Complement Method Research Articles

A parallel Schur complement solver for the solution of the adjoint steady-state lattice Boltzmann equations: application to design optimisation

We study the numerical performance of a parallel Schur complement method towards the solution of the steady-state lattice Boltzmann equations on large meshes as needed for parametric design optimisation problems, in particular topology optimisation. For the gradient-based topology optimisation framework a porosity model is used to continuously transition from fluid to solid. Deriving the sensitivity equations for the lattice Boltzmann method (LBM), we identify the Jacobian of the LBM fixed-point formulation, which poses the key computational challenge in LBM topology optimisation due to its large size. We show that the Schur complement method can be used to decompose the fixed-point Jacobian, making the solution of the LBM sensitivity equations more memory efficient, in particular for 2D problems. The effectiveness of the overall lattice Boltzmann based topology optimisation framework is illustrated with the optimal design of a micro valve, posted by a dual-objective optimisation problem.

Efficient Production of Mice from Embryonic Stem Cells Injected into Four- or Eight-Cell Embryos by Piezo Micromanipulation

The conventional method for producing embryonic stem (ES) cell-derived knockout or transgenic mice involves injection of ES cells into normal, diploid blastocysts followed by several rounds of breeding of resultant chimeras and thus is a time-consuming and inefficient procedure. F0 ES cell pups can also be derived directly from tetraploid embryo complementation, which requires fusion of two-cell embryos. Recently, F0 ES cell pups have been produced by injection of ES cells into eight-cell embryos using a laser-assisted micromanipulation system. We report a simple method for producing F0 ES cell germline-competent mice by piezo injection of ES cells into four- or eight-cell embryos. The efficiency of producing live, transgenic mice by this method is higher than that with the tetraploid blastocyst complementation method. This efficient and economical technique for directly producing F0 ES cell offspring can be applicable in many laboratories for creating genetically manipulated mice using ES cell technology and also for stringent testing of the developmental potency of new ES cell or other types of pluripotent stem cell lines.

Genetically Encoded Bioluminescent Indicator for ERK2 Dimer in Living Cells

In this study, a genetically encoded bioluminescent indicator for ERK2 dimer was developed with the split Renilla luciferase complementation method, in which the formation of ERK2 dimer induces a spontaneous emission of bioluminescence in living cells. In response to extracellular stimuli, such as epidermal growth factor (EGF) or 17beta-estradiol (E2), extracellular signal-regulated kinase 2 (ERK2) is phosphorylated by its upstream kinase MEK, and also phosphorylates its substrates in various regions of the cell, including the nucleus. Phosphorylated ERK2 is led to form its dimer, thereby transporting itself into the nucleus. We demonstrated with the indicator that stimulation with EGF or E2 induces the formation of ERK2 dimer in living MCF-7 cells. The dynamics of this dimer formation was examined and discussed.

MRI study in urinary abnormalities of fetus

Objective To illustrate the important complemental function of MRI in dignosing the urinary abnormalities of the fetus by analyzing MR features.Methods MRI findings in 34 fetal urinary abnormalities were retrospectively analyzed.Results Upper urinary tract dilatation was found in 12 cases: one case presented obstructed right renal dysplasia and was on the follow-up,postnatal MR imaging proved the duplex anomaly in one case,one case showed left PUJO on postnatal US imaging and prepared to surgery,7 cases were normal on postnatal US imaging,2 cases were lost to follow up.Bilateral urinary anomalies were found in 7 cases:Multicystic renal dysplasia(n=3),Combined horseshoe kidney in 2 fetuses and bilateral renal aplasia in one case.Bilateral renal dysplasia was diagnosed in 2 cases,one was still born and proved by autopsy and the ohter was lost to follow up.The case of bilateral renal agenesis displayed the appearance of sirenomelia on general specimen.The case of right renal agenesis associated contralateral kidney dyspalsia(n=1)was lost to follow up.MR imaging showed low signal intensity of lung and oligohydramnios in the bilateral anomalies.Unilateral urinary anomalies was found in 15 cases, including 9 cases of unilateral renal dysplasia.Two fetuses were aborted and 3 fetuses were proved with postnatal US or MR.One was lost to follow up;3 cases were on the follow-up.There were 4 cases of unilateral renal agenesis,two fetuses were aborted and 2 fetuses were proved with postnatal US or MR imaging.The case of ectopic kidney was proved with postnatal US imaging.One case of urachal cyst was aborted without autopsy.In the unilateral anomalies,the volume of amniotic fluid was normal,and the fetal lung presented homogenious high signal intensity.Conclusion As a complemental method,MRI is of great value in displaying and dignosing the urinary abnormalities of fetus.

