An odyssey into local refinement and multilevel preconditioning III: Implementation and numerical experiments

In this paper, we examine a number of additive and multiplicative multilevel iterative methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard multilevel methods are effective for uniform refinement-based discretizations of elliptic equations, they tend to be less effective for algebraic systems, which arise from discretizations on locally refined meshes, losing their optimal behavior in both storage and computational complexity. Our primary focus here is on Bramble, Pasciak, and Xu (BPX)-style additive and multiplicative multilevel preconditioners, and on various stabilizations of the additive and multiplicative hierarchical basis (HB) method, and their use in the local mesh refinement setting. In parts I and II of this trilogy, it was shown that both BPX and wavelet stabilizations of HB have uniformly bounded condition numbers on several classes of locally refined two- and three-dimensional meshes based on fairly standard (and easily implementable) red and red-green mesh refinement algorithms. In this third part of the trilogy, we describe in detail the implementation of these types of algorithms, including detailed discussions of the data structures and traversal algorithms we employ for obtaining optimal storage and computational complexity in our implementations. We show how each of the algorithms can be implemented using standard data types, available in languages such as C and FORTRAN, so that the resulting algorithms have optimal (linear) storage requirements, and so that the resulting multilevel method or preconditioner can be applied with optimal (linear) computational costs. We have successfully used these data structure ideas for both MATLAB and C implementations using the FEtk, an open source finite element software package. We finish the paper with a sequence of numerical experiments illustrating the effectiveness of a number of BPX and stabilized HB variants for several examples requiring local refinement

Robust and parallel scalable iterative solutions for large-scale finite cell analyses

The finite cell method is a highly flexible discretization technique for numerical analysis on domains with complex geometries. By using a non-boundary conforming computational domain that can be easily meshed, automatized computations on a wide range of geometrical models can be performed. Application of the finite cell method, and other immersed methods, to large real-life and industrial problems is often limited due to the conditioning problems associated with these methods. These conditioning problems have caused researchers to resort to direct solution methods, which signifi- cantly limit the maximum size of solvable systems. Iterative solvers are better suited for large-scale computations than their direct counterparts due to their lower memory requirements and suitability for parallel computing. These benefits can, however, only be exploited when systems are properly conditioned. In this contribution we present an Additive-Schwarz type preconditioner that enables efficient and parallel scalable iterative solutions of large-scale multi-level hp-refined finite cell analyses.Comment: 32 pages, 17 figure

Effect of helium pre- or postconditioning on signal transduction kinases in patients undergoing coronary artery bypass graft surgery

Background: The noble gas helium induces pre- and postconditioning in animals and humans. Volatile anesthetics induce cardioprotection in humans undergoing coronary artery bypass graft (CABG) surgery. We hypothesized that helium induces pre-and postconditioning in CABG-patients, affecting signaling molecules protein kinase C-epsilon (PKC-epsilon), p38 mitogen activated protein kinase (p38 MAPK), extracellular signal-regulated kinase 1/2 (ERK-1/2) and heat shock protein 27 (HSP-27) within cardiac tissue, and reducing postoperative troponin levels. Methods: After ethical approval and informed consent, 125 elective patients undergoing CABG surgery were randomised into this prospective, placebo controlled, investigator blinded, parallel arm single-centre study. Helium preconditioning (3 x 5 min of 70 % helium and 30 % oxygen) was applied before aortic cross clamping; postconditioning (15 min of helium) was applied before release of the aortic cross clamp. Signaling molecules were measured in right atrial appendix specimens. Troponin-T was measured at 4, 12, 24 and 48 h postoperatively. Results: Baseline characteristics of all groups were similar. Helium preconditioning did not significantly alter the primary outcome (molecular levels of kinases PKC-e and HSP-27, ratio of activated p38 MAPK or ERK 1/2). Postoperative troponin T was 11 arbitrary units [5, 31; area-under-the-curve (interquartile range)] for controls, and no statistically significant changes were observed after helium preconditioning [He-pre: 11 (6, 18)], helium postconditioning [He-post: 11 (8, 15)], helium pre-and postconditioning [He-PP: 14 (6, 20)] and after sevoflurane preconditioning [APC: 12 (8, 24), p = 0.13]. No adverse effects related to study treatment were observed in this study. Conclusions: No effect was observed of helium preconditioning, postconditioning or the combination thereof on activation of p38 MAPK, ERK 1/2 or levels of HSP27 and PKC-e in the human heart. Helium pre-and postconditioning did not affect postoperative troponin release in patients undergoing CABG surgery

Two Weeks of Ischemic Conditioning Improves Walking Speed and Reduces Neuromuscular Fatigability in Chronic Stroke Survivors

This pilot study examined whether ischemic conditioning (IC), a noninvasive, cost-effective, and easy-to-administer intervention, could improve gait speed and paretic leg muscle function in stroke survivors. We hypothesized that 2 wk of IC training would increase self-selected walking speed, increase paretic muscle strength, and reduce neuromuscular fatigability in chronic stroke survivors. Twenty-two chronic stroke survivors received either IC or IC Sham on their paretic leg every other day for 2 wk (7 total sessions). IC involved 5-min bouts of ischemia, repeated five times, using a cuff inflated to 225 mmHg on the paretic thigh. For IC Sham, the cuff inflation pressure was 10 mmHg. Self-selected walking speed was assessed using the 10-m walk test, and paretic leg knee extensor strength and fatigability were assessed using a Biodex dynamometer. Self-selected walking speed increased in the IC group (0.86 ± 0.21 m/s pretest vs. 1.04 ± 0.22 m/s posttest, means ± SD; P\u3c 0.001) but not in the IC Sham group (0.92 ± 0.47 m/s pretest vs. 0.96 ± 0.46 m/s posttest; P= 0.25). Paretic leg maximum voluntary contractions were unchanged in both groups (103 ± 57 N·m pre-IC vs. 109 ± 65 N·m post-IC; 103 ± 59 N·m pre-IC Sham vs. 108 ± 67 N·m post-IC Sham; P = 0.81); however, participants in the IC group maintained a submaximal isometric contraction longer than participants in the IC Sham group (278 ± 163 s pre-IC vs. 496 ± 313 s post-IC, P = 0.004; 397 ± 203 s pre-IC Sham vs. 355 ± 195 s post-IC Sham; P = 0.46). The results from this pilot study thus indicate that IC training has the potential to improve walking speed and paretic muscle fatigue resistance poststroke