## Abstract

We developed a domain decomposition model reduction method for linear steady-state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equation with random diffusivity and the convection-dominated transport equation with random velocity. We investigated two types of random fields, i.e., colored noises and discrete white noises, both of which can lead to high-dimensional parametric dependence in practice. The motivation is to exploit domain decomposition to reduce the parametric dimension of local problems in subdomains, such that an entire parametric map can be approximated with a small number of expensive partial differential equation (PDE) simulations. The new method combines domain decomposition with model reduction and sparse polynomial approximation, so as to simultaneously handle the high dimensionality and irregular behavior of the PDEs under consideration. The advantages of our method lie in three aspects: (i) online-offline decomposition, i.e., the online cost is independent of the size of the triangle mesh; (ii) sparse approximation of operators involving nonaffine high-dimensional random fields; (iii) an effective strategy to capture irregular behaviors, e.g., sharp transitions of the PDE solutions. Two numerical examples are provided to demonstrate the advantageous performance of our method.

Original language | English |
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Pages (from-to) | A1984-A2011 |

Journal | SIAM Journal on Scientific Computing |

Volume | 41 |

Issue number | 3 |

DOIs | |

State | Published - 2019 |

### Funding

This work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contracts ERKJ259 and ERKJ320; by the U.S. National Science Foundation, Computational Mathematics program under award 1620027; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725. \ast Submitted to the journal's Methods and Algorithms for Scientific Computing section February 13, 2018; accepted for publication (in revised form) April 2, 2019; published electronically June 20, 2019. The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by these rights. http://www.siam.org/journals/sisc/41-3/M117060.html \bfF \bfu \bfn \bfd \bfi \bfn \bfg : ThisworkwassupportedbytheU.S.DepartmentofEnergy,OfficeofScience,Officeof Advanced Scientific Computing Research, Applied Mathematics program under contracts ERKJ259 and ERKJ320; by the U.S. National Science Foundation, Computational Mathematics program under award 1620027; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725.

## Keywords

- Colored noise
- Discrete white noise
- Domain decomposition
- High dimensionality
- Sharp transition
- Uncertainty quantification