤ P¯ , 2 f2 where P¯ = Φ (−β ) ≈ 0.0227 and the functions g , g and c are respectively given by fi 1 1 2 Eqs. (37) – (39). The probabilistic model associated to this problem is shown in Table 3. Table 4 shows the bounds of the augmented space in which the training points are sampled. The surrogate model is built in the unit hypercube following a linear mapping from this augmented space. For the solution of this problem, we start with an initial design of 50 points. The threshold for the ﬁrst stage of enrichment is set to η¯ = 0.30. A total of 60 enrichments points have been added to reach the required accuracy through 6 iterations of 10 points each. The optimization is then initiated starting from d(0) = {6.1, 20.2, 26.9}T which is also the initial design in the benchmark references Chateauneuf and Aoues (2008); Dubourg (2011) and corresponds to the optimal deterministic solution. In the second stage, K = 3 points are added per enrichment, thus leading to 15 additional points in the DoE. Convergence of the CMA-ES algorithm is illustrated in Figure 9 where the evolution of the cost function with respect to the iteration number is shown. The green circles highlight the points that were feasible and improved the current best design. The CMA-ES algorithm is stopped after 150 iterations and the solution locally reﬁned 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: the

text: ﬁrst

text: stage

text: of

text: enrichment

text: is

text: set

text: to

text: η¯

text: A

text: total

text: of

text: enrichments

text: points

text: have

text: been

text: added

text: to

text: reach

text: the

text: required

text: accuracy

text: through

text: iterations

text: of

text: points

text: each.

text: The

text: optimization

text: is

text: then

text: initiated

text: starting

text: from

text: which

text: is

text: also

text: the

text: initial

text: design

text: in

text: the

text: benchmark

text: references

text: Chateauneuf

text: and

text: Aoues

text: Dubourg

text: and

text: corresponds

text: to

text: the

text: optimal

text: deterministic

text: solution.

text: In

text: the

text: second

text: stage,

text: K

text: points

text: are

text: added

text: per

text: enrichment,

text: thus

text: leading

text: to

text: additional

text: points

text: in

text: the

text: DoE.

text: Convergence

text: of

text: the

text: CMA-ES

text: algorithm

text: is

text: illustrated

text: in

text: Figure

text: where

text: the

text: evolution

text: of

text: the

text: cost

text: function

text: with

text: respect

text: to

text: the

text: iteration

text: number

text: is

text: shown.

text: The

text: green

text: circles

text: highlight

text: the

text: points

text: that

text: were

text: feasible

text: and

text: improved

text: the

text: current

text: best

text: design.

text: The

text: CMA-ES

text: algorithm

text: is

text: stopped

text: after

text: iterations

text: and

text: the

text: solution

text: locally

text: reﬁned

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: to FORM approximations as in the so-called reliability index (Enevoldsen and Sorensen, 1994) and performance measure (Tu and Choi, 1997; Tu et al., 1999) approaches (respectively RIA and PMA). Simulation techniques may also be used in the inner loop, as we show in the sequel. The mono-level approach transforms the double-loop problem into a single-loop one by introducing optimality criteria for the FORM problem. Kuschel and Rackwitz (1997) propose an equivalent formulation based on RIA while Agarwal et al. (2007) rather rely on PMA. Finally, the decoupled approaches transform the double-loop into a sequence of deterministic problems. A well-known example is sequential optimization and reliability analysis (SORA) proposed in Du and Chen (2004). As introduced above, all theses methods rely on repeated evaluations of the mechanical models, i.e. during the outer optimization loop and more intensively in the reliability analysis steps. This limits their range of applications to engineering problems of practical interest. This issue is even more dramatic in the design of complex industrial systems which relies on high- ﬁdelity models and henceforth time-consuming simulations. Surrogate modeling, a technique in which the mechanical model is replaced by a well calibrated easy-to-evaluate analytical function, has been extensively used in the past decade to alleviate the computational burden. For instance, support vector machines have been used for structural reliability assessment in Hurtado and Alvarez (2001); Bourinet et al. (2011); Deheeger and Lemaire (2007). Polynomial chaos expansion were considered in Blatman (2009); Blatman and Sudret (2010); Hu and Youn (2011). Kriging (a.k.a. Gaussian process modeling) has been successfully used for reliability analysis in Echard et al. (2011); Picheny et al. (2010); Bichon et al. (2008); Balesdent et al. (2013). For the speciﬁc task of RBDO, conservative surrogate models which rely on Kriging or polynomial response surfaces were considered in Viana et al. (2010); Picheny et al. (2008). From another perspective, Dubourg et al. (2011); Chen et al. (2015); Lee et al. (2011); Li et al. (2016), for instance, have proposed some approaches which rely on locally or globally reﬁned Kriging approximations. Likewise, the present work considers Kriging because it provides not only an approximation of the original mechanical model but also gives a built-in error estimate. This enables adaptive techniques that further reduce the computational cost. In contrast to most of the literature in RBDO, we are not interested in this paper in highly reliable designs, for which the probability of failure has to be computed by one of the methods mentioned above. Our goal is rather to develop a conservative optimal design methodology. To this aim, we ﬁrst introduce a quantile-based design optimization procedure, in which the hard constraints are formulated on quantiles of the performance criteria, instead of target probabil- ities of failure. At each iteration of the design optimization, we therefore evaluate quantiles through Monte Carlo simulation instead of solving a reliability problem. This approach is justi- ﬁed by the ”degree of conservatism” targeted in our applications to car body mass optimization: 4

