University of Birmingham

The use of preconditioned Newton-Krylov methods is in many applications mandatory for computing efficiently the solution of large nonlinear systems of equations. However, the available preconditioners are often sub-optimal, due to the changing nature of the linearized operator. This the case, for instance, for quasi-Newton methods where the Jacobian (and its preconditioner) are kept fixed at each non-linear iteration, with the rate of convergence usually degraded from quadratic to linear. Updated Jacobians, on the other hand require updated preconditioners, which may not be readily available. In this work we introduce an adaptive preconditioning technique based on the Krylov subspace information generated at previous steps in the nonlinear iteration. In particular, we use to advantage an adaptive technique suggested for restarted GMRES to enhance existing preconditioners with information about (almost) invariant subspaces constructed by GMRES at previous stages in the nonlinear iteration. We provide guidelines on the choice of invariant-subspace basis used in the construction of our preconditioner and demonstrate the improved performance on various test problems. As a useful general application we consider the case of augmented systems preconditioned by block triangular matrices based on the structure of the system matrix. We show that a sufficiently good solution involving the primal space operator allows for an efficient application of our adaptive technique restricted to the space of dual variables.

It is known for long that the set of possible compactifications of a topological space (up to homeomorphism) is in order-preserving bijection to "strong inclusion" relations on the lattice of open sets. Since these relations do not refer to points explicitly, this bijection has been generalised to point-free topology (a.k.a. locales). The strong inclusion relations involved are typically "witnessed" relations. For example, the Stone-Cech compactification has a strong inclusion witnessed by real-valued functions. This makes it natural to think of compactification in terms of d-frames, a category invented by Jung and Moshier for bitopological Stone duality. Here, a witnessed strong inclusion is inherent to every object and plays a central role.

We present natural analogues of the topological concepts regularity, normality, complete regularity and compactness in d-frames. Compactification is then a coreflection into the category of d-frames dually equivalent to compact Hausdorff spaces. The category of d-frames has a few surprising features. Among them are:

• The real line with the bitopology of upper and lower semicontinuity admits precisely one compactification, the extended reals.
• Unlike in the category of topological spaces (or locales), there is a coreflection into the subcategory of normal d-frames, and every compactification can be factored as "normalisation" followed by Stone-Cech compactification.

We discuss numerical methods for the solution of the palindromic eigenvalue problem Ax=λ A^Tx, where A is a complex matrix. Such eigenvalue problems occur, for example, in the vibration analysis of rail tracks.

The structure of palindromic eigenvalue problems leads to a symmetry in the spectrum: all eigenvalues occur in reciprocal pairs. The need for preservation of this symmetry in finite precision arithmetic requires the use of structure-preserving numerical methods. In this talk, we explain how such methods can be derived.