Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property

A Unified Analysis of Balancing Domain Decomposition by Constraints for Discontinuous Galerkin Discretizations

The BDDC algorithm is extended to a large class of discontinuous Galerkin (DG) discretizations of second order elliptic problems. An estimate of C(1 + log(H/h))2 is obtained for the condition number of the preconditioned system where C is a constant independent of h or H or large jumps in the coefficient of the problem. Numerical simulations are presented which confirm the theoretical results. A key component for the development and analysis of the BDDC algorithm is a novel perspective presenting the DG discretization as the sum of element-wise “local” bilinear forms. The element-wise perspective allows for a simple unified analysis of a variety of DG methods and leads naturally to the appropriate choice for the subdomain-wise local bilinear forms. Additionally, this new perspective enables a connection to be drawn between the DG discretization and a related continuous finite element discretization to simplify the analysis of the BDDC algorithm.Boeing CompanyMassachusetts Institute of Technology (Zakhartchenko Fellowship

BDDC and FETI-DP under Minimalist Assumptions

The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.Comment: 11 pages, 1 figure, also available at http://www-math.cudenver.edu/ccm/reports

Multispace and Multilevel BDDC

BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory.Comment: 26 pages, 3 figures, 2 tables, 20 references. Formal changes onl

Rescue of Photoreceptor Degeneration by Curcumin in Transgenic Rats with P23H Rhodopsin Mutation

The P23H mutation in the rhodopsin gene causes rhodopsin misfolding, altered trafficking and formation of insoluble aggregates leading to photoreceptor degeneration and autosomal dominant retinitis pigmentosa (RP). There are no effective therapies to treat this condition. Compounds that enhance dissociation of protein aggregates may be of value in developing new treatments for such diseases. Anti-protein aggregating activity of curcumin has been reported earlier. In this study we present that treatment of COS-7 cells expressing mutant rhodopsin with curcumin results in dissociation of mutant protein aggregates and decreases endoplasmic reticulum stress. Furthermore we demonstrate that administration of curcumin to P23H-rhodopsin transgenic rats improves retinal morphology, physiology, gene expression and localization of rhodopsin. Our findings indicate that supplementation of curcumin improves retinal structure and function in P23H-rhodopsin transgenic rats. This data also suggest that curcumin may serve as a potential therapeutic agent in treating RP due to the P23H rhodopsin mutation and perhaps other degenerative diseases caused by protein trafficking defects

\u3cem\u3eRPGRIP1\u3c/em\u3e and Cone-Rod Dystrophy in Dogs

Cone–rod dystrophies (crd) represent a group of progressive inherited blinding diseases characterized by primary dysfunction and loss of cone photoreceptors accompanying or preceding rod death. Recessive crd type 1 was described in dogs associated with an RPGRIP1 exon 2 mutation, but with lack of complete concordance between genotype and phenotype. This review highlights role of the RPGRIP1, a component of complex protein networks, and its function in the primary cilium, and discusses the potential mechanisms of genotype–phenotype discordance observed in dogs with the RPGRIP1 mutation