A Novel Percolative Aggregation Approach for Solving Highly Ill-Conditioned Systems

A key aspect in any algebraic multilevel procedure is to be able to reliably capture the physical behavior behind system coefficients. However, the modeling of complex reservoir scenarios generally involves the computation of extremely discontinuous and nonlinear coefficients that, in turn, compromise the robustness and efficiency of smoothing and coarsening strategies. In addition, the need of dealing with large discretized domains leads to highly ill-conditioned systems that represent a true challenge to any linear solver technology known today. In this work, we exploit the fact that flow trend information can be used as a basis for developing a robust percolative aggregation (PA) two-stage preconditioning method. To this end, we identify coefficient aggregates by a means of an efficient version of the Hoshen-Kopelman percolation algorithm suitable for any flow network structure. By partitioning and reordering unknowns according to these aggregates, we obtain a set of high-conductive blocks followed by a low-conductive block. Diagonal scaling allows for weakening the interaction between these high- and low- conductive blocks plus improving the overall conditioning of the system. The solution of the high-conductive blocks approximates well the solution of the original problem. Remaining sources of errors from this approximation are mainly due to small eigenvalues that are properly eliminated with a deflation step. The combination of the algebraic solution of the high-conductive blocks with deflation is realized as a two-stage preconditioner strategy. The deflation stage is carried out by further isolating the aggregate blocks with a matrix compensation procedure. We validate the performance of the PA approach against ILU preconditioning. Preliminary numerical results indicate that the PA two-stage preconditioning can be used as a promising alternative to employ existing algebraic multilevel methods in a more effective way.