text: to

text: FORM

text: approximations

text: as

text: in

text: the

text: so-called

text: reliability

text: index

text: (Enevoldsen

text: and

text: Sorensen,

text: and

text: performance

text: measure

text: (Tu

text: and

text: Choi,

text: Tu

text: et

text: al.,

text: approaches

text: (respectively

text: RIA

text: and

text: PMA).

text: Simulation

text: techniques

text: may

text: also

text: be

text: used

text: in

text: the

text: inner

text: loop,

text: as

text: we

text: show

text: in

text: the

text: sequel.

text: The

text: mono-level

text: approach

text: transforms

text: the

text: double-loop

text: problem

text: into

text: a

text: single-loop

text: one

text: by

text: introducing

text: optimality

text: criteria

text: for

text: the

text: FORM

text: problem.

text: Kuschel

text: and

text: Rackwitz

text: propose

text: an

text: equivalent

text: formulation

text: based

text: on

text: RIA

text: while

text: Agarwal

text: et

text: al.

text: rather

text: rely

text: on

text: PMA.

text: Finally,

text: the

text: decoupled

text: approaches

text: transform

text: the

text: double-loop

text: into

text: a

text: sequence

text: of

text: deterministic

text: problems.

text: A

text: well-known

text: example

text: is

text: sequential

text: optimization

text: and

text: reliability

text: analysis

text: (SORA)

text: proposed

text: in

text: Du

text: and

text: Chen

text: As

text: introduced

text: above,

text: all

text: theses

text: methods

text: rely

text: on

text: repeated

text: evaluations

text: of

text: the

text: mechanical

text: models,

text: i.e.

text: during

text: the

text: outer

text: optimization

text: loop

text: and

text: more

text: intensively

text: in

text: the

text: reliability

text: analysis

text: steps.

text: This

text: limits

text: their

text: range

text: of

text: applications

text: to

text: engineering

text: problems

text: of

text: practical

text: interest.

text: This

text: issue

text: is

text: even

text: more

text: dramatic

text: in

text: the

text: design

text: of

text: complex

text: industrial

text: systems

text: which

text: relies

text: on

text: high-

text: ﬁdelity

text: models

text: and

text: henceforth

text: time-consuming

text: simulations.

text: Surrogate

text: modeling,

text: a

text: technique

text: in

text: which

text: the

text: mechanical

text: model

text: is

text: replaced

text: by

text: a

text: well

text: calibrated

text: easy-to-evaluate

text: analytical

text: function,

text: has

text: been

text: extensively

text: used

text: in

text: the

text: past

text: decade

text: to

text: alleviate

text: the

text: computational

text: burden.

text: For

text: instance,

text: support

text: vector

text: machines

text: have

text: been

text: used

text: for

text: structural

text: reliability

text: assessment

text: in

text: Hurtado

text: and

text: Alvarez

text: Bourinet

text: et

text: al.

