Estimation of conditional distribution functions from data with additional errors applied to shape optimization

Hansmann, Matthias; Horn, Benjamin M.; Kohler, Michael; Ulbrich, Stefan Article — Published Version Estimation of conditional distribution functions from data with additional errors applied to shape optimization Metrika Provided in Cooperation with: Springer Nature Suggested Citation: Hansmann, Matthias; Horn, Benjamin M.; Kohler, Michael; Ulbrich, Stefan (2021) : Estimation of conditional distribution functions from data with additional errors applied to shape optimization, Metrika, ISSN 1435-926X, Springer, Berlin, Heidelberg, Vol. 85, Iss. 3, pp. 323-343, https://doi.org/10.1007/s00184-021-00831-4 This Version is available at: https://hdl.handle.net/10419/286790 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Metrika (2022) 85:323–343 https://doi.org/10.1007/s00184-021-00831-4 Estimation of conditional distribution functions from data with additional errors applied to shape optimization Matthias Hansmann1·Benjamin M. Horn2·Michael Kohler1·Stefan Ulbrich2 Received: 25 January 2020 / Accepted: 5 July 2021 / Published online: 17 July 2021 © The Author(s) 2021 Abstract We study the problem of estimating conditional distribution functions from data containing additional errors. The only assumption on these errors is that a weighted sum of the absolute errors tends to zero with probability one for sample size tending to infinity. We prove sufficient conditions on the weights (e.g. fulfilled by kernel weights) of a local averaging estimate of the codf, based on data with errors, which ensure strong pointwise consistency. We show that two of the three sufficient conditions on the weights and a weaker version of the third one are also necessary for the spc. We also give sufficient conditions on the weights, which ensure a certain rate of convergence. As an application we estimate the codf of the number of cycles until failure based on data from experimental fatigue tests and use it as objective function in a shape optimization of a component. Keywords Conditional distribution function estimation ·Consistency ·Experimental fatigue tests ·Local averaging estimate ·Shape optimization ·Isogeometric analysis Mathematics Subject Classification 62G05 ·62G20 BMatthias Hansmann [email protected] Benjamin M. Horn [email protected] Michael Kohler [email protected] Stefan Ulbrich [email protected] 1Department of Mathematics, TU Darmstadt, 64289 Darmstadt, Germany 2Department of Mathematics, TU Darmstadt, 64293 Darmstadt, Germany 123 324 M. Hansmann et al. 1 Introduction Let (X,Y)be a random vector, such that Xis Rdvalued and Yis real-valued, with conditional distribution function (codf) F, i.e., F(y,x)=P{Y≤y|X=x}=EI{Y≤y}X=x. One idea to construct estimates, approaching the codf asymptotically for some fixed y∈Rand PX–almost all x∈Rd(where PXis the of Xinduced measure on Rd,Bd, i.e., PX(B)=P(X∈B)for every B∈Bd), is to use an independent and identically distributed (i.i.d.) sample (X1,Y1),…,(Xn,Yn)of (X,Y)to compute a local averaging estimate Fn(y,x)= n  i=1 Wn,i(x)I{Yi≤y}(1) of the codf. Here Wn,i(x)for i=1,...,nare nonnegative weights, which can depend on the samples X1,...,Xn. A commonly used example for those weights are the weights of the so-called kernel estimate, which are defined by Wn,i(x)= Kx−Xi hn n i=1Kx−Xi hn,(2) where 0/0=0 by definition (cf., e.g., Nadaraya (1964) and Watson (1964)). Here hn>0 is the so-called bandwidth and K:Rd→Ris a so-called kernel function, e.g., the so-called naive kernel defined by K(x)=I{||x||≤1}for all x∈Rd. For a fixed y∈R, the estimate introduced in (1) is a special case of an estimate of a regression function m(x)=E{Y|X=x}(with the choice of I{Y≤y}as dependent variable). Thus, all known results on estimates of the regression function do also apply for the corresponding estimates of the codf. The regression estimate corresponding to the one of the codf in (1), i.e., mn(x)= n  i=1 Wn,i(x)·Yi has been seminally considered by Stone (1977). In particular, in Theorem 1 Stone gave necessary and sufficient conditions on the weights, such that the regression estimate mnis weakly consistent in Lrfor every real r≥1, i.e., E|mn(x)−m(x)|rPX(dx)→0(n→∞). 