A Domain Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients

Dr. Guannan Zhang

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 non- affine high-dimensional random fields; (iii) an effective strategy to capture irregular behaviors, e.g., sharp transitions of the PDE solutions. 

Speaker’s Bio:  Dr. Guannan Zhang is a Research Staff in Computational and Applied Mathematics (CAM) Group at Oak Ridge National Laboratory (ORNL). He studied mathematics at Shandong University in China, receiving his Bachelor's degree and Master's degree in 2007 and 2009, respectively. Guannan earned his Ph.D. in computational science at Florida State University in 2012, under the supervision of Prof. Max Gunzburger. He joined ORNL in 2012 as the Householder fellow in the Computational and Applied Mathematics Group within the Computer Science and Mathematics Division. He has been holding a joint faculty appointment with the Department of Mathematics and Statistics at Auburn University since 2014. His research interests include high-dimensional approximation, uncertainty quantification, machine learning and artificial intelligence, stochastic optimization and control, numerical solution of stochastic differential equations, and model reduction for parametrized differential equations. His research webpage is here.

Host:  Eirik Endeve, endevee@ornl.gov

About the Seminar:  The Computational and Applied Math Seminar features talks by invited speakers, local mathematicians, and domain scientists working on problems of mathematical interest. The seminar is held weekly, every Thursday from 3:00pm-4:00pm. If you are interested in giving a seminar, please contact Eirik Endeve, endevee@ornl.gov. To subscribe to the CAM Seminar mailing list, please contact Kasi Arnold, arnoldkl@ornl.gov.  To see the full list of previous and upcoming seminars, go to https://csmd.ornl.gov/events/9/seminars.