BiLQ: An Iterative Method for Nonsymmetric Linear Systems with a Quasi-Minimum Error Property

We introduce an iterative method named BiLQ for solving general square linear systems Ax = b based on the Lanczos biorthogonalization process defined by least-norm subproblems, and that is a natural companion to BiCG and QMR. Whereas the BiCG (Fletcher, 1976), CGS (Sonneveld, 1989) and BiCGSTAB (van der Vorst, 1992) iterates may not exist when the tridiagonal projection of A is singular, BiLQ is reliable on compatible systems even if A is ill-conditioned or rank deficient. As in the symmetric case, the BiCG residual is often smaller than the BiLQ residual and, when the BiCG iterate exists, an inexpensive transfer from the BiLQ iterate is possible. Although the Euclidean norm of the BiLQ error is usually not monotonic, it is monotonic in a different norm that depends on the Lanczos vectors. We establish a similar property for the QMR (Freund and Nachtigal, 1991) residual. BiLQ combines with QMR to take advantage of two initial vectors and solve a system and an adjoint system simultaneously at a cost similar to that of applying either method. We derive an analogous combination of USYMLQ and USYMQR based on the orthogonal tridiagonalization process (Saunders, Simon, and Yip, 1988). The resulting combinations, named BiLQR and TriLQR, may be used to estimate integral functionals involving the solution of a primal and an adjoint system. We compare BiLQR and TriLQR with Minres-qlp on a related augmented system, which performs a comparable amount of work and requires comparable storage. In our experiments, BiLQR terminates earlier than TriLQR and MINRES-QLP in terms of residual and error of the primal and adjoint systems

Spin-fluctuation spectra in magnetic systems: a novel approach based on TDDFT

Magnetism at the micro- and nano-scale level is a well-established research field, by virtue of its relentless technological impact and astounding variety of structures it can shape in condensed-matter systems. The characterization of most of these structures has become possible in the last fifty years thanks to the development and refinement of magnetic spectroscopies, most notably neutron scattering for bulk magnetism, and electron spectroscopies for surfaces and thin films. A fundamental outcome of the most recent experiments is the need to address magnetism in its full non-collinear nature also at the theoretical level, i.e. by treating the magnetization density as a true vector field, allowed to vary its direction at each point in space. This paves the way to the study of chiral topological magnetic structures such as skyrmions, or of the effect of Spin-Orbit Coupling (SOC) on the ground-state con- figuration and on the excited-state dynamics. Handling non-collinearity however, a far-from-trivial task on its own, proves to be particularly demanding in ab-initio calculations, where, at present, it is far from being a standard tool in the study of excited states. In this thesis we shall focus on the development of a method to study the dynamical spin-fluctuations of magnetic systems in a fully non-collinear framework, within Time-Dependent Density Function Theory (TDDFT). The outline of the thesis follows. In Ch. 1 the technological framework and the main experimental findings which have inspired our work are presented; a link between the experiments and the relevant physical quantities, namely the magnetic susceptibility, will also be shown. In Ch. 2 and 3 the theoretical framework in which we move will be introduced, namely Time-Dependent Density Functional Theory (TDDFT) and linear response. In Ch. 4 and Ch. 5 original work is presented: in the former, we devise a computational approach for the study of magnetic excitations via TDDFT, in a fully non-collinear framework. In the latter, we discuss the implementation and compute the spin-wave dispersion for BCC Iron. The final chapter is devoted to the conclusions

Automated Construction of Equivalent Electrical Circuit Models for Electromagnetic Components and Systems

The description of electromagnetic components and systems by electrical circuit models is indispensable for a wide range of applications: In the field of EMC, electrical circuit models are ideally suited for the detection of EMC coupling paths, which are very difficult to track for 3D geometries. In the field of numerical optimization techniques, electrical circuit models offer short simulation times and allow the coupling of the electromagnetic domain to other physical domains. In the field of power electronics, electrical circuit models describe energy dissipation due to parasitic electromagnetic interactions. The construction of an equivalent electrical circuit model is in general cumbersome and less formalized than a description in terms of electromagnetic fields. No general and reliable technique for the automated construction of equivalent electrical circuit models exists. The aim of this thesis is the development of a technique that allows a fully automated construction of equivalent electrical circuit models from 3D geometry information. Instead of constructing the circuit directly from geometry data, our approach consists of reducing a field-theoretical model to an equivalent electrical circuit model. In this way, we exploit the generality of the field-theoretical approach, which can be applied for a wide range of geometries using state-of-the-art simulation techniques. The electromagnetic effects having the largest impact in the frequency range of interest are then used for the construction of the electrical circuit model. The circuit elements can be seen as condensed representations of these field-theoretical processes. The reduction process allows a very direct assessment of the accuracy of the electrical circuit model

A general linear optimization algorithm based upon labeling and factorizing of basic paths on RPM network

A new method is developed for solving linear optimization problems based on the RPM network modeling technique which represents the primal and the corresponding dual models simultaneously upon a single graph. The network structure is used to eliminate the need for explicit logical variables and to provide a graphic tool in analyzing the problem. The new algorithm iterates through a finite number of basic solutions working towards optimality (primal) or towards feasibility (dual). At each iteration a set of critical constraints and basic structural variables are identified to form the current basic path network. A solution for the basic variables is obtained through factorization of the basis and used to update the nonbasic network. If the Kuhn-Tucker conditions are not satisfied, the method proceeds with the next iteration unless an unbounded or infeasible solution is encountered. Under the new scheme, the original data remains unchanged throughout the optimization procedure and round-off errors can be kept to a minimum. Furthermore, the basic paths representation used in factorization reduces computer core requirement and permits direct - addressing of pertinent non-basic node data on disk storage. These features are especially appealing in solving large-scale problems even on limited computer hardware. Since the size of the basis is never greater than the size of the basis required by simplex-type algorithms, the new scheme has an advantageous memory storage requirement. Any basic solution (not necessarily optimum or feasible) can be used as a starting point and multipivoting can accelerate the optimization process. In general, the number of iterations and the amount of operations depends on the sparsity of the constrained matrix and the complexity of the problem. Statistical data based on sample experimental results indicate that the new algorithm, on the average, requires less arithmetic operations and no more iterations to reach the final solution than the simplex-type algorithms