@article {2017, title = {Reduced Basis Methods for Uncertainty Quantification}, journal = {SIAM/ASA Journal on Uncertainty Quantification}, volume = {5}, year = {2017}, month = {08/2017}, pages = {869}, type = {reviewed}, chapter = {813}, abstract = {

In this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babu{\v s}ka, F. Nobile, and R. Tempone,\ SIAM Rev., 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.


Read More:\ http://epubs.siam.org/doi/abs/10.1137/151004550

}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} } @article {2015, title = {Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations}, journal = {Numerische Mathematik, (2015), 36 p. Article in Press}, number = {Numerische Mathematik;}, year = {2015}, publisher = {Springer}, abstract = {In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems.}, doi = {10.1007/s00211-015-0743-4}, url = {http://urania.sissa.it/xmlui/handle/1963/34491}, author = {Gianluigi Rozza and Peng Chen and Alfio Quarteroni} } @article {2014, title = {Comparison between reduced basis and stochastic collocation methods for elliptic problems}, number = {Journal of scientific computing;volume 59; issue 1; pages 187-216;}, year = {2014}, publisher = {Springer}, abstract = {The stochastic collocation method (Babu{\v s}ka et al. in SIAM J Numer Anal 45(3):1005-1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411-2442, 2008a; SIAM J Numer Anal 46(5):2309-2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118-1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289-294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229-275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41-44):3187-3206, 2009; Arch Comput Methods Eng 17:435-454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions O (1) to moderate dimensions O (10) and to high dimensions O (100). The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.}, doi = {10.1007/s10915-013-9764-2}, url = {http://urania.sissa.it/xmlui/handle/1963/34727}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} } @article {2014, title = {A weighted empirical interpolation method: A priori convergence analysis and applications}, number = {ESAIM: Mathematical Modelling and Numerical Analysis;volume 48; issue 4; pages 943-953}, year = {2014}, publisher = {EDP Sciences}, abstract = {We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Lo{\`e}ve expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.}, doi = {10.1051/m2an/2013128}, url = {http://urania.sissa.it/xmlui/handle/1963/35021}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} } @article {10671, title = {Simulation-based uncertainty quantification of human arterial network hemodynamics}, journal = {International Journal Numerical Methods Biomedical Engineering}, number = {SISSA preprint;}, year = {2012}, publisher = {Wiley}, abstract = {This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time.}, keywords = {uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} }