text: Deheeger

text: and

text: Lemaire

text: Polynomial

text: chaos

text: expansion

text: were

text: considered

text: in

text: Blatman

text: Blatman

text: and

text: Sudret

text: Hu

text: and

text: Youn

text: Kriging

text: (a.k.a.

text: Gaussian

text: process

text: modeling)

text: has

text: been

text: successfully

text: used

text: for

text: reliability

text: analysis

text: in

text: Echard

text: et

text: al.

text: Picheny

text: et

text: al.

text: Bichon

text: et

text: al.

text: Balesdent

text: et

text: al.

text: For

text: the

text: speciﬁc

text: task

text: of

text: RBDO,

text: conservative

text: surrogate

text: models

text: which

text: rely

text: on

text: Kriging

text: or

text: polynomial

text: response

text: surfaces

text: were

text: considered

text: in

text: Viana

text: et

text: al.

text: Picheny

text: et

text: al.

text: From

text: another

text: perspective,

text: Dubourg

text: et

text: al.

text: Chen

text: et

text: al.

text: Lee

text: et

text: al.

text: Li

text: et

text: al.

text: for

text: instance,

text: have

text: proposed

text: some

text: approaches

text: which

text: rely

text: on

text: locally

text: or

text: globally

text: reﬁned

text: Kriging

text: approximations.

text: Likewise,

text: the

text: present

text: work

text: considers

text: Kriging

text: because

text: it

text: provides

text: not

text: only

text: an

text: approximation

text: of

text: the

text: original

text: mechanical

text: model

text: but

text: also

text: gives

text: a

text: built-in

text: error

text: estimate.

text: This

text: enables

text: adaptive

text: techniques

text: that

text: further

text: reduce

text: the

text: computational

text: cost.

text: In

text: contrast

text: to

text: most

text: of

text: the

text: literature

text: in

text: RBDO,

text: we

text: are

text: not

text: interested

text: in

text: this

text: paper

text: in

text: highly

text: reliable

text: designs,

text: for

text: which

text: the

text: probability

text: of

text: failure

text: has

text: to

text: be

text: computed

text: by

text: one

text: of

text: the

text: methods

text: mentioned

text: above.

text: Our

text: goal

text: is

text: rather

text: to

text: develop

text: a

text: conservative

text: optimal

text: design

text: methodology.

text: To

text: this

text: aim,

text: we

text: ﬁrst

text: introduce

text: a

text: quantile-based

text: design

text: optimization

text: procedure,

text: in

text: which

text: the

text: hard

text: constraints

text: are

text: formulated

text: on

text: quantiles

text: of

text: the

text: performance

text: criteria,

text: instead

text: of

text: target

text: probabil-

text: ities

text: of

text: failure.

text: At

text: each

text: iteration

text: of

text: the

text: design

text: optimization,

text: we

text: therefore

text: evaluate

text: quantiles

text: through

text: Monte

text: Carlo

text: simulation

text: instead

text: of

text: solving

text: a

text: reliability

text: problem.

text: This

text: approach

text: is

text: justi-

text: ﬁed

text: by

text: the

text: ”degree

text: of

text: conservatism”

text: targeted

text: in

text: our

text: applications

text: to

text: car

text: body

text: mass

text: optimization:


