"Steps towards a fast O(N) approach for direct inversion of linear operators with applications to nonlinear partial differential equations"
Andrew J. Christlieb
Department of Mathematics
Michigan State University
Multi-scale problems in science and engineering require accurate implicit methods for solving partial differential equations. However, in the world of distributed multi-core computing, a key bottleneck is the inversion of matrices. Hence, implicit solutions to partial differential equations have difficulty scaling on these computing platforms. Our goal is to develop a fast O(N) direct approach to the inversion of linear operators in real space. The work is based on the method of lines transpose which combines Green's function methods, successive convolution, fast summation and Rothe's method. The method may also be expressed as an efficient approach for direct evaluation of pseudodifferential operators. Practically speaking, the formulation of the method based on successive convolution can be directly expressed as an O(N) method for computing the resolvent expansion of a pseudodifferential operator in real space. This method has been used to develop an A-stable arbitrary order method for solving the two way wave equation, Maxwell's equations and both linear and nonlinear parabolic problems. We are current working to extend these methods to high-order phase field models.