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Optimal Spectral Approximation Methods for Parameterized PDEs with Application to Flow Problems

Event Type: 
Date and Time: 
Friday, November 3, 2017 - 16:30
Location: 
CTR Conference Room 103
Event Sponsor: 
Parviz Moin, Director of Center for Turbulence Research
Speaker(s): 
Kookjin Lee

We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated in a chosen finite-dimensional subspace. It has been shown that the stochastic Galerkin projection technique typically does not minimize any measure of the solution error. As a remedy for this, we propose a novel stochastic least-squares Petrov–Galerkin (LSPG) method. The proposed method is optimal in the sense that it produces the solution that minimizes a weighted l2-norm of the residual over all solutions in a given finite-dimensional subspace. Moreover, the method can be adapted to minimize the solution error in different weighted l2-norms by simply applying a weighting function within the least-squares formulation. In addition, a goal-oriented semi-norm induced by an output quantity of interest can be minimized by defining a weighting function as a linear functional of the solution. Extensive numerical experiments show that the weighted LSPG methods outperforms other spectral methods in minimizing corresponding target weighted norms.

Bio: 
Kookjin Lee is a Ph.D. candidate in the Computer Science department at the University of Maryland College Park, working with Professor Howard Elman. Kookjin’s primary research interest lies in uncertainty quantification, designing and developing efficient and optimal solution algorithms for parameterized PDEs. He has developed fast iterative low-rank solvers for linear and nonlinear parameterized PDEs, and optimal spectral approximation methods for parameterized PDEs. He received his B.S. and M.S. in Computer Science and Engineering from Seoul National University.