text: Liang, J., Z. Mourelatos, and J. Tu (2004). A single-loop method for reliability-based design optimization. In Proc. DETC’04 ASME 2004 Design engineering technical conferences and computers and information in engineering conference, Sept.28 - Oct. 2, 2004, Salt Lake City, Utah, USA. Madsen, H., S. Krenk, and N. Lind (1986). Methods of structural safety. Prentice Hal, Inc. Englewood Cliﬀs. Marelli, S. and B. Sudret (2014). UQLab: A framework for uncertainty quantiﬁcation in Matlab. In Vulnerability, Uncertainty and Risk, Proc. 2nd Int. Conf. on Vulnerability, Risk and analysis management (ICVRAM2014), Liverpool, United Kingdom, pp. 2554–2563. Melchers, R. (1989). Importance sampling in structural systems. Structural Safety 6, 3–10. Picheny, V., D. Ginsbourger, O. Roustant, R. T. Haftka, and N. H. Kim (2010). Adaptive designs of experiments for accurate approximation of a target region. J. Mech.Design 132(7). Picheny, V., N. H. Kim, R. T. Haftka, and N. V. Queipo (2008). Conservative predictions using surrogate modeling. In Proc. 49th AIAA/AME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials, 7-10 April 2008, Schaumburg, IL. Ranjan, P., D. Bingham, and G. Michailidis (2008). Sequential experiment design for contour estimation from complex computer codes. Technometrics 50(4), 527–541. Roustant, O., D. Ginsbourger, and Y. Deville (2012). DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. J. Stat. Software 51(1), 1–55. Santner, T., B. Williams, and W. Notz (2003). The design and analysis of computer experiments. Springer, New York. Sch¨obi, R. and B. Sudret (2014). PC-Kriging: A new meta-modelling method and its applications to quantile estimation. In J. Li and Y. Zhao (Eds.), Proc. 17th IFIP WG7.5 Conference on Reliability and Optimization of Structural Systems, Huangshan, China. Taylor & Francis. Sch¨obi, R., B. Sudret, and S. Marelli (2016). Rare event estimation using Polynomial-Chaos- Kriging. ASCE-ASME J. Risk Uncertainty Eng. Syst., Part A: Civ. Eng.. D4016002. Schonlau, M., W. J. Welch, and D. R. Jones (1998). Global versus local search in constrained op- timization of computer models. New Developments and Applications in Experimental Design, 11–25. Vol. 34, New Developments and Applications in Experimental Design. 35

text: Liang,

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text: J.,

text: Z.

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text: method

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text: reliability-based

text: design

text: optimization.

text: In

text: Proc.

text: ASME

text: Design

text: engineering

text: technical

text: conferences

text: and

text: computers

text: and

text: information

text: in

text: engineering

text: conference,

text: Oct.

text: Salt

text: Lake

text: City,

text: Utah,

text: USA.

text: Madsen,

text: H.,

text: S.

text: Krenk,

text: and

text: N.

text: Lind

text: Methods

text: of

text: structural

text: safety.

text: Prentice

text: Hal,

text: Inc.

text: Englewood

text: Cliﬀs.

text: Marelli,

text: S.

text: and

text: B.

text: Sudret

text: UQLab:

text: A

text: framework

text: for

text: uncertainty

text: quantiﬁcation

text: in

text: Matlab.

text: In

text: Vulnerability,

text: Uncertainty

text: and

text: Risk,

text: Proc.

text: Int.

text: Conf.

text: on

text: Vulnerability,

text: Risk

text: and

text: analysis

text: management

text: Liverpool,

text: United

text: Kingdom,

text: pp.

text: Melchers,

text: R.

text: Importance

text: sampling

text: in

text: structural

text: systems.

text: Structural

text: Safety

text: Picheny,

text: V.,

text: D.

text: Ginsbourger,

text: O.

text: Roustant,

text: R.

text: T.

text: Haftka,

text: and

text: N.

text: H.

text: Kim

text: Adaptive

text: designs

text: of

text: experiments

text: for

text: accurate

text: approximation

text: of

text: a

text: target

text: region.

text: J.

text: Mech.Design

text: Picheny,

text: V.,

text: N.

text: H.

text: Kim,

text: R.

text: T.

text: Haftka,

text: and

text: N.

text: V.

text: Queipo

text: Conservative

text: predictions

text: using

text: surrogate

text: modeling.

text: In

text: Proc.

text: AIAA/AME/ASCE/AHS/ASC

text: Structures,

text: Structural

text: Dynamics

text: and

text: Materials,

text: April

text: Schaumburg,

text: IL.

text: Ranjan,

text: P.,

text: D.

text: Bingham,

text: and

text: G.

text: Michailidis

text: Sequential

text: experiment

text: design

text: for

text: contour

text: estimation

text: from

text: complex

text: computer

text: codes.

text: Technometrics

text: Roustant,

text: O.,

text: D.

text: Ginsbourger,

text: and

text: Y.

