12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 1 Importance Sampling in the Evaluation and Optimization of Buffered Failure Probability Marwan M. Harajli Graduate Student, Dept. of Civil and Environ. Engineering, University of California, Berkeley, USA R. Tyrrell Rockafellar Emeritus Professor, Dept. of Mathematics, University of Washington, Seattle, USA Johannes O. Royset Associate Professor, Operations Research Dept., Naval Postgraduate School, Monterey, USA ABSTRACT: Engineering design is a process in which a systemโs parameters are selected such that the system meets certain criteria. These criteria vary in nature and may involve such matters as structural strength, implementation cost, architectural considerations, etc. When random variables are part of a system model, an added criterion is usually the failure probability. In this paper, we examine the buffered failure probability as an attractive alternative to the failure probability in design optimization problems. The buffered failure probability is more conservative and possesses properties that make it more convenient to compute and optimize. Since a failure event usually occurs with small probability in structural systems, Monte-Carlo sampling methods require large sample sizes for high accuracy estimates of failure and buffered failure probabilities. We examine importance sampling techniques for efficient evaluation of buffered failure probabilities, and illustrate their use in structural design of two multi-story frames subject to ground motion. We formulate a problem of design optimization as a cost minimization problem subject to buffered failure probability constraints. The problem is solved using importance sampling and a nonlinear optimization algorithm. Uncertainty in loads, material properties and other parameters need to be accounted for in designing modern engineering systems. For that purpose, the choice of how reliability is quantified plays an important role when assessing the feasibility of a certain choice of design parameters. One such quantification is the failure probability, see, e.g. Ditlevsen and Madsen (1996). The buffered failure probability is an alternative presented in Rockafellar and Royset (2010); see also Rockafellar and Royset (2015), which offers computational and practical benefits over the former choice. Design optimization problems with buffered failure probability constraints are in some sense no harder to solve than the underlying deterministic design optimization problems. In particular, if a deterministic design optimization problem is convex, the corresponding stochastic one with buffered failure probability constraints is also convex. This situation is dramatically different than that for failure probability constraints, which involves significant added complications when passing from the deterministic to the stochastic problem. Moreover, the buffered failure probability captures tail behavior more comprehensively than the failure probability and, in fact, incorporates the degree of failure to some extent; see Rockafellar and Royset (2010) for a discussion. Analytical computation of the buffered failure probability is usually not possible. Consequently, numerical sampling techniques are usually used; namely Monte-Carlo Sampling (MCS) methods. However, since failure events may occur with low probability, MCS methods require large sample sizes to estimate the value 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2 of both failure and buffered failure probabilities, and therefore also long computing times. Importance Sampling (IS) is a method that can improve accuracy and require fewer sample points to estimate (buffered) failure probabilities, see Srinivasan (2002). This paper uses buffered failure probability as a quantification of reliability, describes a method of evaluating buffered failure probability constraints by IS, and incorporates the method into the design of a structural system. An example involving two multi-story structural frames subject to ground motion is studied in detail. 