Projected Schur complement method for solving non‐symmetric systems arising from a smooth fictitious domain approach

AbstractThis paper deals with a fast method for solving large‐scale algebraic saddle‐point systems arising from fictitious domain formulations of elliptic boundary value problems. A new variant of the fictitious domain approach is analyzed. Boundary conditions are enforced by control variables introduced on an auxiliary boundary located outside the original domain. This approach has a significantly higher convergence rate; however, the algebraic systems resulting from finite element discretizations are typically non‐symmetric. The presented method is based on the Schur complement reduction. If the stiffness matrix is singular, the reduced system can be formulated again as another saddle‐point problem. Its modification by orthogonal projectors leads to an equation that can be efficiently solved using a projected Krylov subspace method for non‐symmetric operators. For this purpose, the projected variant of the BiCGSTAB algorithm is derived from the non‐projected one. The behavior of the method is illustrated by examples, in which the BiCGSTAB iterations are accelerated by a multigrid strategy. Copyright © 2007 John Wiley & Sons, Ltd.

On the Schur complement form of the Dirichlet-to-Neumann operator

The Dirichlet-to-Neumann operator relates the values of the normal derivative of a scalar field to the values of the field itself on the boundary of a simply connected domain. Although it is easy to prove analytically that the Dirichlet-to-Neumann operator is self-adjoint, the discretization of the pertinent Green’s integral equation by means of the usual Galerkin approach generally results in a non-symmetric matrix representation of that operator. In this paper we remedy this by means of a Hamiltonian Schur complement method and compare it with other symmetrization approaches. It is also shown that the problem of E-wave scattering by a lossy dielectric cylinder can be nicely simplified by means of the Dirichlet-to-Neumann operator. A numerical Galerkin implementation demonstrates the strength and versatility of the present approach.

An assessment of the genetic diversity within Ganoderma strains with AFLP and ITS PCR-RFLP

Ganoderma lucidum is one of the most important medicinal materials and plant pathogens. Because of its specific interhybridization, the genetic background, however, is relatively unclear. It made identification of Ganoderma strains, especially closely related strains difficulty. Amplified fragment length polymorphism (AFLP) using 14 primer combinations and internal transcribed spacer (ITS) PCR-RFLP were used in a comparative study which was designed to investigate the closely related Ganoderma strains genetic relations at molecular level. The analysis of 37 Ganoderma strains showed there were 177 polymorphic AFLP markers and 12 ITS PCR-RFLP markers, and all accessions could be uniquely identified. Among the Ganoderma accessions, similarity coefficients ranged from 0.07692 to 0.99194 in AFLP. The Ganoderma strains formed a tight cluster in nine groups in AFLP whereas seven groups in ITS PCR-RFLP. The cluster analysis revealed that the taxonomical system of subgenus Ganoderma is composed of Sect. Ganoderma and Sect. Phaeonema, and the strain 22 should be a variant form of strain 21. All methods delineated the Ganoderma strains from the different regions seeming to show a greater level of genetic diversity. It indicated that the genotype study at molecular level is a useful complement method to the current classification system of Ganoderma strains based on morphological traits. The congruency of the experiments was analyzed using the biostatistical software DPS V3.01.

Specificity, promiscuity and localization of ARF protein interactions with NCS-1 and phosphatidylinositol-4 kinase-III beta.