text: Deville

text: DiceKriging,

text: DiceOptim:

text: two

text: R

text: packages

text: for

text: the

text: analysis

text: of

text: computer

text: experiments

text: by

text: Kriging-based

text: metamodeling

text: and

text: optimization.

text: J.

text: Stat.

text: Software

text: Santner,

text: T.,

text: B.

text: Williams,

text: and

text: W.

text: Notz

text: The

text: design

text: and

text: analysis

text: of

text: computer

text: experiments.

text: Springer,

text: New

text: York.

text: Sch¨obi,

text: R.

text: and

text: B.

text: Sudret

text: PC-Kriging:

text: A

text: new

text: meta-modelling

text: method

text: and

text: its

text: applications

text: to

text: quantile

text: estimation.

text: In

text: J.

text: Li

text: and

text: Y.

text: Zhao

text: (Eds.),

text: Proc.

text: IFIP

text: Conference

text: on

text: Reliability

text: and

text: Optimization

text: of

text: Structural

text: Systems,

text: Huangshan,

text: China.

text: Taylor

text: Francis.

text: Sch¨obi,

text: R.,

text: B.

text: Sudret,

text: and

text: S.

text: Marelli

text: Rare

text: event

text: estimation

text: using

text: Polynomial-Chaos-

text: Kriging.

text: ASCE-ASME

text: J.

text: Risk

text: Uncertainty

text: Eng.

text: Syst.,

text: Part

text: A:

text: Civ.

text: Eng..

text: Schonlau,

text: M.,

text: W.

text: J.

text: Welch,

text: and

text: D.

text: R.

text: Jones

text: Global

text: versus

text: local

text: search

text: in

text: constrained

text: op-

text: timization

text: of

text: computer

text: models.

text: New

text: Developments

text: and

text: Applications

text: in

text: Experimental

text: Design,

text: Vol.

text: New

text: Developments

text: and

text: Applications

text: in

text: Experimental

text: Design.


text: θ (Koehler and Owen, 1996; Dubourg, 2011). The accuracy of the solution of the optimization problem in Eq. (15) is crucial as it conditions the quality of the Kriging predictor. General-purpose algorithms such as genetic algorithm or BFGS are often used. Available softwares such as DiceKriging (Roustant et al., 2012) in R, UQLab (Marelli and Sudret, 2014; Lataniotis et al., 2015) or ooDace (Couckyut et al., 2013) in MATLAB make use of such algorithms and more generally provide a framework to train a Kriging model. 4 Kriging-based optimization 4.1 Construction of a Kriging model in the augmented reliability space (cid:99) In this section, the optimization problem in Eq. (6) is solved while each performance function M is replaced by a Kriging model M as introduced above. This simply means that the k k performance functions for now on read: (cid:98) (cid:99) g (x, z) = g¯ − M (x, z) , k = {1, . . . , n } . (16) k k k h Computing the quantile with respect to this surrogate model becomes an extremely cheap operation. The expensive part is the initial building of the surrogate model which requires to set and evaluate a design of experiments. Given the possible number of iterations before convergence is achieved, building one surrogate model for each reliability analysis (i.e. the quantile computation for a given design in our case) would be quite cumbersome. Instead we advocate for the use of a single Kriging model as already proposed in other contributions. The idea is to build the surrogate model in a unique space that embeds both the design and random variables. In Kharmanda et al. (2002), this space is called the hybrid design space and is deﬁned as the tensor product between the design and random variables. This needlessly increases the dimension of the space where the surrogate model is built and may be problematic when it comes to space-ﬁlling design of experiments. From another perspective, Au (2005) eﬃciently computes the failure probability in the so-called augmented reliability problem for a given design considering a space where the design variables are artiﬁcially considered as random. Taﬂanidis and Beck (2008) use this augmented reliability problem to construct a stochastic optimization problem. Eventually, Dubourg et al. (2011); Dubourg (2011) propose an augmented reliability space following the ideas in the two above contributions. In these works, the size of the augmented reliability space remains equal to that of the original reliability problem. This is because they consider that the uncertainty in the random variables is simply augmented by the choice of the design points. In practice, the design and environmental variables are treated separately in the 10

text: θ

text: (Koehler

text: and

text: Owen,

text: Dubourg,

text: The

text: accuracy

text: of

text: the

text: solution

text: of

text: the

text: optimization

text: problem

text: in

text: Eq.

