Dr. Miroslav Stoyanov, ORNL
Abstract:
Sparse Grids (SG) is a common tool for applications that require multidimensional function approximation, such as constructing surrogate models in parameter space or solving high-dimensional partial differential equations. From uncertainty quantification to Kinetic models for plasma physics, the key advantage of SG is the mitigation of the computational cost associated with high-dimensional problems, also known as the curse of dimesionality. The grids are constructed from hierarchical superposition of multidimensional tensors of single dimensional basis functions, and the resulting complex structure allows for fine tuning and adapting to the behavior of a specific function with minimal number of degrees of freedom. The linear algebra operations associated with SG resemble algebraic operations using classical Kronecker matrices and Kronecker theory is in turn a powerful tool for tensor linear algebra. The Kronecker structure allows combinations of like-terms and reduction of the overall computational cost by a factor exponential in the number of dimensions. However, the same like-term relationship is lost in a general sparse context.
Several sufficient conditions are presented that enable recovery of the Kronecker cost reduction in a hierarchical sparse-grid context. The analysis considers hierarchy completeness, the direction of information transfer (upward or downward through the hierarchy), and minimal padding strategies. Numerical results demonstrate speedups of up to 100×, even for moderate dimensions, when computing surplus coefficients for sparse-grid interpolation. Greater speedups are obtained when sparse grids are combined with the Discontinuous Galerkin (DG) method for the solution of high-dimensional Partial Differential Equations (PDEs).
Speaker’s Bio:
Dr. Miroslav Stoyanov obtained his Ph.D. in Mathematics from Virginia Tech in 2009. After a postdoctoral assignment at Florida State University, Dr. Stoyanov joined Oak Ridge National Laboratory in 2012, where he has continued his research. His research interests include uncertainty quantification, high-dimensional approximation theory and high-performance computing and the development of mathematical libraries.
Host:
Pablo Seleson, [email protected], 865-323-2859