Almost Block Diagonal Linear Systems: Sequential and Parallel Solution Techniques, and Applications

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright (C) 2000 John Wiley & Sons, Ltd

Extending BACOLI to solve multi-scale problems

The BACOLI package is a numerical software package for solving parabolic partial differential equations in one spatial dimension. It implements a B-spline collocation method for the spatial discretization of a system of partial differential equations. The resultant ordinary differential equations together with the boundary conditions form a system of differential-algebraic equations. The differential-algebraic equations are then solved using the DASSL solver. The BACOLI software package features adaptive error control in the temporal and spatial domains. The estimate of the temporal error is controlled through the DASSL solver. The estimate of the spatial error is controlled based on the difference between two solutions computed in the BACOLI software package. This difference gives an estimation of the error. If this error estimate does not meet the user-supplied tolerance, then the spatial mesh is changed. The BACOLI software package can only solve parabolic partial differential equations that depend on spatial derivatives. In this thesis, the BACOLI software package is modified to solve a broader spectrum of problems. In fact, after some modifications, the extended BACOLI software package can solve systems of parabolic partial differential equations and time-dependent equations that do not depend on spatial derivatives. We apply this extended software package to solve the monodomain model of cardiac electrophysiology. The monodomain model is a multi-scale mathematical model for the evolution of the electrical potential in cardiac tissue that couples the ionic currents at the cellular scale with their propagation at the tissue scale. Because of their local nature, the mathematical models of a single cell have no dependency on spatial derivatives whereas the models at the tissue level do. The heart models considered in our numerical experiments use various cardiac cell models. We find that solving the heart models through the extended BACOLI software package, in some cases, leads to a speed-up in comparison with the Chaste software package, which is a powerful, widely used, and well-respected software package for heart simulation

Parallel numerical solution of ABD and BABD linear systems arising from BVPs

We consider linear systems with coefficient matrices having the ABD or the Bordered ABD (BABD) structures. These systems arise in the discretization of BVPs for ordinary and partial differential equations with separated and non-separated boundary conditions, respectively. We describe the cyclic reduction algorithm for the solution of BABD linear systems which allowed us to write the codes BABDCR and GBABDCR (the latter code is suitable for matrices with a more generic BABD structure). A comparison of the GBABDCR code with respect to the well-known sequential code COLROW on ABD linear systems is then analysed. We report some tests on an OpenMP Fortran 90 parallel version of the GBABDCR code and finally we discuss about the use of GBABDCR inside the BVP code BVP SOLVER