Certified Offline-Free Reduced Basis (COFRB) Methods for Stochastic Differential Equations Driven by Arbitrary Types of Noise

被引：0
作者：
Yong Liu
Tianheng Chen
Yanlai Chen
Chi-Wang Shu
机构：
[1] University of Science and Technology of China,School of Mathematical Sciences
[2] Brown University,Division of Applied Mathematics
[3] University of Massachusetts Dartmouth,Department of Mathematics
来源：
Journal of Scientific Computing | 2019年 / 81卷
关键词：
Reduced basis method; Stochastic PDE; Least squares; Greedy algorithms; Offline-free;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
In this paper, we propose, analyze, and implement a new reduced basis method (RBM) tailored for the linear (ordinary and partial) differential equations driven by arbitrary (i.e. not necessarily Gaussian) types of noise. There are four main ingredients of our algorithm. First, we propose a new space-time-like treatment of time in the numerical schemes for ODEs and PDEs. The second ingredient is an accurate yet efficient compression technique for the spatial component of the space-time snapshots that the RBM is adopting as bases. The third ingredient is a non-conventional “parameterization” of a non-parametric problem. The last is a RBM that is free of any dedicated offline procedure yet is still efficient online. The numerical experiments verify the effectiveness and robustness of our algorithms for both types of differential equations.
引用
收藏
页码：1210 / 1239
页数：29
相关论文
共 77 条
  • [1] Barrault M(2004)An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations C. R. Acad. Sci. Paris 339 667-672
  • [2] Nguyen NC(1993)The proper orthogonal decomposition in the analysis of turbulent flows Ann. Rev. Fluid Mech. 25 539-575
  • [3] Maday Y(2013)Convergence rates of the pod-greedy method ESAIM: Math. Model. Numer. Anal. 47 859-873
  • [4] Patera AT(2011)Convergence rates for greedy algorithms in reduced basis methods SIAM J. Math. Anal. 43 1457-1472
  • [5] Berkooz PHG(1947)The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals Ann. Math. 48 385-392
  • [6] Lumley J(2013)Reduced collocation methods: reduced basis methods in the collocation framework J. Sci. Comput. 55 718-737
  • [7] Bernard Haasdonk(2010)Certified reduced basis methods and output bounds for the harmonic maxwell’s equations SIAM J. Sci. Comput. 32 970-996
  • [8] Binev P(2019)Numerical solutions of stochastic pdes driven by arbitrary type of noise Stoch. Partial Diff. Equ. Anal. Comput. 7 1-39
  • [9] Cohen A(2019)A robust error estimator and a residual-free error indicator for reduced basis methods Comput. Math. Appl. 77 1963-1979
  • [10] Dahmen W(2013)Reduced basis collocation methods for partial differential equations with random coefficients SIAM/ASA J. Uncertain. Quantif. 1 192-217
12345678