123 Estimation of conditional distribution functions from data… 325 These conditions are for example fulfilled by the special choices of the weights of the partitioning, kernel and nearest-neighbor estimate, for details we refer to Chapters 4, 5 and 6 in Györfi et al. (2002). Since the work of Stone (1977), several authors also dealt with strongly pointwise consistency of special regression function estimates for dependent variable Y, which are almost surely bounded by some constant. In case of the estimation of the codf we say that the estimate Fnof the codf Fis strongly pointwise consistent for some fixed y∈R,if Fn(y,x)→F(y,x)a.s.for PX–almost every x.(3) In the context of nonparametric regression, Devroye (1981) showed the strongly pointwise consistency of the kernel regression estimate, presuming that Kis a so-called window kernel and that the bandwidth hnfulfills some mild asymptotic conditions. Greblicki et al. (1984) generalized this consistency result to some broader class of kernels with possibly unbounded support. Stute (1986) also showed a result concerning the uniform pointwise consistency of the kernel estimate of the conditional distribution function. A proof of the strongly pointwise consistency of the partitioning regression estimate can be found in (Györfi et al. 2002, Theorem 25.6.). Györfi (1981a) and Devroye (1981) independently showed results concerning the strongly pointwise consistency of the nearest neighbor regression estimates. Devroye (1982) also gave necessary and sufficient conditions for the strongly pointwise consistency of the nearest neighbor regression estimates. In order to obtain consistency results for all distributions (X,Y)with E|Y|<∞ (so-called universal consistency), some authors considered modified versions of the above mentioned estimates [cf., e.g., Walk (2001), Algoet and Györfi (1999)]. See also (Györfi et al. 2002, Chapter 25) and the literature cited therein for other estimates and further results on the strongly universal pointwise consistency. Rates of convergence in probability for the kernel regression estimate have been obtained in Krzyzak and Pawlak (1987) and in Györfi (1981b) for the nearest neighbor regression estimate. Uniform almost sure rates of convergence for regression estimates have been shown in Härdle et al. (1988) by considering a more general setting of kerneltype estimators of conditional functionals. Optimal global rates of convergence for nonparametric regression estimates have been shown by Stone (1982). Other estimates of the codf have been proposed by Hall et al. (1999), who studied the rate of convergence of a weighted kernel estimator. Cai (2002) showed asymptotic normality of this estimate. Furthermore, Hall and Yao (2005) used a dimension reduction technique to approximate the codf and study the asymptotic properties. Preadjusted local averaging estimates of the codf were proposed by Veraverbeke et al. (2014), who proved results concerning the uniform rate of convergence. So far, only Liero (1989) and Hansmann et al. (2019) studied a local averaging regression estimate with generalized weights and formulated conditions for consistency results. Liero (1989) assumed the weights to have the special form Wn,i(x)=φn,i(x,Xi) n i=1φn,i(x,Xi), 123 326 M. Hansmann et al. where φi,nis a Borel-measurable function on Rd×Rdand does therefore only depend on Xiand formulated conditions that ensure a certain uniformly strong rate of convergence. Hansmann et al. (2019) gave conditions on the above introduced general weights Wn,iof a local averaging regression estimate which imply the strongly universally consistency, i.e. |mn(x)−m(x)|2PX(dx)→0a.s. for all distributions (X,Y)with EY2<∞. To the authors knowledge there is no result so far, which characterizes necessary and sufficient conditions on the above mentioned general weights Wn,i, that ensure the strongly pointwise consistency or a certain rate of convergence in probability of the corresponding local averaging estimate of the conditional distribution function. One of the main goals of this paper is to present these two results. A further aspect investigated in this paper is the consideration of additional errors in the data, which is motivated by an application in the context of shape optimization with respect to the fatigue life of a component. A short overview on the method used to asses the fatigue behaviour will be described in the following section. 