9th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability
ROBUST MOST PROBABLE POINT SEARCH ALGORITHM FOR DIFFICULT PERFORMANCE FUNCTIONS D.S. Riha, B.H. Thacker, J. Kong, and L. Huyse Southwest Research Institute, San Antonio, TX 78238
[email protected],
[email protected],
[email protected] J.S. Kong Korea University, Seoul, South Korea 136-701
[email protected] Abstract The cornerstone of efficient probabilistic analysis methods is the ability to locate the most probable point (MPP) with as few function evaluations as possible. However, locating the MPP may be difficult or impossible in some situations and no single optimization algorithm is guaranteed to converge due to nonlinearity or discontinuities of the response functions, the existence of multiple MPPs, or the presence of bounded random variables (e.g., uniform). The cause of the MPP search failure has been identified for a number of problems and alternate solutions strategies adopted. A cyclic behavior of the MPP location during the search is observed in a majority of failed cases. The authors propose a method based on the autocorrelation function of the MPP location during the search to detect this cyclic behavior. The failure of the search can be detected in as little as two cycles of the search when this form of failure occurs. Once a potential search failure occurs, a more robust yet computationally intensive MPP search algorithm can be used. Approaches for improving the convergence of the second MPP search by using information from the failed search are also presented. This paper includes several example problems demonstrating the approach.
Introduction Physics-based computational models are now routinely used to predict the behavior of complex systems both in new designs and life extension efforts. In many cases, the models can be developed with high fidelity such that accurate representation of model responses can be obtained in the presence of nonlinear loading and material models, multiple physics and complex geometry. Typically, the high fidelity models come at a large computation cost running into hours and days of computer time even on the fastest computers. Since system performance is directly affected by uncertainties associated with the geometry, material parameters, and loads, the development and application of probabilistic analysis methods suitable for use with complex computational models are needed. Fast probability integration (FPI) methods have been relied upon for computing the uncertain response for complex models since they generally provide sufficient accuracy with an acceptable computational effort. Most FPI methods are based on locating the most probable point (MPP) where the MPP is defined as the minimum distance to the failure surface (limit state) in the transformed probability space. FPI methods use optimization algorithms to locate the MPP, which is then used to estimate the probability in the failure region. Many optimization algorithms are available to locate the MPP including some developed specifically for probabilistic analysis such as the Rackwitz-
Riha, D.S., Thacker, B.H., Kong, J., and Huyse, L.
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Fiessler1 (RF) algorithm. Because derivatives are computed during the search, efficiency of the MPP search algorithm is desirable for computationally intensive models. In general, no optimization algorithm is guaranteed to converge due to many factors including highly nonlinear or discontinuous response functions, multiple MPP’s, or problems containing random variables with bounded distributions (e.g., uniform). Another source of error inherent to FPI methods is the approximation of the probability by the limit state in the transformed space.2 To enable rapid probability integration, first and second order models are usually used, which may not be accurate if the exact limit state is highly nonlinear. However, since the majority of the probability is located at the MPP, first and second order expansions about the MPP are usually adequate. While this error can be important, it is irrelevant if the MPP cannot first be located. This paper addresses the failure of the numerical algorithm during location of the MPP. This research is performed using a modified version of the RF method available in the NESSUS probabilistic analysis software.3 An approach has been developed to identify when the MPP search is failing early in the calculation. Secondly, information from the failed search is used as a starting point for a more robust yet computationally intensive search algorithm. The algorithms are presented followed by an example problem. Most Probable Point Search Failure Detection Algorithm The MPP search is typically performed using an optimization algorithm that locates the minimum distance, β, from the origin to the failure surface in a transformed probability (u) space (Figure 1). Many optimization algorithms are available to perform this task such as the RF method, the modified method of feasible directions (MMFD), sequential linear programming (SLP), and sequential quadratic programming (SQP).4 For locating the MPP, NESSUS uses a modified form of the RF method to provide convergence checks in addition to the standard check for convergence of the safety index, β. The first convergence check requires that the MPP is on the limit state by checking the tolerance of specified performance level. The second convergence check requires that the angle between successive MPPs is within a user-specified tolerance to ensure that the MPP is not locating different MPPs with similar β. The convergence criteria for the modified RF algorithm are defined by the following equations
β i − β i −1 ≤ εβ , β i −1
Zi − Z0 Z0
≤ εZ ,
θ = cos −1 α i ⋅ α i −1 ≤ εθ
(1)
The tolerances are specified by ε, Z0 is the limiting value of the performance defining the limit state such that g=Z-Z0=0, θ is the angle between MPP’s, and αi are the direction cosines to the MPP.
