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Abstract In this paper, we give a new short proof of the local quadratic convergence of the Dominant Pole Spectrum Eigensolver (DPSE). Also, we introduce here the Diagonal Dominant Pole Spectrum Eigensolver (DDPSE), another fixed-point method that computes several eigenvalues of a matrix A at a time, which also has local quadratic convergence. From results of some experiments with a large power system model, it is shown that DDPSE can also be used in small-signal stability studies to compute dominant poles of a transfer function of the type c T (A − s I) − 1 b , where s ∈ C , b and c are vectors, by its own or combined with DPSE. Besides DDPSE is also effective in finding low damped modes of a large scale power system model. [ABSTRACT FROM AUTHOR]

We study the ridge regression ( L 2 regularized least squares) problem and its dual, which is also a ridge regression problem. We observe that the optimality conditions describing the primal and dual optimal solutions can be formulated in several different but equivalent ways. The optimality conditions we identify form a linear system involving a structured matrix depending on a single relaxation parameter which we introduce for regularization purposes. This leads to the idea of studying and comparing, in theory and practice, the performance of the fixed point method applied to these reformulations. We compute the optimal relaxation parameters and uncover interesting connections between the complexity bounds of the variants of the fixed point scheme we consider. These connections follow from a close link between the spectral properties of the associated matrices. For instance, some reformulations involve purely imaginary eigenvalues; some involve real eigenvalues and others have all eigenvalues on the complex circle. We show that the deterministic Quartz method—which is a special case of the randomized dual coordinate ascent method with arbitrary sampling recently developed by Qu, Richtárik and Zhang—can be cast in our framework, and achieves the best rate in theory and in numerical experiments among the fixed point methods we study. Remarkably, the method achieves an accelerated convergence rate. Numerical experiments indicate that our main algorithm is competitive with the conjugate gradient method. [ABSTRACT FROM AUTHOR]

Continued advances in computational power and methods have enabled image-based biomechanical modeling to become an important tool in basic science, diagnostic and therapeutic medicine, and medical device design. One of the many challenges of this approach, however, is identification of a stress-free reference configuration based on in vivo images of loaded and often prestrained or residually stressed soft tissues and organs. Fortunately, iterative methods have been proposed to solve this inverse problem, among them Sellier’s method. This method is particularly appealing because it is easy to implement, convergences reasonably fast, and can be coupled to nearly any finite element package. By means of several practical examples, however, we demonstrate that in its original formulation Sellier’s method is not optimally fast and may not converge for problems with large deformations. Fortunately, we can also show that a simple, inexpensive augmentation of Sellier’s method based on Aitken’s delta-squared process can not only ensure convergence but also significantly accelerate the method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the existence of solutions to boundary value problems (BVPs) involving systems of nonlinear first-order ordinary differential equations and two-point, anti-periodic boundary conditions. A new existence result for the BVPs above is obtained by using fixed-point theory. [Copyright &y& Elsevier]

Abstract: This article investigates both basic qualitative and basic quantitative properties of solutions to first- and higher-order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field. Particular focus lies in the: existence; uniqueness; dependency; approximation; and explicit representation, of solutions to nonlinear initial value problems. The main tools used are from modern areas of nonlinear analysis, including: the fixed-point theorems of Banach and Schäfer; the method of successive approximations; a novel definition of measuring distance in metric spaces and normed spaces; and a “separation” of variables technique is introduced to the general time scale setting. The new results compliment and extend those of Stefan Hilger’s seminal paper of 1990. As an application of the new results we present and analyse a simple model from economics, known as the Keynesian–Cross model with “lagged” income, in the general time scale environment. Ideas suggesting further applications and possible new directions for the novel results are also presented. [Copyright &y& Elsevier]

In this paper, we give a formal derivation of several systems of equations for injection moulding. This is done starting from the basic equations for nonisothermal, non-Newtonian flows in a three-dimensional domain. We derive systems for both (T0, p0) and (T1, p1) in the presence of body forces and sources. We find that body forces and sources have a nonlinear effect on the systems. We also derive a nonlinear “Darcy law”. Our formulation includes not only the pressure gradient, but also body forces and sources, which play the role of a nonlinearity. Later, we prove the existence of weak solutions to certain boundary value problems and initial-boundary value problems associated with the resulting equations for (T0,p0) but in a more general mathematical setting. [Copyright &y& Elsevier]