1. IMPORTANCE SAMPLING In this section we explain the concept of IS. 1.1. Definition Given a random variable ๐ with probability density function ๐, the expected value of Y is defined as ๐ธ[๐] = โซ ๐ฆ๐(๐ฆ)๐๐ฆโโโ. Let ๐ฆ1, ๐ฆ2, โฆ , ๐ฆ๐ be realizations of ๐, independently sampled according to f. Then, the expectation of the random variable Y can be approximated by ๐ธ[๐] โ1๐โ๐ฆ๐๐๐=1 Multiplying Eq. (1) by โ(๐ฆ)โ(๐ฆ), where โ is a probability density that is zero only when f is zero, we obtain that ๐ธ[๐] = โซ๐ฆ๐(๐ฆ)โ(๐ฆ)โ(๐ฆ)๐๐ฆโโโ= ๐ธ [๐๐(๐)โ(๐)] where now V is distributed according to h. Thus with ๐ฃ1, ๐ฃ2, โฆ , ๐ฃ๐ sampled from โ we find that ๐ธ[๐] โ1๐โ๐ฃ๐๐(๐ฃ๐)โ(๐ฃ๐)๐๐=1 Appropriate choices of sampling density โ may result in an increased efficiency in estimating the expectation; see for example Asmussen and Glynn (2007). 2. BUFFERED FAILURE PROBABILITY Failure of a structural component is defined by means of a limit-state function ๐(๐ฑ, ๐ฏ), where ๐ฑ is a vector of design variables, and ๐ฏ is a vector of quantities that represent uncertain parameters of the system (e.g. loads and material strength), see Rockafellar and Royset (2010). It is common that the vector of uncertain parameters is modeled as a vector of random variables ๐ =(๐1, ๐2, โฆ , ๐๐) described by a given or estimated joint probability density function. We henceforth denote all random variables using upper case letters and their realizations with lower case ones. For a realization ๐ฏ of ๐, and a choice of design variables ๐ฑ, the system is assumed to be in an unsatisfactory state, i.e., in failure, when ๐(๐ฑ, ๐ฏ) is strictly positive. The event of failure is thus described by {๐(๐ฑ, ๐) >0}. We note that ๐(๐ฑ, ๐) is a random variable for any fixed design x. In this section we define the buffered failure probability and describe methods for assessing whether it is sufficiently small. 2.1. Definition For a fixed x and a given probability level ๐ผ, we recall that the ๐ผ-quantile of ๐(๐ฑ, ๐), denoted by ๐๐ผ(๐ฑ), is the smallest scalar ๐ such that ๐(๐(๐, ๐ฝ) โค ๐) โฅ ๐ผ where ๐ is the probability measure. We define the ๐ผ-superquantile as ?ฬ ?๐ผ(๐ฑ) =11 โ ๐ผโซ๐๐ฝ(๐ฑ)๐๐ฝ1๐ผ 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 3 It is clear that a superquantile is a normalized average of quantiles for a range of probability levels. If the cumulative distribution function of ๐(๐ฑ, ๐) is continuous at ๐๐ผ(๐ฑ), then a superquantile is equivalently expressed by ?ฬ ?๐ผ(๐ฑ) = ๐ธ[๐(๐ฑ, ๐)|๐(๐ฑ, ๐) โฅ ๐๐ผ(๐ฑ) ]. It is clear that in this case, a superquantile is the expected value of ๐(๐ฑ, ๐), conditioned on ๐(๐ฑ, ๐) being greater or equal to the ๐ผ-quantile. Regardless of the cumulative distribution function of ๐(๐ฑ, ๐), a superquantile can also be expressed as the optimal value of a minimization problem (see Rockafellar and Royset, 2010), i.e., ?ฬ ?๐ผ(๐ฅ) = min๐ฆ{๐ฆ + ๐ธ[max {๐(๐ฑ, ๐) โ ๐ฆ, 0}]} (1) Computing a superquantile thus involves finding a scalar y that minimizes a convex function given in terms of an expectation, which is straightforward when the distribution of ๐ is known and the limit-state function can be evaluated relatively easily. If ๐(๐ฑ, ๐) follows a discrete distribution with realizations ๐ฆ1 < ๐ฆ2 < โฏ < ๐ฆ๐ and corresponding probabilities ๐1, ๐2, โฆ , ๐๐, a superquantile obeys the expressions ?ฬ ?๐ผ(๐ฑ) =โ๐๐๐ฆ๐ if ๐ผ = 0๐๐=1 ?ฬ ?๐ผ(๐) =11 โ ๐ผ[(โ๐๐๐๐=1โ ๐ผ)๐ฆ๐+ โ ๐๐๐ฆ๐๐๐=๐+1 ] if โ๐๐ <๐โ1๐=1๐ผ โคโ๐๐๐๐=1< 1 and ?ฬ ?๐ผ(๐ฑ) = ๐ฆ๐ if ๐ผ > 1 โ ๐๐ The buffered failure probability ?ฬ ?(๐ฑ) is defined in terms of a superquantile. Specifically, ?ฬ ?(๐ฑ) = 1 โ ฮฑ where ๐ผ is the probability level such that ?ฬ ?๐ผ(๐ฑ) = 0. Consequently, the buffered failure probability constraint ?ฬ ?(๐ฑ) โค 1 โ ฮฑ is satisfied if and only if ?ฬ ?๐ผ(๐ฑ) โค 0. This equivalence will be utilized when formulating design optimization problems below. 2.2. Superquantile estimates using IS In this subsection we discuss sampling techniques for estimating superquantiles for a given probability level. Moreover, we present a method to compute an estimate of an upper bound of a superquantile and build a confidence interval around it. For probability levels close to 1, the event {๐(๐, ๐ฝ) > ๐ฆ} appearing in Eq. (1) occurs with small probabilities for typical values of y, and the use of IS may become beneficial. After sampling ๐โฒ realizations ๐ฏ1, ๐ฏ2, โฆ , ๐ฏ๐โฒ from a suitable density โ, an estimate for the superquantile becomes ?