Riha, D.S., Thacker, B.H., Kong, J., and Huyse, L.
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fu(u)
Approximate Limit-State Exact Limit-State g(x) = 0
β u1 Most Probable Point (MPP)
u2
Figure 1. Joint probability density function (jpdf), exact and approximate limit-state, and most probable point (MPP) for two random variables in transformed (u) space
The RF algorithm is a Newton optimization-based method and is not guaranteed to converge. However, when the RF method does converge, it generally converges in far fewer function evaluations than other optimization algorithms such as SQP. Therefore, an approach is needed to identify when the RF method fails, ideally as early in the search process as possible. A second goal is to use all information from the failed search as a starting point for the subsequent, more robust yet computationally intensive search algorithm. The authors’ experience has been that most failures of the RF method provide a characteristic cyclic MPP search pattern. To detect this behavior, an algorithm has been developed based on autocorrelation of the minimum distance, β, for steps in the search. The cyclic behavior is identified by calculating the autocorrelation between each of the N iterations: N −k
rk =
∑ (β i =1
i
− β )( β i + k − β ) (4)
N
∑ (β i =1
i
−β)
where k, the lag between search points (iteration steps), indicates if the search algorithm is continually locating the same points. If the autocorrelation is sufficiently large then failure is detected and an alternate solution approach is required. The initial iteration steps are removed from the autocorrelation computations to eliminate the iteration history that does not exhibit the cyclic behavior. The algorithm computes the autocorrelation using increasing starting points. A large correlation (e.g., greater than 0.8) is required to prevent false failure detection for slowly converging sequences. In addition, correlation values are not computed until at least 20 iterations have been completed. Finally, three successive points of the autocorrelation values must exceed the correlation limit for failure to be identified. In NESSUS, the modified RF method is employed first since it is generally the most efficient. If convergence difficulties are encountered, the method is switched to SQP.
Riha, D.S., Thacker, B.H., Kong, J., and Huyse, L.
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MPP Search Restart Useful information about the MPP location is available from the failed search and can be used as a starting point for a second attempt using a more robust but computationally intensive optimization algorithm. The algorithm uses the first point identified in the iteration step that identifies failure based on the autocorrelation function. This point will precede the cyclic behavior has started and is expected to provide a better initial point than the mean values. Example Problem This example demonstrates the MPP failure detection algorithm. The problem is an analytical function given by
z = x1
3.5
− 100 x2 + 50
(4)
The random variable x1 follows a uniform distribution with lower bound 0 and upper bound 100. x2 follows a beta distribution with β=0.5. The problem is defined in NESSUS and solved using the modified RF method to locate the MPP. The modified RF method in NESSUS fails to converge for a limit state value of Z0=4.52. The history of β for the search is shown in Figure 2 and clearly indicates the cyclic search behavior. The autocorrelation for different lag values (number between iteration steps) is also plotted in Figure 2. The correlation is near 1 for a lag of 11 indicating that the MPP search algorithm is locating the same points during the search while not converging. 12
1
10
0.8 0.6 Autocorrelation
Beta
8
6
4
0.4 0.2 0
2
-0.2 -0.4
0 0
20
40
60
80
100
iteration
0
5
10
15
20
25
30
lag
Figure 2. The cyclic behavior is identified for β in this problem (left). The autocorrelation identifies the lag or repeat of points during the search (right). The cycle repeats every 11 steps.
