Reduced Basis Collocation Methods for Partial Differential Equations with Random Coefficients

The sparse grid stochastic collocation method is a new method for solving partial differential equations with random coefficients. However, when the probability space has high dimensionality, the number of points required for accurate collocation solutions can be large, and it may be costly to construct the solution. We show that this process can be made more efficient by combining collocation with reduced basis methods, in which a greedy algorithm is used to identify a reduced problem to which the collocation method can be applied. Because the reduced model is much smaller, costs are reduced significantly. We demonstrate with numerical experiments that this is achieved with essentially no loss of accuracy.

Collaboration with Q. Liao and V. Forstall

Speaker Bio: Howard Elman is a Professor in the Computer Science Department at the University of Maryland, College Park. He received his doctorate in Computer Science from Yale University in 1982. He has had visiting positions at Stanford University, the University of Manchester Institute of Science and Technology, and the University of Oxford. He has been selected as a SIAM Fellow and has served on the editorial boards of SIAM Journal on Scientific Computing, where he was editor-in-chief from 1998-2004, Mathematics of Computation, and Numerical Linear Algebra and Applications. His research concerns numerical solution algorithms for partial differential equations, computational fluid dynamics and sparse matrix problems.

Presentation Slides: PDF

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