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A new sparse grid based method for uncertainty propagation

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Abstract

Current methods for uncertainty propagation suffer from their limitations in providing accurate and efficient solutions to high-dimension problems with interactions of random variables. The sparse grid technique, originally invented for numerical integration and interpolation, is extended to uncertainty propagation in this work to overcome the difficulty. The concept of Sparse Grid Numerical Integration (SGNI) is extended for estimating the first two moments of performance in robust design, while the Sparse Grid Interpolation (SGI) is employed to determine failure probability by interpolating the limit-state function at the Most Probable Point (MPP) in reliability analysis. The proposed methods are demonstrated by high-dimension mathematical examples with notable variate interactions and one multidisciplinary rocket design problem. Results show that the use of sparse grid methods works better than popular counterparts. Furthermore, the automatic sampling, special interpolation process, and dimension-adaptivity feature make SGI more flexible and efficient than using the uniform sample based metamodeling techniques.

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References

  • Barthelmann V, Novak E, Ritter K (2000) High dimensional polynomial interpolation on sparse grids. Adv Comput Math 12:273–288

    Article  MATH  MathSciNet  Google Scholar 

  • Du X, Chen W (2000) Towards a better understanding of modeling feasibility robustness in engineering design. J Mech Des 122:385–394

    Article  Google Scholar 

  • Du X, Chen W (2001) A most probable point-based method for efficient uncertainty analysis. J Design Manuf Autom 4:47–66

    Article  Google Scholar 

  • Du X, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. ASME J Mech Des 126(2):225–233

    Article  Google Scholar 

  • Du X, Sudjianto A, Chen W (2004) An integrated framework for optimization under uncertainty using inverse reliability strategy. J Mech Des 126:562–570

    Article  Google Scholar 

  • Gerstner T, Griebel M (1998) Numerical integration using sparse grids. Number Algorithms 18:209–232

    Article  MATH  MathSciNet  Google Scholar 

  • Gerstner T, Griebel M (2003) Dimension-adaptive tensor-product quadrature. Computing 71:65–87

    Article  MATH  MathSciNet  Google Scholar 

  • Irfan K (2005) Application of Kriging method to structural reliability problems. Struct Saf 27:133–151

    Article  Google Scholar 

  • Jin R, Chen W (2005) An efficient algorithm for constructing optimal design of computer experiment. J Stat Plan Inference 134:268–287

    Article  MATH  MathSciNet  Google Scholar 

  • Klimke A (2006a) Sparse grid interpolation toolbox—user’s guide. IANS report 2006/001, University of Stuttgart

  • Klimke A (2006b) Uncertainty modeling using fuzzy arithmetic and sparse grids. PhD dissertation, Institut für Angewandte Analysis und Numerische Simulation Universität Stuttgart

  • Klimke A, Wohlmuth B (2005a) Piecewise multilinear hierarchical sparse grid interpolation in MATLAB. ACM Trans Math Softw 31:561–579

    Article  MATH  MathSciNet  Google Scholar 

  • Klimke A, Wohlmuth B (2005b) Computing expensive multivariate functions of fuzzy numbers using sparse grids. Fuzzy Sets Syst 154:432–453

    Article  MATH  MathSciNet  Google Scholar 

  • Lee SH, Chen W (2008) A comparative study of uncertainty propagation methods for black-box type functions. Struct Multidiscipl Optim 37:239–253

    Article  MathSciNet  Google Scholar 

  • Lee I, Choi KK, Du L (2006) Alternative methods for reliability-based robust design optimization including dimension reduction method. In: 32nd ASME design automation conference, Philadelphia, PA, 10–13 September 2006

  • Lee I, Choi KK, Du L, Gorsich D (2008a) Dimension reduction method for reliability-based robust design optimization. Comput Struct 86:1550–1562

    Article  Google Scholar 

  • Lee I, Choi KK, Du L, Gorsich D (2008b) Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems. Comput Methods Appl Math 198:14–27

    Google Scholar 

  • Nie J, Ellingwood BR (2000) Directional methods for structural reliability analysis. Struct Saf 22:233–249