1.1 Application in the context of experimental fatigue tests In order to predict the fatigue life of a certain material, we use data from so-called strain-controlled fatigue tests, in which a material sample gets repeatedly elongated by a fixed strain amplitude ε. The repetitions, the so-called number of cycles N, until the material fails are counted and the corresponding stress amplitude τis measured. Repeating this experiment yields data (m) 1,N(m) 1,τ(m) 1,...,(m) lm,N(m) lm,τ(m) lm for each material m. Since the mentioned strain-contolled fatigue tests are very time consuming, we only have 12 data points for the material of interest, which is not enough for a nonparametric estimation of the conditional distribution function of the number of cycles given a certain strain amplitude ε. Thus, we assume the model N(m)(ε)=μ(m)(ε)+σ(m)(ε)·δ(m)(4) to hold. In Sect. 3.1. we will describe a suitable method to estimate μand σby ˆμand ˆσ, respectively, such that we can finally obtain data ˆ δ(m) i=N(m) i−ˆμ(m) i ˆσ(m) i for i=1,...,lm,(5) Due to the assumption in (4) the conditional distribution function of the number of cycles given a strain amplitude εcan be determined by a simple linear transformation 123 Estimation of conditional distribution functions from data… 327 from the distribution function of δ(m). Since we only have available 4 to 35 of the above data samples per material to estimate the distribution function of δ(m), we will use data samples from other materials, that have similar static material properties. To this end we use an estimate of the conditional distribution function, with the vector of five statical material properties (Young’s modulus, the yield limit for 0.2% residual elongation, the tensile strength, the static strength coefficient and the static strain hardening exponent) as covariate X(m)and the samples δ(m) ias dependent variable. More precisely we apply a nonparametric estimate of the codf to the data X(m),δ(m) 1,...,X(m),δ(m) lm:mis a material in our database Furthermore the above data points contain errors in the dependent variable since we only estimated μ(m)and σ(m), which leads to the topic of this paper, where we want to investigate how additional errors in the dependent variable influence an estimate of codf and show theoretical results concerning strongly pointwise consistency and rate of convergence in probability. 1.2 Data with errors Motivated by the application described in the previous subsection, we generalize our mathematical setting and assume that we only have available data X1,¯ Y1,n,…, Xn,¯ Yn,nwith errors in the samples of the dependent variable instead of the i.i.d. data (X1,Y1),…,(Xn,Yn). In our above mentioned application we do not know anything explicitly on the errors ¯ Yi,n−Yi(i=1,...,n). Thus, we are not able to impose a structure on those errors. In particular, we can not assume that those errors have to be random and in case that they are random they do not need to be independent or identically distributed and they do not need to have expectation zero, so estimates for convolution problems (see, e.g., Meister (2009) and the literature cited therein) are not applicable in the context of this paper. But we can assume that with increasing number nof total samples we also get more samples from the strain controlled fatigue tests for each of the materials. Thus, our estimates ˆμ(m)and ˆσ(m)and therefore our data ˆ δ(m) iof (5) get more reliable for all materials m. Consequently, with increasing n, our errors ˆ δ(m) i−δ(m) iget small for all materials m. Since the δ(m) iare the samples of our dependent variable it seems to be a natural idea in our application to assume that the absolute errors between Yi and ¯ Yi,nuniformly converge to zero almost surely, i.e., to assume that max i=1,...,n|Yi−¯ Yi,n|→0a.s. In our theoretical results in Section 2 it will turn out that we only have to assume the weaker condition ηn(x)= n  i=1 Wn,i(x)·Yi−¯ Yi,n→0a.s.for PX–almost every x.