text: is

text: crucial

text: as

text: it

text: conditions

text: the

text: quality

text: of

text: the

text: Kriging

text: predictor.

text: General-purpose

text: algorithms

text: such

text: as

text: genetic

text: algorithm

text: or

text: BFGS

text: are

text: often

text: used.

text: Available

text: softwares

text: such

text: as

text: DiceKriging

text: (Roustant

text: et

text: al.,

text: in

text: R,

text: UQLab

text: (Marelli

text: and

text: Sudret,

text: Lataniotis

text: et

text: al.,

text: or

text: ooDace

text: (Couckyut

text: et

text: al.,

text: in

text: MATLAB

text: make

text: use

text: of

text: such

text: algorithms

text: and

text: more

text: generally

text: provide

text: a

text: framework

text: to

text: train

text: a

text: Kriging

text: model.

label: section

text: Kriging-based

text: optimization

text: Construction

text: of

text: a

text: Kriging

text: model

text: in

text: the

text: augmented

text: reliability

text: space

text: In

text: this

text: section,

text: the

text: optimization

text: problem

text: in

text: Eq.

text: is

text: solved

text: while

text: each

text: performance

text: function

text: M

text: is

text: replaced

text: by

text: a

text: Kriging

text: model

text: M

text: as

text: introduced

text: above.

text: This

text: simply

text: means

text: that

text: the

text: k

text: k

text: performance

text: functions

text: for

text: now

text: on

text: read:

text: g

text: (x,

text: z)

text: g¯

text: M

text: (x,

text: z)

text: k

text: n

text: k

text: k

text: k

text: h

text: Computing

text: the

text: quantile

text: with

text: respect

text: to

text: this

text: surrogate

text: model

text: becomes

text: an

text: extremely

text: cheap

text: operation.

text: The

text: expensive

text: part

text: is

text: the

text: initial

text: building

text: of

text: the

text: surrogate

text: model

text: which

text: requires

text: to

text: set

text: and

text: evaluate

text: a

text: design

text: of

text: experiments.

text: Given

text: the

text: possible

text: number

text: of

text: iterations

text: before

text: convergence

text: is

text: achieved,

text: building

text: one

text: surrogate

text: model

text: for

text: each

text: reliability

text: analysis

text: (i.e.

text: the

text: quantile

text: computation

text: for

text: a

text: given

text: design

text: in

text: our

text: case)

text: would

text: be

text: quite

text: cumbersome.

