Physics-based Pre-conditioners for Large-scale Subsurface Flow Simulation

On The Acceleration Of Ill-Conditioned Linear Systems: A Pod-Based Deflation Method For The Simulation Of Two-Phase Flow

“…We solve a series of problems with the same conditions as the training phase, but with different pressures in the wells, i.e., different rhs (see Algorithm 4). For more details of the POD-based deflation method we refer to Diaz-Cortes et al (2016.…”

“…Therefore, new techniques have to be developed, that together with the usual preconditioned iterative methods can find approximate solutions in a faster way. Recently, Proper Orthogonal Decomposition (POD) methods (Markovinović and Jansen, 2006;Astrid et al, 2011;Pasetto et al, 2017;Diaz-Cortes et al, 2018) and deflation techniques (Vuik et al, 1999(Vuik et al, , 2002Diaz-Cortes et al, 2016) based on system information, have been studied to accelerate iterative methods for large and ill-conditioned problems. An extensive literature exists, with new and innovative ways of approaching deflation and POD methodologies.…”

On POD-based Deflation Vectors for DPCG applied to porous media problems

“…In theory, only one iteration is necessary to reach the solution with the deflated methods. However, the large contrast in the permeability field may require higher accuracy for the snapshots to find the solution with the deflation method, within one iteration, within the imposed tolerance [31,32]. In this case, the first iteration has a relative residual smaller than 10 −10 for the DICCG 4 and DICCG POD 4 methods, therefore, after a first iteration, we have already a good approximation.…”