(E1) 123 328 M. Hansmann et al. where Wn,iare the weights of the local averaging estimate above. Note also that our set-up is triangular, which is the necessary in our application since the estimates ˆμ(m)and ˆσ(m)can change with the number of data points nand can therefore lead in (5) to completely new samples with errors of the random variable δ(m). Errors, for which an (average) sum of the (squared) absolute errors tends to zero (as in (E1)), have been recently considered in the context of nonparametric regression with random design (cf., Kohler (2006), Fromkorth and Kohler (2011)), nonparametric regression with fixed design (cf., Furer et al. (2013), Furer and Kohler (2015)), (conditional) quantile estimation (cf., Hansmann and Kohler (2017) and Hansmann and Kohler (2019)), density estimation (cf., Felber et al. (2015)) and distribution estimation (cf.,Bottetal.(2013)). Since we do not assume anything on the nature of the errors besides that they are pointwise asymptotically negligible in the sense that (E1) holds, it seems to be a natural idea to ignore them completely and to try to use the same estimates as in the case that an independent and identically distributed sample is given. 1.3 Main Results In Theorem 2.1 we present sufficient conditions on the weights and prove that these conditions ensure that the estimate ¯ Fn(y,x)applied to data X1,¯ Y1,n,…,Xn,¯ Yn,n with errors in the samples of the dependent variable is pointwise consistent in the sense that it approaches the interval Fy−,x,F(y,x)=⎡ ⎣lim ε→0, ε>0 F(y−ε, x),F(y,x)⎤ ⎦ for PX–almost all xasymptotically, presumed that the errors fulfill (E1). As we will show in Corollary 2.1, these assumptions on the weights are for example fulfilled by the weights of the kernel estimate. We also show that two of the three sufficient conditions and a weaker version of the third condition are also necessary for the above pointwise consistency (see Theorem 2.2). We also investigate the rate of convergence of the estimate ¯ Fnand present conditions on the weights, which ensure for ¯ Fna pointwise rate of convergence in probability of rn(x)+ηn(x) (see Theorem 2.3), where rn(x)is some deterministic rate fulfilling for PX–almost every x∈Rdrn(x)→0asn→∞, and where ηn(x)is defined in (E1). We also present an application to simulated and real data (see Section 3). In the real data application we use the considered method to estimate the distribution function of the numbers of cycles until failure in the context of fatigue behavior of steel under cyclic loading. This estimate is utilized as the objective in a shape optimization procedure, 123 Estimation of conditional distribution functions from data… 329 which is embedded in an algorithm-based product development approach to determine an optimal profile geometry with respect to the fatigue behavior. 1.4 Notation Throughout this paper the following notation is used: We write Un=OP(Vn)if the nonnegative random variables Unand Vnsatisfy lim c→∞lim sup n→∞ P{Un>c·Vn}=0. The sets of natural positive, natural nonnegative and real numbers are denoted by N, N0and R, respectively. We write →Pas an abbreviation for convergence in probability and IAfor the indicator function of the set A. We denote the Euclidean Norm on Rd by ||·||.Forz∈Rand a set A⊆R, we define the distance from zto Aas dist (z,A):= inf a∈A|z−a|. Furthermore, we write for the left-sided limit of a function G Gy−=lim ε→0, ε>0 G(y−ε) 1.5 Outline The outline of the paper is as follows: The main results are formulated in Section 2 and proven in the supplemental material. In Section 3, we present an application to simulated and real data. 2 Main results Let ¯ Fn(y,x)= n  i=1 Wn,i(x)I¯ Yi,n≤y(6) be a local averaging estimate of the codf F(y,x)corresponding to the data with errors X1,¯ Y1,n,…,Xn,¯ Yn,n. 2.1 Consistency First of all, we give sufficient conditions on the sequence of weights Wn,i, such that the estimate ¯ Fnis pointwise consistent for all distributions of (X,Y)and all y∈R. 123 330 M. Hansmann et al. The following result holds, which will be proven in Section S2 in the supplemental material. Theorem 2.1 Let (X,Y),(X1,Y1),(X2,Y2)... be i.i.d. Rd×R-valued random vectors and let Wn,i(x):= Wn,i(x,X1,...,Xn)x∈Rdbe nonnegative weights, which fulfill (A1) n i=1Wn,i(x)→1a.s.for PX-almost every x ∈Rd, (A2) for every Borel-measurable set B ∈Bd n  i=1 Wn,i(x)I{Xi∈B}−I{x∈B}→0a.s.for PX-almost every x ∈Rd, (A3) log (n)·n i=1Wn,i(x)2→0a.s.for PX-almost every x ∈Rd. Furthermore let ¯ Y1,n,..., ¯ Yn,nbe random variables, which fulfill (E1) and let ¯ Fnbe the local averaging estimate defined in (6) with weights Wn,i. Then ¯ Fnis pointwise consistent in the sense that for all y ∈R dist ¯ Fn(y,x),Fy−,x,F(y,x)→0a.s.for PX–almost every x.