An accelerated subspaces recycling strategy for the deflation of parametric linear systems based on model order reduction

Abstract

Krylov subspace recycling has been extensively used to facilitate the solution of sequences of linear systems by constructing a deflation subspace and accelerating the convergence of a corresponding iterative solver. However, most existing techniques update the recycled subspace sequentially for each system, thus inducing a potentially high computational cost. In that context, this work proposes a method to accelerate the above procedure for the case of multiresolution analyses of affine parametric systems, by decoupling the solution of the system from the construction of the recycled subspace. The proposed method follows the projection based Model Order Reduction (MOR) logic that splits operations into an offline and an online phase and therefore proves particularly beneficial in case of affine parametrizations that involve multiple affine coefficients. In the offline phase of the method an especially tailored version of the Automatic Krylov subspaces Recycling (AKR) algorithm (Panagiotopoulos et al., 2021), proposed within this work and denoted as AKR-D, is employed. In brief, AKR-D constructs a high quality recycling basis W for a predefined parameter interval Ψ by targeting a desirable convergence rate at an automatically generated set of parameter values Ω⊂Ψ. The upfront construction of W enables an a-priori Galerkin projection of the affinely described full parametric system to yield a reduced order model (ROM). This ROM can be leveraged in the online phase of the proposed recycling method to accelerate the construction of the corresponding projector P onto W=span{W}, whose implicit assembly is required in most recycling based deflation schemes. The effectiveness of the proposed strategy is investigated in terms of algorithmic complexity and is demonstrated on a densely parametrized system arising within the boundary integral solver of the Helmholtz equation and a random sparse parametrization example.