×

A flexible uncertainty propagation framework for general multiphysics systems. (English) Zbl 1348.60101

Summary: We present an efficient “module-based” uncertainty propagation framework for stochastic multiphysics systems. Our proposed framework facilitates modeling flexibility and independence within each module of a stochastic multiphysics solver, and extends the framework, previously introduced by the authors, to general (nonlinear) multiphysics systems via generic restriction and prolongation operators, which transform between the global and local polynomial chaos representations of the input/output data. Moreover, these operators preserve the convergence rate of monolithic generalized-polynomial-chaos-based methods, and lower the rate of growth of the overall computational costs w.r.t. the global dimension. Following a discussion of these mathematical and algorithmic details, we implement our framework on a test problem, and demonstrate its superior computational performance over an implementation of the standard monolithic framework.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
65C50 Other computational problems in probability (MSC2010)

Software:

LSQR; COMSOL; CRAIG; OPQ
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. A. Felippa and K. C. Park, {\it Model Based Partitioned Simulation of Coupled Systems}, Springer, Vienna, 2009, pp. 171-216.
[2] K. C. Park and C. A. Felippa, {\it Partitioned analysis of coupled systems}, Comput. Methods Transient Anal., 1 (1983), pp. 157-219. · Zbl 0546.73063
[3] J. U. Schluter, X. Wu, E. van der Weide, S. Hahn, J. J. Alonso, and H. Pitsch, {\it Multi-code Simulations: A Generalized Coupling Approach}. AIAA paper, 2005-4997, 2005.
[4] G. Casella and C. Robert, {\it Monte Carlo Statistical Methods}. Springer, New York, 1999. · Zbl 0935.62005
[5] D. Xiu and G. E. Karniadakis, {\it The Wiener-Askey polynomial chaos for stochastic differential equations}, SIAM J. Sci. Comput., 24 (2002), pp. 619-644. · Zbl 1014.65004
[6] R. Ghanem and P. D. Spanos, {\it Polynomial chaos in stochastic finite elements}, J. Appl. Mech., 57 (1990), pp. 197-202. · Zbl 0729.73290
[7] S. Hosder, R. W. Walters, and R. Perez, {\it A Non-Intrusive Polynomial Chaos Method for Uncertainty Propagation in CFD Simulations}, AIAA paper, 2006-891 (2006).
[8] I. Babuska, R. Tempone, and G. E. Zouraris, {\it Galerkin finite element approximations of stochastic elliptic partial differential equations}, SIAM J. Numer. Anal., 42 (2004), pp. 800-825. · Zbl 1080.65003
[9] X. Chen, B. Ng, Y. Sun, and C. Tong, {\it A flexible uncertainty quantification method for linearly coupled multi-physics systems}, J. Comput. Phys., 248 (2013), pp. 383-401. · Zbl 1349.60112
[10] J. Degroote, K. J. Bathe, and J. Vierendeels, {\it Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction}, Comput. Struct., 87 (2009), pp. 793-801.
[11] R. W. Pryor, {\it Multiphysics Modeling Using COMSOL: A First Principles Approach}, Jones & Bartlett Learning, Sudbury, MA, 2011.
[12] H. G. Matthies, R. Niekamp, and J. Steindorf, {\it Algorithms for strong coupling procedures}, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 2028-2049. · Zbl 1142.74050
[13] A. Ben-Israel, {\it A Newton-Raphson method for the solution of systems of equations}, J. Math. Anal. Appl., 15 (1966), pp. 243-252. · Zbl 0139.10301
[14] J. J. More, {\it The Levenberg-Marquardt algorithm: Implementation and theory}, in Numerical Analysis, Springer, Berlin, 1978, pp. 105-116. · Zbl 0372.65022
