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Sparse Representations in Stochastic Mechanics

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Computational Methods in Stochastic Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 22))

Abstract

Computational approaches to systems involving random fields or stochastic processes have to discretise these fields or processes. This produces– when compared to the deterministic case – many variables in the computation, resulting in a very high-dimensional problem. Based on the conviction that the essential stochastic properties of the system are close to some – albeit unknown – lower dimensional manifold, one may try to approximate the response of the system by a data-sparse representation.

The basis for this sparse representation has to be found in the course of the computation. One first approach is to exploit the natural tensor product structure between basis vectors describing the physical/deterministic behaviour and a basis describing the stochastic response. There are two steps involved here: one is to find a good basis for the physical description, and the other to find/compute a good basis for the stochastic part. One well-known example is the Karhunen-Loève expansion, resulting from the eigenvalue analysis of the covariance. One problem is of course that the covariance of the response is not known beforehand. We will discuss on how to approximate the basis along with the solution.

The singular value decomposition, which is very closely related to the Karhunen-Loève expansion, is optimal in that it uses the minimal number of dyadic products. Furthermore, the stochastic part of this product is itself again naturally an element of a tensor product with potentially many factors, containing functions of just one random variable. This fact can be additionally exploited, and also be used to obtain an adaptive approximation of the stochastic part.

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References

  1. Acharjee, S., Zabaras, N.: A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs. Int. J. Numer. Meth. Eng. 66, 1934–1954 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Acharjee, S., Zabaras, N.: A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes. Comput. Struct. 85, 244–254 (2007)

    Article  Google Scholar 

  3. Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer, Berlin 2007

    MATH  Google Scholar 

  4. Babuška, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42, 800–825 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Babuška, I., Tempone, R., Zouraris, G.E.: Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comp. Meth. Appl. Mech. Eng. 194, 1251–1294 (2005)

    Article  MATH  Google Scholar 

  6. Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45, 1005–1034 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Baroth, J., Bodé, L., Bressolette, Ph., Fogli, M.: SFE method using Hermite polynomials: An approach for solving nonlinear mechanical problems with uncertain parameters. Comp. Meth. Appl. Mech. Eng. 195, 6479–6501 (2006)

    Article  MATH  Google Scholar 

  8. Baroth, J., Bressolette, Ph., Chauvière, C., Fogli, M.: An efficient SFE method using Lagrange polynomials: Application to nonlinear mechanical problems with uncertain parameters. Comp. Meth. Appl. Mech. Eng. 196, 4419–4429 (2007)

    Article  MATH  Google Scholar 

  9. Blatman, G., Sudret, B.: Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. C. R. Mécanique 336, 518–523 (2008)

    Article  MATH  Google Scholar 

  10. Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numerica 7, 1–49 (1998)

    Article  MathSciNet  Google Scholar 

  11. Christakos, G.: Random Field Models in Earth Sciences. Academic Press, SanDiego (1992)

    Google Scholar 

  12. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  13. Courant, R. Hilbert, D.: Methods of Mathematical Physics. Wiley, Chichester (1989)

    Google Scholar 

  14. Doostan, A., Ghanem, R.G., Red-Horse, J.: Stochastic model reduction for chaos representations. Comp. Meth. Appl. Mech. Eng. 196, 3951–3966 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Frauenfelder, Ph., Schwab, Chr., Todor, R.A.: Finite elements for elliptic problems with stochastic coefficients. Comp. Meth. Appl. Mech. Eng. 194, 205–228 (2005)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Ghanem, R.: Stochastic finite elements for heterogeneous media with multiple random non-Gaussian properties. ASCE J. Eng. Mech. 125, 24–40 (1999)

    Google Scholar 

  18. Ghanem. R., Kruger, R.: Numerical solution of spectral stochastic finite element systems. Comp. Methods Appl. Mech. Eng. 129, 289–303 (1999)

    Google Scholar 

  19. Ghanem, R.G., Pellissetti, M.F.: Adaptive data refinement in the spectral stochastic finite element method. Commun. Numer. Meth. Eng. 18, 141–151 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hackbusch, W., Khoromskij, B.N., Tyrtyshnikov, E.N.: Approximate iterations for structured matrices. Numer. Math. 109, 365–383 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Jardak, M., Su, C.-H., Karniadakis, G.E.: Spectral polynomial chaos solutions of the stochastic advection equation. SIAM J. Sci. Comput. 17, 319–338 (2002)

