2 research outputs found
Hierarchical Interpolative Factorization Preconditioner for Parabolic Equations
This note proposes an efficient preconditioner for solving linear and
semi-linear parabolic equations. With the Crank-Nicholson time stepping method,
the algebraic system of equations at each time step is solved with the
conjugate gradient method, preconditioned with hierarchical interpolative
factorization. Stiffness matrices arising in the discretization of parabolic
equations typically have large condition numbers, and therefore preconditioning
becomes essential, especially for large time steps. We propose to use the
hierarchical interpolative factorization as the preconditioning for the
conjugate gradient iteration. Computed only once, the hierarchical
interpolative factorization offers an efficient and accurate approximate
inverse of the linear system. As a result, the preconditioned conjugate
gradient iteration converges in a small number of iterations. Compared to other
classical exact and approximate factorizations such as Cholesky or incomplete
Cholesky, the hierarchical interpolative factorization can be computed in
linear time and the application of its inverse has linear complexity. Numerical
experiments demonstrate the performance of the method and the reduction of
conjugate gradient iterations
Parallel Skeletonization for Integral Equations in Evolving Multiply-Connected Domains
This paper presents a general method for applying hierarchical matrix
skeletonization factorizations to the numerical solution of boundary integral
equations with possibly rank-deficient integral operators. Rank-deficient
operators arise in boundary integral approaches to elliptic partial
differential equations with multiple boundary components, such as in the case
of multiple vesicles in a viscous fluid flow. Our generalized skeletonization
factorization retains the locality property afforded by the "proxy point
method," and allows for a parallelized implementation where different
processors work on different parts of the boundary simultaneously. Further,
when the boundary undergoes local geometric perturbations (such as movement of
an interior hole), the factorization can be recomputed efficiently with respect
to the number of modified discretization nodes. We present an application that
leverages a parallel implementation of skeletonization with updates in a shape
optimization regime.Comment: 31 pages, 18 figure