text: Instead

text: we

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text: 14 Liu et al. (IRDCs) have very low luminosity-to-mass ratios since they have not yet formed massive stars. Furthermore, classical Hii regions are luminous, but have very little as- sociated molecular gas. Therefore, for sources at diﬀerent evolutionary stages, the current (or observed) star for- (cid:48) mation eﬃciencies probed by L /M or L /L IR clump IR mol should be very diverse at clump scale as also suggested by Stephens et al. (2016). Our results in Figure 8 indi- (cid:48) cate that the slopes of L –L correlations (or cur- TIR mol rent star formation law) become more shallow as clumps evolve. (cid:48) Is the bimodal behavior found in the L –L corre- TIR mol lations aﬀected by the assuming of the same size between dense gas (size ) and dust (size ). More evolved mol dust (higher L/M) sources likely also have higher tempera- ture, which could be more eﬃcient to excite the dense gas tracer lines comparing to the cooler sources in the earlier stages (Molinari et al. 2016). So the late stage sources might have higher size /size ratios than the mol dust sources in the early stages. We investigate this eﬀect with HCN (4-3). In extreme case, the line luminosity of a point source will be overestimated by a factor of (352 + 222)/222 = 3.5 if we take the median continuum (cid:48)(cid:48) size of 35 as the line emission size. If all the sources in the low L/M (< L/M >= 29) subsample are point source like in HCN (4-3) emission, the HCN (4-3) line luminosity in panel (f) of Figure 8 should be system- ∼ atically overestimated by a factor of 3.5 and the cor- (cid:48) responding intercept of L –L should be underesti- ∼ TIR mol mated by 0.55. However, even in this extreme case, the ∼ intercept ( 3.89) of the low L/M (< L/M >= 29 sub- ∼ sample is still lower than that ( 4.27) of the high L/M (< L/M >= 107 subsample. Therefore, we argue that (cid:48) the bimodal behavior in the L –L correlations is TIR mol not greatly aﬀected by size assumption. Future mapping observations will test this hypotheses. In pan(cid:48) els (c) and (d) o(cid:48)f Figure 9, we investigate the Fig. 10.— Upper panel: Clump masses vs. luminosities of CS LTIR–L CS and LTIR–L HCN correlations for sources (7-6). Lower panels: Clump masses vs. luminosities of HCN (4- with symmetric line proﬁles (“G”) and asymmetric line 3). The red line represent least squares ﬁts for the whole sample. proﬁles (“B+R”). In general, considering the errors, the The blue line represents least squares ﬁts for clumps with infrared luminosities larger than 103 L(cid:12). correlations for sources with diﬀerent line proﬁles do not show signiﬁcant diﬀerences. (cid:48) (cid:48) 4.3. Comparison between diﬀerent tracers L mol correlations. The parameters of the LTIR–L mol correlations for J=2-1, J=5-4 of CS and J=1-0, J=3-2 In Figure 10, we present the correlations between of HCN were taken from Wu et al. (2010). It should clump masses and line luminosities for HCN (4-3) and (cid:48) (cid:48) be noted that Wu et al. (2010) only ﬁtted the corre- CS (7-6). The M –L and M –L corre- clump CS clump HCN lations for clumps with infrared luminosities larger than lations are very tight and sublinear. For comparison, we 104.5 L(cid:12), where the initial mass function is fully sampled. also investigate the correlations between clump masses Therefore, to compare with their results, we also ﬁtted and line luminosities for J=2-1, J=5-4, J=7-6 of CS and (cid:48) the L –L correlations of CS (7-6) and HCN (4-3) J=1-0, J=3-2 of HCN. The line luminosities are from Wu TIR mol for clumps with infrared luminosities larger than 104.5 L(cid:12) et al. (2010). The clump masses were calculated with in our sample. The derived slopes of CS (7-6) and HCN 350 µm continuum emission (Mueller et al. 2002). As (4-3) are 0.72(0.05) and 0.74(0.06), respectively, which shown in Figure 11, the clump masses are also strongly are smaller than the corresponding slopes for all clumps correlated with line luminosities of these transitions. The with infrared luminosities larger than 103 L(cid:12). In general, slopes (α), intercepts (β) and correlation coeﬃcients (R) (cid:48) as shown in Table 4, the L –L correlations steepen of the correlations were summarized in Table 4. For CS TIR mol for line transitions with lower upper energies and lower (7-6), the correlations for the sample of Wu et al. (2010) eﬀective excitation densities. Especially for HCN (4-3) and our sample are consistent with similar slopes. In (cid:48) and CS (7-6), their L –L correlations are appar- general, considering the errors, the correlations for lower TIR mol ently sublinear. Narayanan et al. (2008) predicted sim- transitions like J=2-1, J=5-4 of CS and J=1-0, J=3-2 ilar trend for external galaxies. However, the slopes of of HCN are close to linear. While for the other higher HCN transitions at clump scale (this work) or at galaxy transitions, the correlations are sublinear. scale (Zhang et al. 2014) are all much higher than those In Table 4, we also present the parameters of the L - TIR predicted by Narayanan et al. (2008). Narayanan et ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: Liu

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text: the

text: Fig.

text: Upper

text: panel:

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text: masses

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text: luminosities

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text: CS

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text: HCN

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text: panels:

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text: luminosities

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text: (“G”)

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text: the

text: correlations

text: between

text: clump

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text: 16 Liu et al. Fig. 12.— (a). clump mass from (Mueller et al. 2002) vs. virial mass derived from HCN (3-2) correlations for the sample of Wu et al. (2010). (b). clump mass derived from 1.1 mm continuum vs. virial mass derived from CS (7-6) correlations for our sample. The red lines represent the best power law ﬁts. The black dashed lines represent linear correlations. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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