The widening gap between organ availability and need is resulting in a worldwide crisis, particularly concerning kidney transplantation. Regenerative medicine options are becoming increasingly advanced and are taking advantage of progress in novel manufacturing techniques, including 3D bioprinting, to deliver potentially viable alternatives. Cell-integrated and wearable artificial kidneys aim to create convenient and efficient systems of filtration and restore elements of immunoregulatory function. Whilst preliminary clinical trials demonstrated promise, manufacturing and trial design issues and identification of suitable and sustainable cell sources have shown that more development is required for market progression. Tissue engineering and advances in biomanufacturing techniques offer potential solutions for organ shortages; however, due to the complex kidney structure, previous attempts have fallen short. With the recent development and progression of 3D bioprinting, cell positioning and resolution of material deposition in organ manufacture have never seen greater control. Cell sources for constructing kidney building blocks and populating both biologic and artificial scaffolds and matrices have been identified, but in vitro culturing and/or differentiation, in addition to maintaining phenotype and viability during and after lengthy and immature manufacturing processes, presents additional problems. For all techniques, significant process barriers, clinical pathway identification for translation of models to humans, scaffold material availability, and long-term biocompatibility need to be addressed prior to clinical realisation.

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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).

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