[15] M. J. D. Powell, {\it Approximation Theory and Methods}, Cambridge University Press, Cambridge, 1981. · Zbl 0453.41001
[16] W. Gautschi, {\it Orthogonal Polynomials: Computation and Approximation}, Oxford University Press, Oxford, 2004. · Zbl 1130.42300
[17] G. H. Golub and J. H. Welsch, {\it Calculation of Gauss quadrature rules}, Math. Comp., 23 (1969), pp. 221-230. · Zbl 0179.21901
[18] D. Xiu and G. E. Karniadakis, {\it Modeling uncertainty in flow simulations via generalized polynomial chaos}, J. Comput. Phys., 187 (2003), pp. 137-167. · Zbl 1047.76111
[19] R. H. Cameron and W. T. Martin, {\it The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals}, Ann. of Math. (2), 48 (1947), pp. 385-392. · Zbl 0029.14302
[20] O. Roderick, M. Anitescu, and P. Fischer, {\it Polynomial regression approaches using derivative information for uncertainty quantification}, Nucl. Sci. Eng., 164 (2010), pp. 122-139.
[21] A. Doostan and G. Iaccarino, {\it A least-squares approximation of partial differential equations with high-dimensional random inputs}, J. Comput. Phys., 228 (2009), pp. 4332-4345. · Zbl 1167.65322
[22] E. Phipps, H. C. Edwards, J. Hu, and J. T. Ostien, {\it Exploring emerging manycore architectures for uncertainty quantification through embedded stochastic Galerkin methods}, Int. J. Comput. Math., 91 (2014), pp. 707-729. · Zbl 1297.65011
[23] T. Gerstner and M. Griebel, {\it Numerical integration using sparse grids}, Numer. Algorithms, 18 (1998), pp. 209-232. · Zbl 0921.65022
[24] P. G. Constantine, A. Doostan, and G. Iaccarino, {\it A hybrid collocation/Galerkin scheme for convective heat transfer problems with stochastic boundary conditions}, Internat. J. Numer. Methods Engrg., 80 (2009), pp. 868-880. · Zbl 1176.76107
[25] J. Massaguer and J. Zahn, {\it Cellular convection in a stratified atmosphere}, Astron. Astrophys., 87 (1980), pp. 315-327.
[26] L. Kantha and C. Clayson, {\it Numerical Models of Oceans and Oceanic Processes}, Academic Press, San Diego, 2000. · Zbl 0989.76002
[27] D. De Vahl, {\it Laminar natural convection in an enclosed rectangular cavity}. Int. J. Heat Mass Transf., 11 (1968), pp. 1675-1693.
[28] D. LeVeque, {\it Finite Volume Methods for Hyperbolic Problems}. Cambridge University Press, Cambridge, 2002. · Zbl 1010.65040
[29] P. Roache, {\it Code verification by the method of manufactured solutions}. J. Fluids Engrg., 124 (2002), pp. 4-10.
[30] M. Loeve, {\it Probability Theory. Foundations. Random Sequences}, Van Nostrand, New York, 1955. · Zbl 0066.10903
[31] C. C. Paige and M. A. Saunders, {\it LSQR: An algorithm for sparse linear equations and sparse least squares}, ACM Trans. Math. Software, 8 (1982), pp. 43-71. · Zbl 0478.65016
[32] C. E. Powell and H. C. Elman, {\it Block-diagonal preconditioning for spectral stochastic finite-element systems}, IMA J. Numer. Anal., 29 (2009), pp. 350-375. · Zbl 1169.65007
[33] O. P. Le Maitre, O. M. Knio, H. N. Najm, and R. G. Ghanem, {\it Uncertainty propagation using Wiener-Haar expansions}, J. Comput. Phys., 197 (2004), pp. 28-57. · Zbl 1052.65114
[34] X. Wan and G. E. Karniadakis, {\it Multi-element generalized polynomial chaos for arbitrary probability measures}, SIAM J. Sci. Comput., 28 (2006), pp. 901-928. · Zbl 1128.65009
[35] X. Chen, J. Connors, C. H. Tong, and Y. Sun, {\it A Flexible Method to Calculate the Distributions of Discretization Errors in Operator-Split Codes with Stochastic Noise in Problem Data}, Technical report LLNL-TR-649085, Lawrence Livermore National Laboratory, Livermore, CA, 2014.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.