(7) In the following corollary, which will be proven in Section S2 in the supplemental material, we formulate sufficient conditions for the pointwise strongly consistency of the kernel estimate of the codf, defined by the weights in (2). Corollary 2.1 Let (X,Y),(X1,Y1),(X2,Y2)... be i.i.d. Rd×R-valued random vectors. Assume that K is the naive kernel and that the bandwidth hn>0fulfills hn→0and n ·hd n/log (n)→∞ (n→∞ ).(8) Furthermore, let ¯ Y1,n,..., ¯ Yn,nbe random variables, which fulfill n  i=1Yi−¯ Yi,n·Kx−Xi hn n i=1Kx−Xi hn→0a.s.for PX–almost every x.(9) Let Wn,ibe the weights of the kernel estimate with kernel K and bandwidth hn. Then the kernel estimate ¯ Fnof the codf as defined in (6) is pointwise consistent in the sense that for all y ∈R dist ¯ Fn(y,x),Fy−,x,F(y,x)→0a.s.for PX–almost every x. Remark 2.1 Analogous results can be shown for estimates of the codf corresponding to the partitioning and nearest neighbor weights, assuming the conditions from Theorems 25.6. and 25.17., respectively, in Györfi et al. (2002). Furthermore, Corollary 2.1 can be extended to a more general class of kernels, which has been considered by Greblicki et al. (1984). 123 Estimation of conditional distribution functions from data… 337 for each material m, which usually needs more samples. So we augmented our data points per material mby 100 artificial ones as in Furer and Kohler (2015): At first, we interpolate the squared deviations Y(k) ifor each material k= mon a grid of 100 equidistant strain amplitudes ε. In order to generate an artificial data point at a fixed grid point, we also use interpolated values from materials, that are similar to the material m, assuming that similar materials yield similar fatigue behavior. Observe that we use the whole database consisting of 132 materials in order to obtain more interpolated values and to improve the statistical power of our estimation. The similarity is measured using 5 static material properties, namely Young’s modulus, the yield limit for 0.2% residual elongation, the tensile strength, the static strength coefficient and the static strain hardening exponent. In order to ensure that we only use interpolated values from those materials that have similar static material properties, we apply the Nadaraya-Watson kernel regression estimates with the static material properties as covariate and the interpolated data as dependent variable. In this way we obtain 100 artificial data points (one at each grid point) per material m. Finally, the estimation of the standard deviation σ(m)(ε)is done by weighting the NadarayaWatson kernel regression estimates applied to the real and the artificial data of the squared deviations as dependent variable and the corresponding ε-values as covariate. Thus, we finally determine the data samples ˆ δ(m) i=N(m) i−ˆμ(m) i ˆσ(m) i for i=1,...,lm of the random variables δ(m)for each material m. Notice that these samples contain errors because we only estimated μ(m)(ε)and σ(m)(ε). Since only 12 and 8 of the above data samples for the two material states of HC 480 LA are available, we also use data samples from other materials of the database, that have similar static material properties (with the same justification as above), in order to estimate the codf of δ(m). This consideration of similar materials in the estimation of the conditional distribution function is done by using the kernel estimate of the codf with the static material properties as covariate Xiand the data samples of δ(m)for all 122 materials mas the dependent variable. The bandwidth hof the kernel weights is determined by a crossvalidation of the corresponding regression estimate as described in the beginning of this section. As described in Sect. 1.2 it can be assumed that (E1) holds for the errors ˆ δ(m) i−δ(m) i. Thus, evaluating the mentioned estimate of the codf at the static material properties x= X(m)of some material mleads to an estimate ˆ Gδ(m)of the codf of δ(m)(see Theorem 2.2 for a theoretical justification). However, this estimate ˆ Gδ(m)can be transformed to an estimate ˆ F(m)of the codf of N(m)given a strain amplitude εby ˆ FN(m)(y,ε )=ˆ Gδ(m)y−ˆμ(m)(ε) ˆσ(m)(ε).