On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations

AbstractThis work is motivated by the preconditioned iterative solution of linear systems that arise from the discretization of uniformly elliptic partial differential equations. Iterative methods with bounds independent of the discretization are possible only if the preconditioning strategy is based upon equivalent operators. The operators A, B: W → V are said to be V norm equivalent if ∥Au∥v∥Bu∥v is bounded above and below by positive constants for u ϵ D, where D is “sufficiently dense.” If A is V norm equivalent to B, then for certain discretization strategies one can use B to construct a preconditioned iterative scheme for the approximate solution of the problem Au = F. The iteration will require an amount of work that is at most a constant times the work required to approximately solve the problem Bû = \̂tf to reduce the V norm of the error by a fixed factor. This paper develops the theory of equivalent operators on Hubert spaces. Then, the theory is applied to uniformly elliptic operators. Both the strong and weak forms are considered. Finally, finite element and finite difference discretizations are examined

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

ENGAGING CLINICIANS IN A PRE-IMPLEMENTATION ASSESSMENT OF THE WOMEN & PERSON-EMPOWERED COMMUNITY ACCESS FOR REPRODUCTIVE EQUITY (WE CARE) INTERVENTION

Objectives: To assess clinicians’ perspectives on WE CARE (an emergency department (ED) family planning counseling and referral intervention that uses an online health tool and community health workers) to inform intervention design for implementation. Methods: We conducted one-on-one, semi-structured interviews with Emergency Medicine, Family Medicine, and Obstetrics & Gynecology clinicians until thematic saturation. The Consolidated Framework for Implementation Research (CFIR) informed the interview guide and was used to code all transcripts. A CFIR expert conducted an external coding audit. Results: We interviewed 30 clinicians (female (77%), ED staff (47%), white (63%), and attending physicians (43%)). WE CARE was highly acceptable. Dominant CFIR domains include: (1) Clinicians suggested Design Quality and Packaging modifications, particularly the referral processes, to promote successful implementation; (2) transportation and insurance were essential Patient Needs and Resources; (3) WE CARE was Compatible with the Value of “no missed opportunity” to help patients; (4) Compatibility with Work Processes – WE CARE posed scheduling and reimbursement challenges to clinics; (5) Clinicians expressed concerns about an ED Culture of reproductive health frustrations, resistance to change, and competing priorities. Others identified the ED “safety net” culture and long wait times as assets to the intervention; (6) WE CARE had a significant Relative Advantage over the status quo. A few clinicians identified more advantageous alternatives (e.g., WE CARE in the clinic, home, or community settings); (7) Engaging Key Stakeholders throughout the hospital was a critical implementation element. Conclusions: Clinicians contextualized several implementation constructs relevant to designing and implementing an ED family planning intervention

Methionine restriction improves renal insulin signalling in aged kidneys.

Dietary methionine restriction (MR) leads to loss of adiposity, improved insulin sensitivity and lifespan extension. The possibility that dietary MR can protect the kidney from age-associated deterioration has not been addressed. Aged (10-month old) male and female mice were placed on a MR (0.172% methionine) or control diet (0.86% methionine) for 8-weeks and blood glucose, renal insulin signalling, and gene expression were assessed. Methionine restriction lead to decreased blood glucose levels compared to control-fed mice, and enhanced insulin-stimulated phosphorylation of PKB/Akt and S6 in kidneys, indicative of improved glucose homeostasis. Increased expression of lipogenic genes and downregulation of PEPCK were observed, suggesting that kidneys from MR-fed animals are more insulin sensitive. Interestingly, renal gene expression of the mitochondrial uncoupling protein UCP1 was upregulated in MR-fed animals, as were the anti-ageing and renoprotective genes Sirt1, FGF21, klotho, and β-klotho. This was associated with alterations in renal histology trending towards reduced frequency of proximal tubule intersections containing vacuoles in mice that had been on dietary MR for 190days compared to control-fed mice, which exhibited a pre-diabetic status. Our results indicate that dietary MR may offer therapeutic potential in ameliorating the renal functional decline related to ageing and other disorders associated with metabolic dysfunction by enhancing renal insulin sensitivity and renoprotective gene expression