    MATH  MathSciNet  Google Scholar 

  22. Keese, A.: A review of recent developments in the numerical solution of stochastic PDEs (stochastic finite elements). Informatikbericht 2003-6, Institute of Scientific Computing, Department of Mathematics and Computer Science, Technische Universität Braunschweig, Brunswick (2003) http://opus.tu-bs.de/opus/volltexte/2003/504/

  23. Keese, A., Matthies, H.G.: Parallel solution of stochastic PDEs. Proc. Appl. Math. Mech. 2, 485–486 (2003)

    Article  MATH  Google Scholar 

  24. Keese, A., Matthies, H.G.: Numerical methods and Smolyak quadrature for nonlinear stochastic partial differential equations. Informatikbericht 2003-5, Institute of Scientific Computing, Department of Mathematics and Computer Science, Technische Universität Braunschweig, Brunswick (2003) http://opus.tu-bs.de/opus/volltexte/2003/471/

  25. Keese, A., Matthies, H.G.: Hierarchical parallelisation for the solution of stochastic finite element equations. Comput. Struct. 83, 1033–1047 (2005)

    Article  MathSciNet  Google Scholar 

  26. Kolda, T.G., O‘Leary, D.P., Nazareth, L.: BFGS with update skipping and varying memory. SIAM J. Optim. 8, 1060–1083 (1998)

    Google Scholar 

  27. Krée, P., Soize, C.: Mathematics of Random Phenomena. Reidel, D. Dordrecht (1986)

    Google Scholar 

  28. Le Maître, O.P., Najm, H.N., Ghanem, R.G., Knio, O.M.: Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comp. Phys. 197, 502–531 (2004)

    Article  MATH  Google Scholar 

  29. Matthies, H.G., Brenner, C.E., Bucher, C.G., GuedesSoares, C.: Uncertainties in probabilistic numerical analysis of structures and solids – stochastic finite elements. Struct. Saf. 19, 283–336 (1997)

    Google Scholar 

  30. Matthies, H.G., Bucher, C.G.: Finite elements for stochastic media problems. Comp. Meth. Appl. Mech. Eng. 168, 3–17 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Matthies, H.G., Keese, A.: Multilevel solvers for the analysis of stochastic systems. In: Bathe, K.-J. (ed.) Computational Fluid and Solid Mechanics – Proceedings of the 1st MIT Conference pp. 1620–1622. Elsevier, Amsterdam (2001)

    Google Scholar 

  32. Matthies, H.G., Keese, A.: Fast solvers for the white noise analysis of stochastic systems. Proc. Appl. Math. Mech. 1, 456–457 (2002)

    Article  MATH  Google Scholar 

  33. Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comp. Meth. Appl. Mech. Eng. 194, 1295–1331 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Matthies, H.G.: Computational aspects of probability in non-linear mechanics. In: Ibrahimbegović A., Brank, B. (eds.) Engineering Structures under Extreme Conditions. Multi-Physics and Multi-scale Computer Models in Non-linear Analysis and Optimal Design of Engineering Structures under Extreme Conditions. NATO Science Series III: Computer and System Sciences Vol.194, IOS Press, Amsterdam (2005)

    Google Scholar 

  35. Matthies, H.G.: Quantifying uncertainty: modern computational representation of probability and applications. In: Ibrahimbegović, A. Kožar, I. (eds.) Extreme Man-Made and Natural Hazards in Dynamics of Structures. In: Proceedings of the NATO Advanced Research Workshop PST.ARW981641. ISBN 953-6953-12-9. Sveučilišna knjižica, Rijeka (2006)

    Google Scholar 

  36. Matthies, H.G.: Stochastic finite elements: Computational approaches to stochastic partial differential equations. Z. Angew. Math. Mech. (ZAMM) 88, 849–873 (2008)

