Using constraint preconditioners with regularized saddle-point problems

The problem of finding good preconditioners for the numerical solution of a certain important class of indefinite linear systems is considered. These systems are of a 2 by 2 block (KKT) structure in which the (2,2) block (denoted by -C) is assumed to be nonzero. In Constraint preconditioning for indefinite linear systems , SIAM J. Matrix Anal. Appl., 21 (2000), Keller, Gould and Wathen introduced the idea of using constraint preconditioners that have a specific 2 by 2 block structure for the case of C being zero. We shall give results concerning the spectrum and form of the eigenvectors when a preconditioner of the form considered by Keller, Gould and Wathen is used but the system we wish to solve may have C \neq 0 . In particular, the results presented here indicate clustering of eigenvalues and, hence, faster convergence of Krylov subspace iterative methods when the entries of C are small; such situations arise naturally in interior point methods for optimization and we present results for such problems which validate our conclusions.\ud \ud The first author's work was supported by the OUCL Doctorial Training Accoun

Error estimation in reduced basis method for systems with time-varying and nonlinear boundary conditions

Many physical phenomena, such as mass transport and heat transfer, are modeled by systems of partial differential equations with time-varying and nonlinear boundary conditions. Control inputs and disturbances typically affect the system dynamics at the boundaries and a correct numerical implementation of boundary conditions is therefore crucial. However, numerical simulations of high-order discretized partial differential equations are often too computationally expensive for real-time and many-query analysis. For this reason, model complexity reduction is essential. In this paper, it is shown that the classical reduced basis method is unable to incorporate time-varying and nonlinear boundary conditions. To address this issue, it is shown that, by using a modified surrogate formulation of the reduced basis ansatz combined with a feedback interconnection and a input-related term, the effects of the boundary conditions are accurately described in the reduced-order model. The results are compared with the classical reduced basis method. Unlike the classical method, the modified ansatz incorporates boundary conditions without generating unphysical results at the boundaries. Moreover, a new approximation of the bound and a new estimate for the error induced by model reduction are introduced. The effectiveness of the error measures is studied through simulation case studies and a comparison with existing error bounds and estimates is provided. The proposed approximate error bound gives a finite bound of the actual error, unlike existing error bounds that grow exponentially over time. Finally, the proposed error estimate is more accurate than existing error estimates

Error estimates for model order reduction of Burgers' equation

Burgers' equation is a nonlinear scalar partial differential equation, commonly used as a testbed for model order reduction techniques and error estimates. Model order reduction of the parameterized Burgers' equation is commonly done by using the reduced basis method. In this method, an error estimate plays a crucial rule in both accelerating the offline phase and quantifying the error induced after reduction in the online phase. In this study, we introduce two new estimates for this reduction error. The first error estimate is based on a Lur'e-type model formulation of the system obtained after the full-discretization of Burgers' equation. The second error estimate is built upon snapshots generated in the offline phase of the reduced basis method. The second error estimate is applicable to a wider range of systems compared to the first error estimate. Results reveal that when conditions for the error estimates are satisfied, the error estimates are accurate and work efficiently in terms of computational effort

Proximal adductor avulsions are rarely isolated but usually involve injury to the PLAC and pectineus: Descriptive MRI findings in 145 athletes

Purpose The purpose of the study is to review the MRI findings in a cohort of athletes who sustained acute traumatic avulsions of the adductor longus fibrocartilaginous entheses, and to investigate related injuries namely the pyramidalis- anterior pubic ligament - adductor longus complex (PLAC). Associated muscle and soft tissue injuries were also assessed. Methods The MRIs were reviewed for a partial or complete avulsion of the adductor longus fibrocartilage, as well as continuity or separation of the adductor longus from the pyramidalis. The presence of a concurrent partial pectineus tear was noted. Demographic data was analysed. Linear and logistic regression was used to examine associations between injuries. Results The mean age was 32.5 (SD 10.9). The pyramidalis was absent in 3 of 145 patients. 85 of 145 athletes were professional and 52 competed in the football Premier League. 132 had complete avulsions and 13 partial. The adductor longus was in continuity with pyramidalis in 55 athletes, partially separated in seven and completely in 81 athletes. 48 athletes with a PLAC injury had a partial pectineus avulsion. Six types of PLAC injuries patterns were identified. Associated rectus abdominis injuries were rare and only occurred in five patients (3.5%). Conclusion The proximal adductor longus forms part of the PLAC and is rarely an isolated injury. The term PLAC injury is more appropriate term. MRI imaging should assess all the anatomical components of the PLAC post-injury, allowing recognition of the differentpatterns of injury

Genome-wide search for breast cancer linkage in large Icelandic non-BRCA1/2 families

Abstract Introduction: A significant proportion of high-risk breast cancer families are not explained by mutations in known genes. Recent genome-wide searches (GWS) have not revealed any single major locus reminiscent of BRCA1 and BRCA2, indicating that still unidentified genes may explain relatively few families each or interact in a way obscure to linkage analyses. This has drawn attention to possible benefits of studying populations where genetic heterogeneity might be reduced. We thus performed a GWS for linkage on nine Icelandic multiple-case non-BRCA1/2 families of desirable size for mapping highly penetrant loci. To follow up suggestive loci, an additional 13 families from other Nordic countries were genotyped for selected markers. Methods: GWS was performed using 811 microsatellite markers providing about five centiMorgan (cM) resolution. Multipoint logarithm of odds (LOD) scores were calculated using parametric and nonparametric methods. For selected markers and cases, tumour tissue was compared to normal tissue to look for allelic loss indicative of a tumour suppressor gene. Results: The three highest signals were located at chromosomes 6q, 2p and 14q. One family contributed suggestive LOD scores (LOD 2.63 to 3.03, dominant model) at all these regions, without consistent evidence of a tumour suppressor gene. Haplotypes in nine affected family members mapped the loci to 2p23.2 to p21, 6q14.2 to q23.2 and 14q21.3 to q24.3. No evidence of a highly penetrant locus was found among the remaining families. The heterogeneity LOD (HLOD) at the 6q, 2p and 14q loci in all families was 3.27, 1.66 and 1.24, respectively. The subset of 13 Nordic families showed supportive HLODs at chromosome 6q (ranging from 0.34 to 1.37 by country subset). The 2p and 14q loci overlap with regions indicated by large families in previous GWS studies of breast cancer. Conclusions: Chromosomes 2p, 6q and 14q are candidate sites for genes contributing together to high breast cancer risk. A polygenic model is supported, suggesting the joint effect of genes in contributing to breast cancer risk to be rather common in non-BRCA1/2 families. For genetic counselling it would seem important to resolve the mode of genetic interaction

Iterative Solution of Linear Systems in Circuit Simulation

An overview is given of iterative techniques for the solution of linear systems which occur during the simulation of electronic circuits. In developing a suitable method, several characteristics of electronic circuits have been used. The ordering of the unknowns is based on the observation that two types exist, namely currents and voltages. Furthermore, the linear systems are of a hierarchical structure which is quite different from what is found in discretized partial differential equations. Methods have been developed which make use of the aforementioned characteristics, and which are very suitable for the solution of large linear systems