    Article  Google Scholar 

  • Novak E, Ritter K (1996) High dimensional integration of smooth functions over cubes. J Numer Math 75:79–97

    Article  MATH  MathSciNet  Google Scholar 

  • Novak E, Ritter K (1999) Simple cubature formulas with high polynomial exactness. Constr Approx 15:499–522

    Article  MATH  MathSciNet  Google Scholar 

  • Rackwitz R (2001) Reliability analysis—a review and some perspectives. Struct Saf 23:365–395

    Article  Google Scholar 

  • Rahman S, Wei D (2006) A univariate approximation at most probable point for higher-order reliability analysis. Int J Solids Struct 43:2820–2839

    Article  MATH  Google Scholar 

  • Rahman S, Xu H (2004) A univariate dimension–reduction method for multidimensional integration in stochastic mechanics. Probab Eng Mech 19:393–408

    Article  Google Scholar 

  • Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472

    Article  MATH  MathSciNet  Google Scholar 

  • Seo HS, Kwak BM (2002) Efficient statistical tolerance analysis for general distributions using three-point information. Int J Prod Res 40:931–944

    Article  MATH  Google Scholar 

  • Smolyak SA (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov Math, Dokl 4:240–243

    Google Scholar 

  • Tao J, Zeng X, Cai W, Su Y, Zhou D, Chiang C (2007) Stochastic sparse-grid collocation algorithm (SSCA) for periodic steady-state analysis of nonlinear system with process variations. In: Proc ASPDAC 5A-5, Jan 2007, pp 474–479

  • Wang G, Wang L, Shan Q (2005) Reliability assessment using discriminative sampling and metamodeling. SAE Transact 114:291–300

    Google Scholar 

  • Wu Y (1994) Computational methods for efficient structural reliability and reliability sensitivity analysis. AIAA J 32:1717–1723

    Article  MATH  Google Scholar 

  • Xiu D, Hesthaven J (2005) High order collocation method for differential equations with random inputs. SIAM J Sci Comput 27:1118–1139

    Article  MATH  MathSciNet  Google Scholar 

  • Xu H, Rahman S (2004) A generalized dimension–reduction method for multidimensional integration in stochastic mechanics. Int J Numer Methods Eng 61:1992–2019

    Article  MATH  Google Scholar 

  • Xu H, Rahman S (2005) Decomposition methods for structural reliability analysis. Probab Eng Mech 20:239–250

    Article  Google Scholar 

  • Youn B, Xi Z, Wells L, Wang P (2006) Enhanced dimension reduction (eDR) method for sensitivity-free uncertainty quantification. In: 11th AIAA/ISSMO multidisciplinary analysis and optimization conference, Portsmouth, Virginia, 6–8 September 2006

  • Youn B, Xi Z, Wang P (2008) Eigenvector dimension reduction (EDR) method for sensitivity-free probability analysis. Struct Multidiscipl Optim 37:13–28

    Article  MathSciNet  Google Scholar 

  • Zhao L, Yang S (2008) Application of MDO to conceptual design of WAF rockets. 7th ASMO-UK/ISSMO International Conference on Engineering Design Optimization, Bath, UK, 7–8 July, 2008

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Authors and Affiliations

  1. School of Aerospace Science and Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Fenfen Xiong & Shuxing Yang

  2. Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd Tech B224, Evanston, IL, 60208, USA

    Fenfen Xiong & Wei Chen

  3. Theoretical & Applied Mechanics, Northwestern University, Evanston, IL, 60208, USA

    Steven Greene

  4. Bank of America, Phoenix, AZ, 85004, USA

    Ying Xiong

Authors
  1. Fenfen Xiong
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  2. Steven Greene
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  3. Wei Chen
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  4. Ying Xiong
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  5. Shuxing Yang
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Correspondence to Wei Chen.

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Xiong, F., Greene, S., Chen, W. et al. A new sparse grid based method for uncertainty propagation. Struct Multidisc Optim 41, 335–349 (2010). https://doi.org/10.1007/s00158-009-0441-x

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  • DOI: https://doi.org/10.1007/s00158-009-0441-x

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