This issue of Numerical Linear Algebra with Applications is dedicated to the 70th birthday of Owe Axelsson, a long-time collaborator and friend. It contains eight invited papers from a list of collaborators, students, friends, and authors who have gotten to know Owe throughout the years.
No matter how long a list I write containing Owe's research contributions, there is always the danger that it may be incomplete, so, I have mentioned a few contributions that have personally in uenced or impressed me the most.
I begin with the block-incomplete factorization (or block-ILU) methods, proposed in the early-to-mid 1980s, together with algorithms for banded approximations to inverses of banded matrices, such as the famous ABI algorithm. It is my opinion, that the block-ILU methods are still the most robust methods for solving 2-d elliptic partial di erential equations (or PDEs), since they are algebraic in nature and can gradually improve their performance by increasing the halfbandwidth in the factorization process.
Another area, where Owe has made a signiΓΏcant impact, is on 'two-level' preconditioning methods. In his pioneering work with Ivar Gustafsson, in the early 1980s, they paved the road that led to a new class of preconditioners that were later proven to be the ΓΏrst 'regularity-free' optimal order preconditioning methods, independent of the jumps of the PDE coe cients. The latter is the case since those can be viewed as algebraically stabilized (based on proper polynomials) hierarchical basis methods. These methods are nowadays known in the literature as the algebraic multi-level iteration (AMLI) preconditioning methods.
Owe's research was also in uential in the derivation and the analysis of the generalized conjugate gradient (GCG) methods, their versions with variable-step preconditioners, as well as, the CG methods for non-linear problems. All of these and other topics are well represented in his monograph on Iterative Solution Methods published by Cambridge University Press in 1994.
All of Owe's achievements make him a leading authority in numerical linear algebra, particularly in preconditioned iterative methods.
A key to his success is that Owe has constantly kept many international contacts. For many years he has had strong ties with Czech and Bulgarian mathematicians, and has interacted with a number of Russia's leading computational mathematics centres. Through his numerous projects and grants, both local and European, he has supported many visiting scientists, post-docs and students. He organized a number of conferences in Nijmegen, The Netherlands, (PCGM'89, AMLI'96, PRISM'97, IMMB'98, PRISM'01, PMOCCO'02) that have attracted many researchers from Europe and all over the world. Last, but not least, Owe is