In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard LU or LDL T factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from the approach based on the Sherman-Morrison formula [18]. In contrast to the RIF algorithm [11], the direct and inverse factors here direc...

Some algorithmic aspects of systems of PDEs based simulations can be better clarified by means of symbolic computation techniques. This is very important since numerical simulations heavily rely on solving systems of PDEs. For the large-scale problems we deal with in today’s standard applications, it is necessary to rely on iterative Krylov methods that are scalable (i.e., weakly dependent on t...

Computing Symplectic Eigenpairs of Symmetric Positive-Definite Matrices via Trace Minimization and Riemannian Optimization

We experimentally examine the performance of preconditioners based on entries of the symmetric positive definite part and small subspace solvers for linear system of equations obtained from the high-order compact discretization of convection-diffusion equations. Numerical results are described to illustrate that the preconditioned GMRES algorithm converges in a reasonable number of iterations.

A novel method for estimating a field of orientation distribution functions (ODF) from a given set of DW-MR images is presented. We model the ODF by Cartesian tensor basis using a parametrization that explicitly enforces the positive definite property to the computed ODF. The computed Cartesian tensors, dubbed Cartesian Tensor-ODF (CT-ODF), are symmetric positive definite tensors whose coeffici...

In this paper we consider a strongly coupled (monolithic) fluid structure interaction framework for incompressible flow, as opposed to a loosely coupled (partitioned) method. This requires solving a single linear system that combines the unknown velocities of the structure with the unknown pressures of the fluid. In our previous work, we were able to obtain a symmetric formulation of this coupl...