An Automatic Krylov subspaces Recycling technique for the construction of a global solution basis of non-affine parametric linear systems

Recycling of Krylov subspaces is often used to obtain an augmentation subspace in the context of iterative algorithms for the solution of sequences of linear systems. However, it still remains difficult to quantify the effect of subspaces recycling and thus to determine the dimension of the subspaces to be recycled targeting a specific accuracy. In that context, this work proposes the Automatic Krylov subspaces Recycling algorithm (AKR) that automates the selection of Krylov subspaces to be recycled and generates a basis that can provide sufficiently accurate approximations of the solution for a parametric system on a predefined interval Ψ . The constructed basis is employed as a Galerkin projection basis for a model order reduction (MOR) scheme in the context of non-affine parametric systems. In the offline phase of the MOR scheme, AKR constructs a projection subspace W by sampling Krylov subspaces at an iteratively built set of parameter values Ω . Keeping a balance between the solution accuracy and the memory required, the algorithm, apart from guaranteeing a predefined residual level r tol , also permits the predetermination of a threshold regarding the maximum memory employed. Nevertheless, following the unpreconditioned Krylov methods effectiveness criteria, the proposed technique proves to be efficient for systems with relatively clustered eigenvalues such as the ones encountered in the conventional Boundary Element Method. The performance of the proposed AKR algorithm is assessed in comparison with an alternative version of the reduced basis method, which is based on the same assumptions as the AKR and is specifically designed to provide a good benchmark. These techniques are deployed for a randomly generated complex system and an acoustic BEM system. The advantage of employing AKR is demonstrated as fewer system assemblies are required for the construction of the projection basis.