02500nas a2200157 4500008004100000245005700041210005700098260001200155300000800167490000600175520201600181100001502197700002202212700002102234856008702255 2017 eng d00aReduced Basis Methods for Uncertainty Quantification0 aReduced Basis Methods for Uncertainty Quantification c08/2017 a8690 v53 a

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. Babuška, 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

1 aChen, Peng1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://math.sissa.it/publication/reduced-basis-methods-uncertainty-quantification01516nas a2200133 4500008004100000245012100041210006900162260001300231520102900244100002101273700001501294700002201309856005101331 2015 en d00aMultilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations0 aMultilevel and weighted reduced basis method for stochastic opti bSpringer3 aIn 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.1 aRozza, Gianluigi1 aChen, Peng1 aQuarteroni, Alfio uhttp://urania.sissa.it/xmlui/handle/1963/3449102162nas a2200133 4500008004100000245009400041210006900135260001300204520170200217100001501919700002201934700002101956856005101977 2014 en d00aComparison between reduced basis and stochastic collocation methods for elliptic problems0 aComparison between reduced basis and stochastic collocation meth bSpringer3 aThe stochastic collocation method (Babuška 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.1 aChen, Peng1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttp://urania.sissa.it/xmlui/handle/1963/3472701557nas a2200133 4500008004100000245009400041210006900135260001700204520109300221100001501314700002201329700002101351856005101372 2014 en d00aA weighted empirical interpolation method: A priori convergence analysis and applications0 aweighted empirical interpolation method A priori convergence ana bEDP Sciences3 aWe 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-Loève 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.1 aChen, Peng1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttp://urania.sissa.it/xmlui/handle/1963/3502101505nas a2200145 4500008004100000245009700041210006900138300001600207490000700223520095300230100001501183700002201198700002101220856011801241 2013 eng d00aStochastic optimal robin boundary control problems of advection-dominated elliptic equations0 aStochastic optimal robin boundary control problems of advectiond a2700–27220 v513 aIn this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided.1 aChen, Peng1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://math.sissa.it/publication/stochastic-optimal-robin-boundary-control-problems-advection-dominated-elliptic01371nas a2200145 4500008004100000245010300041210006900144300001600213490000700229520080500236100001501041700002201056700002101078856012601099 2013 eng d00aA weighted reduced basis method for elliptic partial differential equations with random input data0 aweighted reduced basis method for elliptic partial differential  a3163–31850 v513 aIn this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.1 aChen, Peng1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://math.sissa.it/publication/weighted-reduced-basis-method-elliptic-partial-differential-equations-random-input-data01576nas a2200145 4500008004100000245008700041210006900128260001000197520093400207653011301141100001501254700002201269700002101291856011801312 2012 en d00aSimulation-based uncertainty quantification of human arterial network hemodynamics0 aSimulationbased uncertainty quantification of human arterial net bWiley3 aThis 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.10auncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction1 aChen, Peng1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://math.sissa.it/publication/simulation-based-uncertainty-quantification-human-arterial-network-hemodynamics