# Reduced Basis Techniques for Stochastic Problems

@article{Boyaval2010ReducedBT, title={Reduced Basis Techniques for Stochastic Problems}, author={S{\'e}bastien Boyaval and C. Le Bris and Tony Leli{\`e}vre and Yvon Maday and Ngoc Cuong Nguyen and Anthony T. Patera}, journal={Archives of Computational Methods in Engineering}, year={2010}, volume={17}, pages={435-454} }

We report here on the recent application of a now classical general reduction technique, the Reduced-Basis (RB) approach initiated by C. Prud’homme et al. in J. Fluids Eng. 124(1), 70–80, 2002, to the specific context of differential equations with random coefficients. After an elementary presentation of the approach, we review two contributions of the authors: in Comput. Methods Appl. Mech. Eng. 198(41–44), 3187–3206, 2009, which presents the application of the RB approach for the…

## 120 Citations

### A reduced-basis method for input-output uncertainty propagation in stochastic PDEs

- Computer Science
- 2013

This work proposes a non-intrusive reduced-basis method for the rapid and reliable evaluation of the statistics of linear functionals of stochastic PDEs and develops the reduced basis for not only the primal problem, but also the adjoint problem.

### Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems

- Computer ScienceJ. Sci. Comput.
- 2014

The main result stemming from this 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 stochastics problems when considering computational costs.

### A fast Monte–Carlo method with a reduced basis of control variates applied to uncertainty propagation and Bayesian estimation

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### A reduced basis approach for some weakly stochastic multiscale problems

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In this paper, a multiscale problem arising in material science is considered. The problem involves a random coefficient which is assumed to be a perturbation of a deterministic coefficient, in a…

### Adaptive reduced basis strategy dedicated to the solution of nonstationary stochastic thermal problems

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### Multiscale Problems in Materials Science: A Mathematical Approach to the Role of Uncertainty

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Abstract : The bottom line of this work is to develop affordable numerical methods in the context of stochastic homogenization. Many partial differential equations of materials science indeed involve…

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- Computer Science, MathematicsJ. Comput. Phys.
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### Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations

- Computer ScienceNumerische Mathematik
- 2016

A multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations is developed and it is proved the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data is proved.

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