(17) 123 338 M. Hansmann et al. Fig. 1 Estimated conditional distribution function ˆ FN(m)(Nmin,ε )for Nmin =50,000 for the material HC 480 LA in as received and linear flow split state This estimate of the conditional distribution function is evaluated at y= Nmin =50,000 numbers of cycles to obtain an approximate probability of a failure before Nmin number of cycles for a fixed strain amplitude ε.InFig.1, we illustrated this estimated probability ˆ FN(m)(Nmin,ε )for the considered material HC 480 LA in as-received and linear flow split state and ε∈[0%,1%]. Here the strain amplitude ε is given proportional to the length of the material sample used in the experiments. As expected, ˆ FN(m)(Nmin,ε )is increasing in ε. Since we also needed the derivative of ˆ FN(m)(Nmin,ε )w.r.t. ε, we interpolated the function ˆ FN(m)(Nmin,ε )by a piece-wise cubic smoothing spline, using 200 equidistant data points of εand corresponding function values. 3.2.2 Fatigue Strength Shape Optimization The former presented estimate of the failure probability ˆ FN(m)(Nmin,·)is in the following applied to the shape optimization of a multichambered profile. Our aim is to find the optimal geometry for a specific load scenario and a given starting geometry under certain design constraints, to reach minimal failure probability, as defined above. In order to calculate the failure probability of every point of the profile, we model the physical behavior of the considered geometry under applied loads at each point. For this purpose we describe the mechanical system in terms of the linear elasticity equations, for further details on the elasticity equations we refer to the supplement material Section S1. For numerical treatment of the elasticity equations are discretized in the 123 Estimation of conditional distribution functions from data… 339 sense of isogeometric analysis. Thus, the discretized linear elasticity equations are denoted by Ah(u)yh=bh(u). The discretization by methods of the isogeometric approach is explained in detail in the supplement material. 3.2.3 Shape Optimization In this section we briefly describe the shape optimization problem governed by the linear elasticity problem as defined in the supplement material Section S1. The finite dimensional shape optimization problem can be written as min Jh(u,yh) s.t. u∈Uad,Ah(u)yh=bh(u). The design variables are denoted by u∈Rn, where n∈Nis the dimension of the design space. By yh∈R˜nthe displacement is described, which is determined by the linear elasticity equations Ah(u)yh=bh(u). The number of the control points of the isogeometric mesh is denoted by ˜n∈N. An introduction to shape optimization is given in Haslinger and Mäkinen (2003). Since the elasticity problem has a unique solution, we define a Lipschitz continuous solution operator u→ yh(u)such that the reduced form of the objective function can be written as jh(u):= Jh(u,yh(u)). The corresponding shape gradient gh(u)can be efficiently determined by the adjoint approach as described in Hinze et al. (2009). The reduced shape optimization is stated as min jh(u) s.t. u∈Uad. Here the set of admissible designs Uad ⊂Rnis defined by design constraints, for example angle or length restrictions. In this work we use the accumulated failure probability as objective function jh(u):=  i∈{1,...,˜n}ˆ FN(m)(Nmin,¯ε([yh(u)]i), u), as defined above, see (17), with Nmin =50,000 fixed, where the sum is calculated over all control points (coefficients of the basis functions in yhabove). If we define the failure of the whole profile by the failure of one of its parts, the accumulated failure probability over all parts is a (discretized) upper