ADP-ribosylation factor (ARF) proteins are involved in multiple intracellular vesicular transport pathways. Most studies have focused on the functions of ARF1 or ARF6 and little is known about the remaining ARF isoforms. Although the mammalian ARF proteins share a high degree of sequence identity, recent evidence has indicated that they may control distinct trafficking steps within cells. A unanswered issue is the degree of specificity of ARF family members for different interacting proteins. To investigate potential functional differences between the human ARF proteins, we have examined the localization of all human ARF isoforms and their interactions with two ARF1 binding proteins, neuronal calcium sensor-1 (NCS-1) and phosphatidylinositol-4 kinase-IIIbeta (PI4Kbeta). Use of a fluorescent protein fragment complementation method showed direct interactions between ARFs 1, 3, 5 and 6 with NCS-1 but at different intracellular locations in live HeLa cells. Photobleaching experiments indicated that complementation did not detect dynamic changes in protein interactions over short-time scales. A more specific interaction between ARFs 1/3 and PI4Kbeta was observed. Consistent with these latter findings ARF1 but not ARF5 or 6 enhanced the stimulatory effect of PI4Kbeta on regulated exocytosis, suggesting a specific role for class-I ARFs in the regulation of PI4Kbeta.

Production of Identical Mouse Twins and a Triplet with Predicted Gender

The aim of this study was to develop a method to generate identical twins and triplets with predicted gender. As a first step toward that aim, single blastomeres obtained from EGFP expressing eight-cell stage embryos and either diploid or tetraploid host embryos were used to compose chimera. We could follow the fate of EGFP expressing diploid blastomere derived cells in 3.5- and 4.5-day-old chimera embryos in vitro. We found that the diploid blastomere-derived cells had significantly higher chance to contribute to the inner cell mass if tetraploid host embryos were applied. After that, we developed a quick and reliable multiplex PCR strategy for sex diagnosis from single blastomeres by simultaneous amplification of the homologous ZFX and ZFY genes. By composed chimeras using single blastomeres, derived from sexed eight-cell stage embryos and a tetraploid host embryo, we could get preplanned sex newborns, wholly derived from these blastomeres. Among these mice, identical twins and a triplet were identified by microsatellite analysis. Unlike clones produced by nuclear transfer, these mice are identical at both the nuclear as well as mitochondrial DNA level. Therefore, the tetraploid embryo complementation method to produce monozygotic twins and triplets could be a valuable tool both in biomedical and agricultural applications.

AtMTM1, a novel mitochondrial protein, may be involved in activation of the manganese-containing superoxide dismutase in Arabidopsis

Mtm1p is essential for the posttranslational activation of manganese-containing superoxide dismutase (SOD2) in Saccharomyces cerevisiae; however, whether the same holds true for Arabidopsis thaliana is unknown. In this study, by using the yeast mtm1 mutant complementation method, we identified a putative MTM gene (AtMTM1, At4g27940) that is necessary for SOD2 activation. Further, analysis of SOD activity revealed that an SOD2 defect is rescued in the yeast mutant Y07288 harboring the AtMTM1 gene. Related mRNA-level analysis showed the AtMTM1 gene is induced by paraquat but not by hydrogen peroxide, which indicates that this gene is related to the superoxide scavenger SOD. In addition, an AtMTM1::GFP fusion construct was transiently expressed in the protoplasts, and it was localized to the mitochondria. Furthermore, sequence deletion analysis of AtMTM1 revealed that the code region (amino acid (aa) 60-198) of Mtm1p plays an important role in localization of the protein to the mitochondria. Regulation of AtMTM1 gene expression was analyzed using a fusion construct of the 1,766 bp AtMTM1 promoter and the GUS (beta-glucuronidase) reporter gene. The screen identified GUS reporter gene expression in the developing cotyledons, leaves, roots, stems, and flowers but not in the siliques. Our results suggest that AtMTM1 encodes a mitochondrial protein that may be playing an important role in activation of MnSOD1 in Arabidopsis.

Elementary linear algebra for advanced spectral problems

We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators of mathematical physics.