ฬ ?๐ผ(๐ฑ) โ min๐ฆ ๐ฆ + 1๐โฒ11 โ ๐ผโmax(๐(๐ฑ, ๐ฏ๐) โ ๐ฆ, 0) ๐(๐ฏ๐)โ(๐ฏ๐)๐โฒ๐=1 (2) which is easily computed using a linear programming algorithm or specialized procedures. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 4 Let ๐ฆโ denote a scalar that minimizes the expression in Eq. (2). Next we obtain ๐โฒโฒ sample point from h, namely ๐ฏ1, ๐ฏ2, โฆ , ๐ฏ๐โฒโฒ, and define ๐ข๐๐โฒ = ๐ฆโ +11 โ ๐ผmax(๐(๐ฑ, ๐ฏ๐) โ ๐ฆโ, 0) ๐(๐ฏ๐)โ(๐ฏ๐) Since the expectation of this expression is an upper bound on ?ฬ ?๐ผ(๐ฑ), due to the fact that ๐ฆโ might not minimize the right-hand side in Eq. (1), it can be seen that ๐๐โฒ =1๐โฒโฒโ๐ข๐๐โฒ๐โฒโฒ๐=1 is an approximate upper bound on ?ฬ ?๐ผ(๐ฑ). For large ๐โฒโฒ and an independent sample ๐ฏ1, ๐ฏ2, โฆ , ๐ฏ๐โฒโฒ, it is clear that ๐๐โฒ is approximately normal, making the computation of confidence interval straightforward. With sample variance ๐2 โ1๐โฒโฒ โ 1โ(๐ข๐๐โฒ โ ๐๐โฒ)2๐โฒโฒ๐=1 we obtain, for example, an approximate 95% confidence interval for an upper bound on the ๐ผ-superquantile as [๐๐โฒ โ 1.96๐, ๐๐โฒ + 1.96๐]. 3. SYSTEM DESIGN In this section we present a mathematical formulation of a design problem incorporating uncertainty and also give an algorithm for designing a system as well as assessing the resulting design. Given a system described by a limit-state function ๐, we seek a design ๐ฑ that minimize a cost function ๐ถ, which typically depends on the design. Moreover, we would like the buffered failure probability of the design to be no larger than 1 โ ๐ผ. Consequently, we seek to solve the optimization problem min๐โ๐๐ถ(๐) ๐ . ๐ก. ?ฬ ?(๐ฑ) โค 1 โ ๐ผ where X is subset of the space of design parameters considered admissible. For example, X could incorporate bounds on the design parameters that ensure a practical design. In view of the equivalence between buffered failure probability constraints and superquantile constraints, we find that this problem is identical to min๐โ๐๐ถ(๐) ๐ . ๐ก. ?ฬ ?๐ผ(๐ฑ) โค 0 Moreover, from Eq. (1), this simplifies to the problem min๐โ๐๐ถ(๐) ๐ . ๐ก. ๐ง0 +11 โ ๐ผ๐ธ[max{๐(๐, ๐ฝ) โ ๐ง0, 0}] โค 0 where ๐ง0 is an auxiliary variable corresponding to y in Eq. (1). Relying on IS and the sample ๐ฏ1, ๐ฏ2, โฆ , ๐ฏ๐ from h, an approximation of this problem takes the form min๐โ๐๐ถ(๐) ๐ . ๐ก. ๐ง0 +1๐(1 โ ๐ผ)โmax{๐(๐ฑ, ๐ฏ๐) โ ๐ง0, 0}๐(๐ฏ๐)โ(๐ฏ๐)๐๐=1โค 0 This problem can be rewritten as min๐โ๐๐ถ(๐) ๐ . ๐ก. ๐ง0 +1๐(1 โ ๐ผ)โ๐ง๐๐(๐ฏ๐)โ(๐ฏ๐)๐๐=1โค 0 ๐(๐ฑ, ๐ฏ๐) โ ๐ง0 โค ๐ง๐ ๐ = 1,2, โฆ . , ๐ ๐ง๐ โฅ 0 ๐ = 1,2, โฆ , ๐ . (3) Here, ๐ง1, โฆ , ๐ง๐ are auxiliary optimization variables; see Rockafellar and Royset (2010). The design and assessment process can be summarized as follows: ๏ท Select a suitable sampling density h. ๏ท Sample independently from h and obtain the realizations ๐ฏ1, ๐ฏ2, โฆ , ๐ฏ๐. ๏ท Solve Eq. (3) using an optimization solver to obtain a solution ?ฬ?. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 5 ๏ท Sample independently from h and obtain realizations ๐ฏ1, ๐ฏ2, โฆ , ๐ฏ๐โฒ, where ๐โฒ does not need to be particularly large. ๏ท Solve for y in Eq. (2) and obtain ๐ฆโ. ๏ท Sample independently from h and obtain ๐ฏ1, ๐ฏ2, โฆ , ๐ฏ๐โฒโฒ. Since these realizations are only going to be used to obtain average values, and not to be used in any optimization formulation, ๐โฒโฒ can be large. ๏ท Obtain the upper bound estimate ๐๐โฒ and build a confidence interval around it as described in subsection 2.2. Figure 1: Structures subject to ground motion and consequently the risk of pounding. 