Preface to BIT 53:2

A sizable part of this regular issue deals with large systems of equations, and specially to those arising from approximation and data analysis applications. By tradition, we describe a problem solving chain, starting with building a mathematical model, then discretizing this, later solving a linear system and finally displaying the result. And normally this division of labor is an advantage! I take the matrix and device an algorithm that gets the most out of it. But the origin of the matrix tells me quite a lot about how it is expected to behave. The matrix is not just a matrix, it may have come from a discretized PDE or from an image reconstruction task. These issues are discussed in several of the papers here. These are the papers: In the first paper, James Baglama and Lothar Reichel discuss block Lanczos bidiagonalization, the first part of an iterative method to compute the singular value decomposition of a large rectangular matrix. They use Leja shifts to accelerate the convergence of an implicit restart procedure. Mario Bebendorf, Matthias Bollhofer, and Michael Bratsch represent a large matrix as a hierarchical matrix. Approximation with low rank blocks need less computations. Constraints that preserve invariant subspaces are derived. The approach is useful as a preconditioner for iterative conjugate gradient multigrid solution of PDE problems. Tomas Bjork, Anders Szepessy, Raul Tempone, and Georgios Zouraris study Monte Carlo Euler approximations of Heath-Jarrow-Morton term structure financial models. Weak convergence estimates of the underlying Ito stochastic differential equation are derived. Several numerical experiments are reported.