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Stochastic Collocation on Unstructured Multivariate Meshes

  • Akil Narayan (a1) and Tao Zhou (a2)
Abstract
Abstract

Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification (UQ) community. Techniques for least-squares regularization, compressive sampling recovery, and interpolatory reconstruction are becoming standard tools used in a variety of applications. Selection of a collocation mesh is frequently a challenge, but methods that construct geometrically unstructured collocation meshes have shown great potential due to attractive theoretical properties and direct, simple generation and implementation. We investigate properties of these meshes, presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.

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Corresponding author
*Corresponding author. Email addresses: akil@sci.utah.edu (A. Narayan), tzhou@lsec.cc.ac.cn (T. Zhou)
References
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[1]Babuska I., Nobile F., and Tempone R.. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Review, 52(2):317355, January 2010.
[2]Barthelmann V., Novak E., and Ritter K.. High dimensional polynomial interpolation on sparse grids. Advances in Computational Mathematics, 12(4):273288, March 2000.
[3]Bellman R. E.. Adaptive control processes: a guided tour. Princeton University Press, 1961.
[4]Bendito E., Carmona A., Encinas A.M., and Gesto J.M.. Estimation of Fekete points. Journal of Computational Physics, 225(2):23542376, August 2007.
[5]Berman R., Boucksom S., and Nyström D.. Fekete points and convergence towards equilibrium measures on complex manifolds. Acta Mathematica, 207(1):127, 2011.
[6]Bieri M. and Schwab C.. Sparse high order FEM for elliptic sPDEs. Computer Methods in Applied Mechanics and Engineering, 198(13-14):11491170, March 2009.
[7]Binev P., Cohen A., Dahmen W., and DeVore R.. Universal algorithms for learning theory. ii. piecewise polynomial functions. Constr. Approx., 2(26):127152, 2007.
[8]Binev P., Cohen A., Dahmen W., Devore R. A., and N V.. Temlyakov. Universal algorithms for learning theory part I: Piecewise constant functions. Journal of Machine Learning Research, 6:12971321, 2005.
[9]Blatman G. and Sudret B.. Adaptive sparse polynomial chaos expansion based on least angle regression. Journal of Computational Physics, 230(6):23452367, March 2011.
[10]Bloom T., Bos L., Christensen C., and Levenberg N.. Polynomial interpolation of holomorphic functions in inline-graphic and inline-graphicn. Rocky Mountain Journal of Mathematics, 22(2):441470, June 1992.
[11]Bloom T. and Levenberg N.. Weighted pluripotential theory in $c^n$. American Journal of Mathematics, 125(1):57103, February 2003.
[12]Bos L.. Near optimal location of points for Lagrange interpolation in several variables. PhD thesis, University of Toronto, 1981.
[13]Bos L., Caliari M., Marchi S. De, Vianello M., and Xu Y.. Bivariate lagrange interpolation at the padua points: The generating curve approach. Journal of Approximation Theory, 143(1):1525, November 2006.
[14]Bos L., Marchi S. De, Sommariva A., and Vianello M.. Computing multivariate fekete and leja points by numerical linear algebra. SIAM Journal on Numerical Analysis, 48(5):1984, 2010.
[15]Bos L. P. and Levenberg N.. On the calculation of approximate fekete points: the univariate case. Electronic Transactions on Numerical Analysis, 30:377397, 2008.
[16]Bourgain J., Dilworth S., Ford K., Konyagin S., and Kutzarova D.. Explicit constructions of RIP matrices and related problems. Duke Mathematical Journal, 159(1):145185, July 2011.
[17]Breidt J., Butler T., and Estep D.. A measure-theoretic computational method for inverse sensitivity problems i: Method and analysis. SIAM Journal on Numerical Analysis, 49(5):18361859, 2011.
[18]Brutman L.. On the lebesgue function for polynomial interpolation. SIAM Journal on Numerical Analysis, 15(4):694, 1978.
[19]Bungartz H.-J. and Griebel M.. Sparse grids. Acta Numerica, 13(-1):147269, 2004.
[20]Burkardt J. and Eldred M.. Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. In 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition. American Institute of Aeronautics and Astronautics, 2009.
[21]Burkardt J., Gunzburger M., and Lee H.-C.. POD and CVT-based reduced-ordermoeling of navier-stokes flows. Computer Methods in Applied Mechanics and Engineering, 196(1-3):337355, December 2006.