4. EXAMPLE DESIGN PROBLEM In this section we present an example concerned with the design of two multi-story structures subject to random ground motion. The obtained design is assessed using IS. 4.1. Problem Definition We consider two multi-story, single-bay frames with the same story height, separated by a distance ๐ of 20 inches, which are both subject to an unknown dynamic load. The structures have five and three stories, and the lateral inertial of each story is 0.25 kip.s2/in for both frames; see Figure 1. We assume that all columns in a frame have the same stiffness, and that all beams are rigid. The goal is to design the frames by choosing values for the column stiffness for each frame such that the buffered failure probability of pounding between the two frames is no more than 0.1. 4.1.1. Stochastic Load The system is subject to a ground acceleration assumed to be white noise of duration 20s with uncertain stationary amplitude. The amplitude is modeled by a lognormal distribution with mean 0.4 G and standard deviation 0.05 G. We adopt a time discretization with timestep of 0.02 seconds. The random vector V therefore comprise 1000 independent lognormal random variables of this kind. The assumed load is a simple model of an earthquake ground motion here used for illustration. 4.1.2. Limit State Function We let ๐ฑ = (๐1, ๐2) be the vector of design parameters, with ๐1 representing the stiffness of each column in each story of the leftmost (taller) frame in Figure 1 and ๐2 representing those of the right frame. With ๐ข๐(๐ก,๐) being the displacement of the mth story of Frame i at time t (with displacements to the right taken as positive), we define the limit-state function ๐(๐, ๐) = โ min๐ก,๐=1,2,3{ ๐ + ๐ข2(๐ก,๐) โ ๐ข1(๐ก,๐)} We use central differences to solve the dynamic system with time step 0.02 and ignore the inaccuracies this introduces in the evaluation of displacements. Likewise, the minimization over time is taken over the 1000 discretized time points. 4.1.3. Cost Function Since it is reasonable to assume that construction cost is proportional to stiffness, we adopt the cost function ๐ถ(๐) = 5๐1 + 3๐2. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 6 4.1.4. Computational Setup As sampling distribution in IS, we use a lognormal distribution of mean 0.488 G, and standard deviation 0.05 G. The design process discussed in Section 3 is implemented in MATLAB, using โfminconโ to solve all optimization problems. Following the notation established in Section 3, we let ๐, ๐โฒ, and ๐โฒโฒ be 40, 40, and 500 respectively. Computations are carried out on a laptop with 8.00 GB RAM and an Intelยฎ Coreโข i7-4700MQ CPU @2.40GHz. 4.2. Design Results We solve Eq. (3) with IS and n = 40, which takes approximately 25 minutes of fmincon solver time, including repeated evaluations of the dynamical system. We obtain a column stiffness of 57.44 kips/in for Frame 1, and 105.93 kips/in for Frame 2. 4.3. Design Assessment Results Although the obtained design satisfies all constraints in Eq. (3), the sample of only n = 40 results in an approximation of the buffered failure probability. We next make an assessment of the actual buffered failure probability. Using the explicit formula for the case of discrete distributions and a sample size of around 1.5 million, we obtain a 0.9-superquantile of -0.71, which is essentially exact. This shows that the 0.9-superquantile is below zero and, consequently, the buffered failure probability is below the required 0.1. However, such a large sample size is excessively costly in practice and we would like to reach the same conclusion using a much smaller sample size. Using a sample size ๐โฒ = 40 and ๐โฒโฒ = 500, we obtain an approximate 95% confidence interval of an upper bound of the 0.9-superquantile as [โ0.078,0.039] Ideally, we would have liked to have a design with a confidence interval that is entirely below zero. However, one cannot expect to achieve this for small sample sizes due to sampling error. In fact, if the sample sizes ๐โฒ and ๐โฒโฒ had been increased, then such a conclusion would be obtained. 