The number of function evaluations to locate the MPP for 13 levels are listed in Table 1 for SQP and the RF/SQP approach after failure detection. The SQP method requires 2032 function evaluations for all levels. The failure detection of RF and subsequent SQP solution required 979 function evaluations. The failure detection algorithm identified
Riha, D.S., Thacker, B.H., Kong, J., and Huyse, L.
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failure for the first three levels. Once failure was detected, the SQP algorithm was used to locate the MPP. For the first level, an additional 39 function evaluations using SQP were required to locate the MPP using the starting point from the failure detection algorithm. The lower number of function evaluations as compared to SQP alone (133) shows the usefulness of the failed solution points in restarting the search. Minimizing the number of function evaluations becomes more important as the computational cost of each function evaluation increases. The limit states in the transformed space for different performance levels are shown in Figure 3 along with the MPPs computed using the RF algorithm with failure detection. The limit states corresponding to the left tail of the distribution of Z are located in the upper left quadrant of the figure. The steep gradient between limits states is evident in this region and is the most likely cause of failure of the RF method. In addition, the RF algorithm converged to a slightly incorrect MPP for level 4. The error is attributed to inaccurate gradient approximations of the response function for the RF method. This level was examined using the exact gradients for the response function and resulted in cyclic behavior and subsequent solution using SQP.
RF with Failure Detection Level
Z0
RF
SQP
Total Failure Detection
SQP
1
2.55
120
39
159
Cyclic behavior detected iteration 40, lag is 4
133
2
13.72
132
76
208
Cyclic behavior detected iteration 20, lag is 5
117
3
25.26
162
81
243
Cyclic behavior detected iteration 54, lag is 7
190
4
36.71
105
0
105
119
5
48.22
114
0
114
102
6
3201.27
33
0
33
71
7
883980
6
0
6
28
8
6917134
12
0
12
22
9
9654818
12
0
12
39
10
9965170
12
0
12
60
11
9996544
12
0
12
388
12
9999699
12
0
12
726
13
9999996
51
0
51
37
Totals
783 196 979 Table 1. Function evaluations to locate the MPP.
Riha, D.S., Thacker, B.H., Kong, J., and Huyse, L.
2032
5
u2
4
2
0
u1
-2
-4 -4
-2
0
2
4
Figure 3. Limit state functions in the transformed space indicate a steep gradient. The MPP for each level is verified graphically.
Conclusions
A most probable point search failure detection algorithm has been developed to identify when the modified Rackwitz-Feissler (RF) algorithm fails to locate the MPP early in the search process. The points from the failed search can be used to start a second search and has been demonstrated to improve efficiency. The efficiency of the combined failure detection and second search approach will be problem specific and governed by the nonlinearity of the limit state and errors introduced in the transformations to the standard normal space and gradient approximations. Acknowledgements This effort was supported by the Southwest Research Institute Advisory Council for Research. The authors would like to acknowledge the efforts of Jason Pleming for his assistance in performing several of these analyses. References 1
Madsen, H. O., Krenk, S., and Lind, N. C., Methods of Structural Safety, Prentice-Hall, Inc., New Jersey, 1986.
2
B.H. Thacker, D.S. Riha, H.R. Millwater, M.P. Enright, 2001, “Errors and Uncertainties in Probabilistic Engineering Analysis,” 42nd Structures, Structural Dynamics, and Materials Conference, AIAA 20011239, Seattle, WA.
3
NESSUS User’s manual, Version 8, Southwest Research Institute, 2004.
4
Faravelli, L., “Response Surface Approach for Reliability Analysis,” J. of Eng. Mech., 115(12), 1989.
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