[22]Caliari M., Marchi S. De, and Vianello M.. Bivariate polynomial interpolation on the square at new nodal sets. Applied Mathematics and Computation, 165(2):261274, June 2005.
[23]Candès E., Romberg J., and Tao T.. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 8(59):12071223, 2006.
[24]Candès E. and Tao T.. Stable signal recovery from incomplete and inaccurate measurements. IEEE Trans. Inform. Theory, 12(51):42034215, 2005.
[25]Candes E.J. and Tao T.. Decoding by linear programming. IEEE Transactions on Information Theory, 51(12):42034215, December 2005.
[26]Candes E.J. and Tao T.. Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Transactions on Information Theory, 52(12):54065425, December 2006.
[27]Chkifa A., Cohen A., Migliorati G., Nobile F., and Tempone R.. Discrete least squares polynomial approximation with random evaluations-application to parametric and stochastic elliptic pdes. EPFL, MATHICSE Technical Report, 35/2013.
[28]Choi S.-K., Grandhi R. V., Canfield R. A., and Pettit C.L.. Polynomial chaos expansion with Latin hypercube sampling for estimating response variability. AIAA Journal, 42(6):11911198, 2004.
[29]Cohen A., Dahmen W., and DeVore R.A.. Compressed sensing and best k-term approximation. J. Amer. Math. Soc., 22(1):211231, 2009.
[30]Cohen A., Davenport M. A., and Leviatan D.. On the stability and accuracy of least squares approximations. Foundations of Computational Mathematics, 13(5):819834, 2013.
[31]Cohen A., DeVore R., and Schwab C.. Convergence rates of best n-term galerkin approximations for a class of elliptic spdes. Found. Comp. Math., 6(10):615646, 2010.
[32]Curtis P.. n-parameter families and best approximation. Pacific Journal of Mathematics, 9(4):10131027, December 1959.
[33]Boor C. de. Gauss elimination by segments and multivariate polynomial interpolation. In Proceedings of the conference on Approximation and computation: a fetschrift in honor of Walter Gautschi: a fetschrift in honor of Walter Gautschi, pages 1122, West Lafayette, Indiana, United States, 1994. Birkhauser Boston Inc.
[34]Boor C. de and Ron A.. Computational aspects of polynomial interpolation in several variables. Mathematics of Computation, 58(198):705727, April 1992.
[35]Boor C. de and Ron A.. The least solution for the polynomial interpolation problem. Mathematische Zeitschrift, 210(1):347378, December 1992.
[36]Deane A. E., Kevrekidis I. G., Karniadakis G. E., and Orszag S. A.. Low-dimensional mdoels for complex geometry flows: Application to grooved channels and circular cylinders. Physics of Fluids A: Fluid Dynamics (1989-1993), 3(10):23372354, October 1991.
[37]Debusschere B. J., Najm H. N., Matta A., Knio O. M., Ghanem R. G., and Le Maitre O. P.. Protein labeling reactions in electrochemical microchannel flow: Numerical simulation and uncertainty propagation. Physics of Fluids, 15:2238, 2003.
[38]DeVore R. A.. Deterministic constructions of compressed sensing matrices. Journal of Complexity, 23(4-6):918925, August 2007.
[39]Donoho D.L.. Compressed sensing. IEEE Trans. Inform. Theory, 4(52):12891306, 2006.
[40]Doostan A. and Owhadi H.. A non-adapted sparse approximation of pdes with stochastic inputs. J. Comput. Phys., 8(230):30153034, 2011.
[41]Ernst O.G., Mugler A., Starkloff H.-J., and Ullmann E.. On the convergence of generalized polynomial chaos expansions. ESAIM: Mathematical Modelling and Numerical Analysis, 46(02):317339, 2012.
[42]Gao Z. and Zhou T.. Choice of nodal sets for least square polynomial chaos method with application to uncertainty quantification. Communications in Computational Physics, 16:365381, 2014.
[43]Gerstner T. and Griebel M.. Numerical integration using sparse grids. Numerical Algorithms, 18(3):209232, January 1998.
[44]Ghanem R. G. and Spanos P. D.Stochastic finite elements: a spectral approach. Springer-Verlag New York, Inc., 1991.
[45]GÃijnttner R.. Evaluation of lebesgue constants. SIAM Journal on Numerical Analysis, 17(4):512520, August 1980.
