Preconditioned Recycling Krylov subspace methods for self-adjoint problems

The authors propose a recycling Krylov subspace method for the solution of a sequence of self-adjoint linear systems. Such problems appear, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with arbitrary inner products and arbitrary self-adjoint positive-definite preconditioners whose inverse can be computed with high accuracy. Numerical experiments with nonlinear Schr\"odinger equations indicate a substantial decrease in computation time when recycling is used

A framework for deflated and augmented Krylov subspace methods

We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) "removes" certain parts from the operator making it singular, while augmentation adds a subspace to the Krylov subspace (often the one that is generated by the singular operator); in contrast, preconditioning changes the spectrum of the operator without making it singular. Deflation and augmentation have been used in a variety of methods and settings. Typically, deflation is combined with augmentation to compensate for the singularity of the operator, but both techniques can be applied separately. We introduce a framework of Krylov subspace methods that satisfy a Galerkin condition. It includes the families of orthogonal residual (OR) and minimal residual (MR) methods. We show that in this framework augmentation can be achieved either explicitly or, equivalently, implicitly by projecting the residuals appropriately and correcting the approximate solutions in a final step. We study conditions for a breakdown of the deflated methods, and we show several possibilities to avoid such breakdowns for the deflated MINRES method. Numerical experiments illustrate properties of different variants of deflated MINRES analyzed in this paper.Comment: 24 pages, 3 figure

Size limits of magnetic-domain engineering in continuous in-plane exchange-bias prototype films

Gaul A, Emmrich D, Ueltzhöffer T, et al. Size limits of magnetic-domain engineering in continuous in-plane exchange-bias prototype films. Beilstein Journal of Nanotechnology. 2018;9:2968-2979.Background: The application of superparamagnetic particles as biomolecular transporters in microfluidic systems for lab-on-a-chip applications crucially depends on the ability to control their motion. One approach for magnetic-particle motion control is the superposition of static magnetic stray field landscapes (MFLs) with dynamically varying external fields. These MFLs may emerge from magnetic domains engineered both in shape and in their local anisotropies. Motion control of smaller beads does necessarily need smaller magnetic patterns, i.e., MFLs varying on smaller lateral scales. The achievable size limit of engineered magnetic domains depends on the magnetic patterning method and on the magnetic anisotropies of the material system. Smallest patterns are expected to be in the range of the domain wall width of the particular material system. To explore these limits a patterning technology is needed with a spatial resolution significantly smaller than the domain wall width. Results: We demonstrate the application of a helium ion microscope with a beam diameter of 8 nm as a mask-less method for local domain patterning of magnetic thin-film systems. For a prototypical in-plane exchange-bias system the domain wall width has been investigated as a function of the angle between unidirectional anisotropy and domain wall. By shrinking the domain size of periodic domain stripes, we analyzed the influence of domain wall overlap on the domain stability. Finally, by changing the geometry of artificial two-dimensional domains, the influence of domain wall overlap and domain wall geometry on the ultimate domain size in the chosen system was analyzed. Conclusion: The application of a helium ion microscope for magnetic patterning has been shown. It allowed for exploring the fundamental limits of domain engineering in an in-plane exchange-bias thin film as a prototypical system. For two-dimensional domains the limit depends on the domain geometry. The relative orientation between domain wall and anisotropy axes is a crucial parameter and therefore influences the achievable minimum domain size dramatically

Recyclende Krylov-Unterraumverfahren für Folgen linearer Gleichungssysteme : Analyse und Anwendungen

Der LaTeX-Code sowie der Source-Code aller Experimente ist unter https://github.com/andrenarchy/phdthesis verfügbar.In vielen Anwendungen ist das Lösen einer Folge linearer Gleichungssysteme mit sich verändernden Matrizen und rechten Seiten erforderlich. Diese Dissertation widmet sich der Analyse und den Anwendungen von Recycling in Krylov-Unterraumverfahren um solche Folgen effizient zu lösen. Im Rahmen der Arbeit werden sowohl die Wohldefiniertheit von CG, MINRES und GMRES mit deflation als auch deren Verhältnis zu Methoden mit augmentation untersucht. Zudem werden die Auswirkungen von Störungen auf Projektionen, Spektren von projizierten Matrizen sowie Krylov-Unterraumverfahren im Allgemeinen studiert. Für Verfahren mit deflation führen die Untersuchungen auf Konvergenzschranken, die bei der automatischen Auswahl von Recycling-Daten als Entscheidungshilfe benutzt werden können. Eine neuartige Konvergenzschranke basiert auf approximativen Krylov-Unterräumen und verschafft auch im Falle nicht-normaler Matrizen wertvolle Informationen über das Konvergenzverhalten von GMRES. Numerische Experimente mit nichtlinearen Schrödinger-Gleichungen zeigen, dass die in der Arbeit hergeleiteten automatischen Recyclingstrategien die Gesamtrechenzeit um bis zu 40% senken.In several applications, one needs to solve a sequence of linear systems with changing matrices and right hand sides. This thesis concentrates on the analysis and application of recycling Krylov subspace methods for solving such sequences efficiently. The well-definedness of deflated CG, MINRES and GMRES methods and their relationship to augmentation is analyzed. Furthermore, the effects of perturbations on projections, spectra of deflated operators and Krylov subspace methods are studied. The analysis leads to convergence bounds for deflated methods which provide guidance for the automatic selection of recycling data. A novel approach is based on approximate Krylov subspaces and also gives valuable insight in the case of non-normal operators. Numerical experiments with nonlinear Schrödinger equations show that the overall time consumption is reduced by up to 40% by the derived automatic recycling strategies

Meshes and initial data for 2D and 3D experiments with PyNosh

<p>Data files used in Experiments with PyNosh.</p

Modification of the saturation magnetization of exchange bias thin film systems upon light-ion bombardment

Huckfeldt H, Gaul A, Müglich ND, et al. Modification of the saturation magnetization of exchange bias thin film systems upon light-ion bombardment. JOURNAL OF PHYSICS-CONDENSED MATTER. 2017;29(12): 125801.The magnetic modification of exchange bias materials by 'ion bombardment induced magnetic patterning' has been established more than a decade ago. To understand these experimental findings several theoretical models were introduced. Few investigations, however, did focus on magnetic property modifications caused by effects of ion bombardment in the ferromagnetic layer. In the present study, the structural changes occurring under ion bombardment were investigated by Monte-Carlo simulations and in experiments. A strong reduction of the saturation magnetization scaling linearly with increasing ion doses is observed and our findings suggest that it is correlated to the swelling of the layer material based on helium implantation and vacancy creation