    Google Scholar 

  37. Nobile, F., Tempone, R., Webster, C.G.: Sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309–2345 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  38. Nouy, A.: A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comp. Meth. Appl. Mech. Eng. 196, 4521–4537 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Novak, E. Ritter, K.: The curse of dimension and a universal method for numerical integration. In: Nürnberger, G., Schmidt, J.W., Walz, G. (eds.) Multivariate Approximation and Splines, ISNM pp.177–188. Birkhäuser, Basel (1997)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  41. Papadrakakis, M., Papadopoulos, V.: Robust and efficient methods for stochastic finite element analysis using Monte Carlo simulation. Comp. Meth. Appl. Mech. Eng. 134, 325–340 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  42. Pellissetti, M. Ghanem, R.: Iterative solution of systems of linear equations arising in the context of stochastic finite elements. Adv. Eng. Softw. 31, 607–616 (2000)

    Google Scholar 

  43. Petras, K.: Fast calculation of coefficients in the Smolyak algorithm. Numer. Algorithms 26, 93–109 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  44. Roman, L.J., Sarkis, M.: Stochastic Galerkin method for elliptic SPDEs: A white noise approach. Discrete Continuous Dyn. Syst.–Series B (DCDS-B) 6, 941–955 (2006)

    Google Scholar 

  45. Schuëller, G.I.: A state-of-the-art report on computational stochastic mechanics. Prob. Eng. Mech. 14, 197–321 (1997)

    Article  Google Scholar 

  46. Schuëller, G.I.: Recent developments in structural computational stochastic mechanics. In: Topping, B.H.V. (ed.) Computational Mechanics for the Twenty-First Century pp. 281–310. Saxe-Coburg Publications, Edinburgh (2000)

    Google Scholar 

  47. Schuëller, G.I., Spanos, P.D. (eds.): Monte Carlo Simulation. Balkema, Rotterdam (2001)

    Google Scholar 

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

    Google Scholar 

  49. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Wellesley-Cambridge Press, Wellesley (1988)

    Google Scholar 

  50. Sudret, B., DerKiureghian, A.: Stochastic finite element methods and reliability. A state-of-the-art report. Report UCB/SEMM-2000/08, Department of Civil & Environmental Engineering, University of California, Berkeley (2000)

    Google Scholar 

  51. Wan, X., Karniadakis, G.E.: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comp. Phys. 209, 617–642 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  53. Xiu, D. Karniadakis, G.E.: Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comp. Meth. Appl. Mech. Eng. 191, 4927–4948 (2002)

    Google Scholar 

  54. Xu, X.F.: A multiscale stochastic finite element method on elliptic problems involving uncertainties. Comp. Meth. Appl. Mech. Eng. 196, 2723–2736 (2007)

    Article  Google Scholar 

  55. Zander, E., Matthies, H.G.: Tensor product methods for stochastic problems. Proc. Appl. Math. Mech. (PAMM) 7, 2040067–2040068 (2008)

    Google Scholar 

  56. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, 5th edn. Butterwort-Heinemann, Oxford (2000)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Scientific Computing, TU Braunschweig, D-38092, Braunschweig, Germany

    Hermann G. Matthies & Elmar Zander

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  1. Hermann G. Matthies
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  2. Elmar Zander
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Correspondence to Elmar Zander .

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Editors and Affiliations

  1. Inst. Structural Analysis &, Seismic Research, National Technical Univ. Athen, Iroon Polytechniou Str. 9, Athens, 157 80, Greece

    Manolis Papadrakakis

  2. , Intitute of Structural Analysis and Seis, National Technical University of Athens, Iroon Polytechniou, Zografou Campus 9, Athens, 15780, Greece

    George Stefanou

  3. , Institute of Structural Analysis and Se, National Technical University of Athens, Iroon Polytechniou, Zografou Campus 9, Athens, 15780, Greece

    Vissarion Papadopoulos

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Matthies, H.G., Zander, E. (2011). Sparse Representations in Stochastic Mechanics. In: Papadrakakis, M., Stefanou, G., Papadopoulos, V. (eds) Computational Methods in Stochastic Dynamics. Computational Methods in Applied Sciences, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9987-7_13

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  • DOI: https://doi.org/10.1007/978-90-481-9987-7_13

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