[46]Gyørfi L., Kohler M., Krzyz˙ak A., and Walk H.. A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics, Springer-Verlag, Berlin, 2002.
[47]Hosder S., Walters R. W., and Balch M.. Point-collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics. AIAA Journal, 48(12):27212730, December 2010.
[48]Iwen M. A.. Simple deterministically constructible rip matrices with sublinear fourier sampling requirements. 43rd Annual Conference on Information Sciences and Systems (CISS), Baltimore, MD, 2009.
[49]Iwen M. A.. Combinatorial sublinear-time fourier algorithms. Foundations of Computational Mathematics, 3(10):303338, 2010.
[50]Kennedy M. C. and O’Hagan A.. Bayesian calibration of computer models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63(3):425464, January 2001.
[51]Klimeck M.. Pluripotential Theory. Oxford University Press, Oxford, 1991.
[52]Levenberg N.. Weighted pluripotential theory results of berman-boucksom. arXiv:1010.4035, October 2010.
[53]Levenberg N.. Ten lectures on weighted pluripotential theory. Dolomites Research Notes on Approximation, 5:159, 2012.
[54]Lubinsky D.. A survey of weighted polynomial approximation with exponential weights. Surveys in Approximation Theory, 3:1105, 2007.
[55]Eldred M.. Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design. In 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Structures, Structural Dynamics, and Materials and Co-located Conferences. American Institute of Aeronautics and Astronautics, May 2009.
[56]Mairhuber J.C.. On haar’s theorem concerning chebychev approximation problems having unique solutions. Proceedings of the American Mathematical Society, 7(4):609, August 1956.
[57]Marzouk Y. and Xiu D.. A stochastic collocation approach to bayesian inference in inverse problems. Communications in Computational Physics, 6(4):826847, October 2009.
[58]Mathelin L. and Gallivan K. A.. A compressed sensing approach for partial differential equations with random input data. J. Comput. Phys., 12(4):919954, 2012.
[59]Matjila D. M.. Bounds for lebesgue functions for freud weights. Journal of Approximation Theory, 79(3):385406, December 1994.
[60]Matjila D. M.. Bounds for the weighted lebesgue functions for freud weights on a larger interval. Journal of Computational and Applied Mathematics, 65(1-3):293298, December 1995.
[61]Migliorati G., Nobile F., Schwerin E., and Tempone R.. Analysis of the discrete l 2 projection on polynomial spaces with random evaluations. Foundations of Computational Mathematics, doi:10.1007/s10208-013-9186-4, 2014.
[62]Najm H.. Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Annual review of fluid mechanics, 41(1):3552, 2009.
[63]Narayan A., Jackman J., and Zhou T.. A Christoffel function wighted least squares algrithm for collocation approximations. ArXiv: 1412.4305, 2014.
[64]Narayan A. and Jakeman J.. Adaptive leja sparse grid constructions for stochastic collocation and high-dimensional approximation. SIAM Journal on Scientific Computing, 36(6):A2952A2983, 2014.
[65]Narayan A. and Xiu D.. Stochastic collocation methods on unstructured grids in high dimensions via interpolation. SIAM Journal on Scientific Computing, 34(3):A1729A1752, June 2012.
[66]Nobile F., Tempone R., and Webster C. G.. An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM Journal on Numerical Analysis, 46(5):24112442, January 2008.
[67]Patera A. T. and Rozza G.. Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT, version 1.0 edition, 2007.
[68]Penga J., Hampton J., and Doostan A.. A weighted ℓ1-minimization approach for sparse polynomial chaos expansions. Journal of Computational Physics, 267(0):92111, 2014.
[69]PoÃñtte G., DesprÃl’s B., and Lucor D.. Uncertainty quantification for systems of conservation laws. Journal of Computational Physics, 228(7):24432467, April 2009.
[70]Prudhomme C., Rovas D. V., Veroy K., Machiels L., Maday Y., Patera A. T., and Turinici G.. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. Journal of Fluids Engineering, 124(1):70, 2002.
[71]Pulch R. and Xiu D.. Generalised polynomial chaos for a class of linear conservation laws. Journal of Scientific Computing, 51(2):293312, May 2012.
[72]Rauhut H.. Compressive sensing and structured random matrices. Theoretical Foundations and Numerical Methods for Sparse Recovery, Fornasier M. (Ed.) Berlin, New York (DE GRUYTER), pages 192, 2010.