Parallel Segregated Schur Complement Methods for Fluid Density Functional Theories

Numerical formulations of density functional theories for inhomogeneous fluids (fluid-DFTs) require the solution of large systems of equations with many degrees of freedom (DOF) per node on a computational grid. Historically, solvers for these problems have used simple Picard iterations across DOF or, more recently, fully coupled general algebraic techniques. In this paper we look at fluid-DFTs from a fresh perspective, retaining a fully coupled formulation but segregating variables for the purpose of introducing Schur complement formulations and specialized preconditioners. By viewing fluid-DFTs from this perspective, we develop a mathematical framework and a collection of solution algorithms that have a dramatic impact on the robustness, performance, and scalability of the implicit equations generated by fluid-DFTs.

Self-similar groups, operator algebras and Schur complement

In the first part of the article we introduce $C$*-algebras associated to self-similar groups and study their properties and relations to known algebras. The algebras are constructed as subalgebras of the Cuntz-Pimsner algebra (and its homomorphic images) associated with the self-similarity of the group. We study such properties as nuclearity, simplicity and Morita equivalence with algebras related to solenoids. &nbsp The second part deals with Schur complement transformations of elements of self-similar algebras. We study the properties of such transformations and apply them to the spectral problem for Markov type elements in self-similar $C$*-algebras. This is related to the spectral problem of the discrete Laplace operator on groups and graphs. Application of the Schur complement method in many situations reduces the spectral problem to study of invariant sets (very often of the type of a strange attractor'') of a multidimensional rational transformation. A number of illustrating examples is provided. Finally, we observe a relation between Schur complement transformations and Bartholdi-Kaimanovich-Virag transformations of random walks on self-similar groups.

Static load balancing applied to Schur complement method

A finite element method often leads to large sparse symmetric and positive definite systems of linear equations. We consider parallel solvers based on the Schur complement method on homogeneous parallel machines with distributed memory. A finite element mesh is partitioned by graph partitioning. Such partitioning results in submeshes with similar numbers of elements and, consequently, submatrices of similar sizes. The submatrices are partially factorised. The time spent on the partial factorisation can be different, i.e., disbalanced, because methods exploiting the sparsity of submatrices are used. This paper proposes a Quality Balancing heuristic that modifies classic mesh partitioning so that the partial factorisation times are balanced, which saves overall computation time, especially for time dependent mechanical and nonstationary transport problems.

Neumann–Neumann–Schur complement methods for Fekete spectral elements

For the iterative solution of the Schur complement system associated with the discretization of an elliptic problem by means of a triangular spectral element method (TSEM), Neumann-Neumann (NN) type preconditioners are constructed and studied. The TSEM approximation, based on Fekete nodes, is a generalization to non-tensorial elements of the classical Gauss-Lobatto-Legendre quadrilateral spec- tral elements. Numerical experiments show that the TSEM Schur complement condition number grows linearly with the polynomial approximation degree, N, and quadratically with the inverse of the mesh size, h. NN preconditioners for the Schur complement allow to reduce the N-dependence of the condition number, by solving local Neumann problems on each spectral element, and to eliminate the h-dependence if an additional coarse solver is employed. Numerical results indicate that, in spite of the more severe ill-conditioning, the condition number of the TSEM preconditioned operator satisfies the same bound as that of the standard SEM, i.e., Ch −2 (1 + logN) 2 for one-level NN preconditioning and C(1 + logN) 2 for two-level Balancing Neumann-Neumann (BNN) preconditioning.

Time-dependent Maxwell’s equations with charges in singular geometries

This paper is devoted to the solution of the instationary Maxwell equations with charges. The geometry of the domain can be singular, in the sense that its boundary can include reentrant corners or edges. The difficulties arise from the fact that those geometrical singularities generate, in their neighborhood, strong electromagnetic fields. The time-dependency of the divergence of the electric field, is addressed. To tackle this problem, some new theoretical and practical results are presented, on curl-free singular fields, and on singular fields with L 2 (non-vanishing) divergence. The method, which allows to compute the instationary electromagnetic field, is based on a splitting of the spaces of solutions into a two-term direct sum. First, the subspace of regular fields: it coincides with the whole space of solutions, provided that the domain is either convex, or with a smooth boundary. Second, a singular subspace, defined and characterized via the singularities of the Laplace operator. Several numerical examples are presented, to illustrate the mathematical framework. This paper is the generalization of the singular complement method.