bound on the failure probability of 123 340 M. Hansmann et al. Fig. 2 Applied load scenario. For simplicity the bending radii are neglected in this draft. The right side of the profile is clamped at the bottom and the top. A uniform surface load q1∈R3, with q1=70N,is applied with 45◦to the surface at the upper left part and a load q2∈R3, with q2=70N,is applied with 45◦to the lower left side of the profile. The load scenario is constant in the third dimension q1 q2 the whole profile and thus a reasonable objective function. It can be shown, that the objective function is nonconvex with respect to the design. The main principal strain ¯ε can be determined by calculating the maximal eigenvalue with respect to the absolute value of the linearized strain tensor ε,byCardano’s formula. The optimization is done using a sequential quadratic programming method (SQP) [see, e.g., Nocedal and Wright (2006)]. 3.2.4 Numerical Result We apply the above described methods to perform a shape optimization of a threechambered profile with respect to fatigue strength. Therefore, we assume a static load scenario, as shown in Fig. 2. The profile is clamped at the boundary on the right-hand side. Additionally, there are surface loads applied at the upper and lower left of the geometry. The loads act on the surface at an angle of 45◦. The geometry is modeled as a tricubic NURBS solid, with 25,920 degrees of freedom and 1350 elements. The outer dimensions are 50 cm ×50 cm ×2 cm. To reduce the need of numerous additional constraints, we applied a parametrization with only twelve degrees of freedom. For this purpose, we subdivide the profile into four parts and determine the barycentric coordinates of each control point. As constraints, we consider an upper bound on the total volume, and we fix the volume of the Neumann and Dirichlet boundaries. For technical reasons, we also add a minimal bound for the volume of each element to circumvent negative element volumes. After 58 iterations with 375 function evaluations, the SQP method found the solution depict in Fig. 3. The accumulated failure probability could be reduced about almost 53.58%. The used SQP method is the standard MATLAB R2018a implementation. Additionally, we compare the result to the optimization with respect to the compliance jh(u):= (yh(u))M hfh+M hqh, where fhand qhare the discretized volume force and surface load acting on the geometry, respectively, and M hand M hare the mass matrices of the interior and 123 Estimation of conditional distribution functions from data… 341 Fig. 3 Starting solution (left) compared to the optimal geometries with respect to the accumulated failure probability (middle) and the compliance (right). The color represents the von Mises stress in MPa. The displacement is neglected. The accumulated failure probabilities of the optimal solutions could be reduced about 53.58% (middle) or 37.82% (right), respectively boundary of the considered geometry, respectively. In this case the compliance could be reduced about 58.34% after 29 iterations and 106 function evaluations. The optimal solution is visualized in Fig. 3. The accumulated failure probability of this geometry is reduced about 37.82% compared to the starting solution. We see that in general the optimization of the accumulated failure probability can not be replaced by the classical compliance optimization. All the calculations are performed on an Intel Core i7-4790 CPU with 3.60 GHz and 16 GB RAM. The used software was Mathworks MATLAB R2018a running in single thread mode. Acknowledgements The authors would like to thank the German Research Foundation (DFG) for funding this project within the Collaborative Research Centre 666. The authors would also like to thank an associate editor and a referee for their helpful comments. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest. Supplementary Materials The supplement contains additional information on the linear elasticity equations and the isogeometric approach complementing the information in Chapter 3.2 and furthermore the proofs of the Theorems and the Corollaries. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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