[73]Rauhut H. and Ward R.. Sparse Legendre expansions via ℓ1-minimization. Journal of Approximation Theory, 164(5):517533, May 2012.
[74]Hosder S., Walters R., and Balch M.. Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. In 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Structures, Structural Dynamics, and Materials and Co-located Conferences. American Institute of Aeronautics and Astronautics, April 2007.
[75]Saff E. and Totik V.. Logarithmic Potentials with External Fields. Springer, Berlin, 1997.
[76]Sommariva A. and Vianello M.. Computing approximate fekete points by QR factorizations of vandermonde matrices. Computers&Mathematics with Applications, 57(8):13241336, April 2009.
[77]Szabados J.. Weighted lagrange and hermite-fejér interpolation on the real line. Journal of Inequalities and Applications, 1997(2):481267, 1997.
[78]Szegö G.. Orthogonal Polynomials. American Mathematical Society, Providence, RI, 1975.
[79]Tang T. and Zhou T.. On discrete least square projection in unbounded domain with random evaluations and its application to parametric uncertainty quantification. SIAM Journal on Scientific Computing, 36(5):A2272A2295, 2014.
[80]Tang T. and Zhou T.. Recent developments in high order numerical methods for uncertainty quantification (in Chinese). Sciences in China: Mathematics, 45(6), 2015.
[81]Tartakovsky D. M. and Xiu D.. Stochastic analysis of transport in tubes with rough walls. Journal of Computational Physics, 217(1):248259, September 2006.
[82]Taylor M. A., Wingate B. A., and Vincent R. E.. An algorithm for computing fekete points in the triangle. SIAM Journal on Numerical Analysis, 38(5):17071720, 2000.
[83]Todor R.A. and Schwab C.. Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal., 2(27):232261, 2007.
[84]Trefethen L. N. and A J.. Weideman C.. Two results on polynomial interpolation in equally spaced points. Journal of Approximation Theory, 65(3):247260, June 1991.
[85]Berg E. van den and Friedlander M.. Spgl1: A solver for large-scale sparse reconstruction. http://www.cs.ubc.ca/labs/scl/spgl1, 2007.
[86]Warburton T.. An explicit construction of interpolation nodes on the simplex. Journal of Engineering Mathematics, 56(3):247262, November 2006.
[87]Weil A.. On some exponential sums. Proceedings of the National Academy of Sciences of the United States of America, 34(5):204207, May 1948. PMID: 16578290 PMCID: PMC1079093.
[88]Xiu D.. Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys, pages 293309, 2007.
[89]Xiu D.. Fast numerical methods for stochastic computations: A review. Communications in Computational Physics, 5(2-4):242272, 2009.
[90]Xiu D.. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, July 2010.
[91]Xiu D. and Hesthaven Jan S.. High-order collocation methods for differential equations with random inputs. SIAM Journal on Scientific Computing, 27(3):11181139, January 2005.
[92]Xiu D. and Karniadakis G. E.. The wiener-askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing, 24(2):619644, January 2002.
[93]Xiu D. and Karniadakis G. E.. Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics, 187(1):137167, May 2003.
[94]Xu Z.. Deterministic sampling of sparse trigonometric polynomials. Journal of Complexity, 27(2):133140, April 2011.
[95]Xu Z. and Zhou T.. On sparse interpolation and the design of deterministic interpolation points. SIAM Journal on Scientific Computing, 36(4):A1752A1769, 2014.
[96]Yan L., Guo L., and Xiu D.. Stochastic collocation algorithms using ℓ1-minimization. International Journal for Uncertainty Quantification, 2(3):279293, 2012.
[97]Yang J. and Zhang Y.. Alternating direction algorithms for ℓ1-problems in compressive sensing. SIAM Journal on Scientific Computing, 33(1):250278, January 2011.
[98]Yang X. and Karniadakis G. E.. Reweighted ℓ1 minimization method for stochastic elliptic differential equations. Journal of Computational Physics, 248:87108, September 2013.
[99]Yin W., Osher S., Goldfarb D., and Darbon J.. Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing. SIAM J. Imaging Sciences, 1(1):143168, 2008.
[100]Zhou T., Narayan A., and Xiu D.. Weighted discrete least-squares polynomial approximation using randomized quadratures. submitted, 2014.
[101]Zhou T., Narayan A., and Xu Z.. Multivariate discrete least-squares approximations with a new type of collocation grid. SIAM Journal on Scientific Computing, 36(5):A2401A2422, January 2014.
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