5. CONCLUSIONS We have presented an algorithm for the design and assessment of a system with stochastic parameters. The approach relies on the buffered failure probability to quantify reliability, which facilitates solution of the resulting design optimization problem using standard optimization solvers. Importance sampling provides hope that such design optimization can be carried with a relatively small sample size. In the design of two structural frames subject to ground motion, we find that a feasible design is obtained with as little as 40 samples. Subsequent assessment of that design using 540 samples provides near certainty that the design satisfies the required reliability level. ACKNOWLEDGEMENT This work is supported in part by AFOSR. Additional support for the third author is obtained from DARPA. 6. REFERENCES Asmussen, Soren, & P. W. Glynn (2007). Stochastic simulation algorithms and analysis. New York: Springer. Ditlevsen, O. & H. O. Madsen (1996). Structural Reliability methods. New York: Wiley. Rockafellar, R. T. & J. O. Royset (2010). On buffered failure probability in design and optimization of structures. Reliability Engineering & Systems Safety 95, 499-510. Rockafellar R.T & J.O. Royset (2015). Engineering Decisions under Risk-Averseness. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, to appear. Srinivasan, R. (2002). Importance Sampling. Berlin: Springer.
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International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)
Importance sampling in the evaluation and optimization of buffered failure probability Harajli, Marwan M.; Rockafellar, R. Tyrrell; Royset, Johannes O. Jul 31, 2015
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Title | Importance sampling in the evaluation and optimization of buffered failure probability |
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Harajli, Marwan M. Rockafellar, R. Tyrrell Royset, Johannes O. |
Contributor | International Conference on Applications of Statistics and Probability (12th : 2015 : Vancouver, B.C.) |
Date Issued | 2015-07 |
Description | Engineering design is a process in which a systemโs parameters are selected such that the system meets certain criteria. These criteria vary in nature and may involve such matters as structural strength, implementation cost, architectural considerations, etc. When random variables are part of a system model, an added criterion is usually the failure probability. In this paper, we examine the buffered failure probability as an attractive alternative to the failure probability in design optimization problems. The buffered failure probability is more conservative and possesses properties that make it more convenient to compute and optimize. Since a failure event usually occurs with small probability in structural systems, Monte-Carlo sampling methods require large sample sizes for high accuracy estimates of failure and buffered failure probabilities. We examine importance sampling techniques for efficient evaluation of buffered failure probabilities, and illustrate their use in structural design of two multi-story frames subject to ground motion. We formulate a problem of design optimization as a cost minimization problem subject to buffered failure probability constraints. The problem is solved using importance sampling and a nonlinear optimization algorithm. |
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Conference Paper |
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Text |
Language | eng |
Notes | This collection contains the proceedings of ICASP12, the 12th International Conference on Applications of Statistics and Probability in Civil Engineering held in Vancouver, Canada on July 12-15, 2015. Abstracts were peer-reviewed and authors of accepted abstracts were invited to submit full papers. Also full papers were peer reviewed. The editor for this collection is Professor Terje Haukaas, Department of Civil Engineering, UBC Vancouver. |
Date Available | 2015-05-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0076214 |
URI | http://hdl.handle.net/2429/53377 |
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Non UBC |
Citation | Haukaas, T. (Ed.) (2015). Proceedings of the 12th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP12), Vancouver, Canada, July 12-15. |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
Aggregated Source Repository | DSpace |
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