Augmented unprojected Krylov subspace methods from an alternative view of an existing framework

Abstract

Augmented Krylov subspace methods aid in accelerating the convergence of a standard Krylov subspace method by including additional vectors in the search space. These methods are commonly used in High-Performance Computing (HPC) applications to considerably reduce the number of matrix vector products required when building a basis for the Krylov subspace. In a recent survey on subspace recycling iterative methods [Soodhalter et al, GAMM-Mitt. 2020], a framework was presented which describes a wide class of such augmented methods. The framework describes these methods broadly in a two step process.Step one involves solving a projected problem via a standard Krylov subspace, or other projection method, and step two then performs an additional projection into the augmentation subspace. In this work we show that the projected problem one must solve in step one has an equivalent unprojected formulation. We then show how this observation allows the framework to be adapted to describe unprojected augmented Krylov subspace methods. We demonstrate this with two examples. We first show one can recover the R 3 GMRES algorithm from this view of the framework, and then we use the framework to derive the first unprojected augmented Full Orthogonalization Method (FOM). This method allows one to recycle information from previous system solves, and we thus denote it as unproj rFOM (unprojected recycled FOM).

Krylov Subspace Recycling For Matrix Functions

“…The proof of this equivalence is a trivial extension of the proof shown in [5] to the shifted case and is thus not repeated here. We now propose to solve the unprojected problem using the unprojected, shift-invariant Krylov subspace K j (A, b).…”

“…One way to overcome this difficulty is to use the observation made in [5] where it was shown that the projected problem (5.3) has an unprojected equivalent given by Find t j (σ) ∈ V j as an approximate solution corresponding to shift σ for the shifted linear system (σI…”

“…One can use this general framework to derive any new augmented unprojected Krylov subspace approximation for solving families of shifted system (3.5) using appropriate choices of the subspaces V j , U and U while keeping V j = K j (A, b). The choice V j = AK(A, b) and U = AU leads to an unprojected augmented GMRES-type method for shifted systems, while the choice V j = K(A, b) and U = U leads to an unprojected augmented FOM-type method which can be viewed as an extension of the unprojected augmented FOM method derived in [5] to the case of shifted systems of the form (3.5). Since we are dealing with matrix functions, where the standard Arnoldi approximation is of FOM-type, we now focus on FOM type methods and make these choices.…”