A parallel time–space least-squares spectral element solver for incompressible flow problems

The least-squares spectral element method (LSQ-SEM) is a very promising numerical method which combines the least-squares formulation with the spectral element method. In this article, a parallel space–time implementation of a least-squares spectral element method for incompressible flows is presented. In the space–time formulation, time is included as an extra dimension, providing an efficient framework for handling time dependent independent multidimensional problems. The algebraic solver is based on the Shur complement method. Numerical experiments are performed for showing the capability and computational efficiency of the solver. Besides, the scalability of the code is tested.

QPSchur: A dual, active-set, Schur-complement method for large-scale and structured convex quadratic programming

We describe an active-set, dual-feasible Schur-complement method for quadratic programming (QP) with positive definite Hessians. The formulation of the QP being solved is general and flexible, and is appropriate for many different application areas. Moreover, the specialized structure of the QP is abstracted away behind a fixed KKT matrix called K o and other problem matrices, which naturally leads to an object-oriented software implementation. Updates to the working set of active inequality constraints are facilitated using a dense Schur complement, which we expect to remain small. Here, the dual Schur complement method requires the projected Hessian to be positive definite for every working set considered by the algorithm. Therefore, this method is not appropriate for all QPs. While the Schur complement approach to linear algebra is very flexible with respect to allowing exploitation of problem structure, it is not as numerically stable as approaches using a QR factorization. However, we show that the use of fixed-precision iterative refinement helps to dramatically improve the numerical stability of this Schur complement algorithm. The use of the object-oriented QP solver implementation is demonstrated on two different application areas with specializations in each area; large-scale model predictive control (MPC) and reduced-space successive quadratic programming (with several different representations for the reduced Hessian). These results demonstrate that the QP solver can exploit application-specific structure in a computationally efficient and fairly robust manner as compared to other QP solver implementations.

On the efficiency and quality of numerical solutions in CFD problems using the interface strip preconditioner for domain decomposition methods

AbstractIn this paper, some pathologies found for simple tests solved by means of preconditioned full iterative schemes are presented. According to these results (Sections 4 and 5), the accuracy deterioration observed should be considered as a warning for the final application given to these solutions. Even though it is well known that full iterative solvers are not the best selection for comparison, they were chosen because they are widely used by the computational fluid dynamic (CFD) community for a diversity of complex fluid dynamics applications.FEM simulated solutions are compared with analytical solutions or measured data for problems that have been considered as ‘benchmarks’ in the CFD literature. For this purpose, the study of the solution obtained via parallelized iterative methods that have been extensively used (e.g. conjugate gradients (CG), GMRes global iteration and its variants, ‘overlapping’ and ‘non‐overlapping’ additive Schwarz domain decomposition schemes) in CFD computations and those obtained with the new interface strip preconditioner (J. Comput. Meth. Sci. Engng 2003; Int. J. Numer. Meth. Engng 2005; 62(13):1873–1894) for the Schur complement method is carried out. The idea is to present the new solver as an alternative to obtain more accurate and faster solutions in the context of monolithic and non‐monolithic schemes applied to a internal/external viscous compressible/incompressible flows around bodies of complex shapes.Therefore, the target of this work is to show how the reliability of CFD codes is affected by the solver selection and why domain decomposition methods should be viewed not only as a more efficient strategy, but also to guarantee the solution quality. Copyright © 2006 John Wiley & Sons, Ltd.

Thermodynamic properties of ferrimagnetic spin chains in the presence of a magnetic field

We have implemented three approaches to describe the thermodynamic properties of ferrimagnetic ($S=5/2, s=2$) spin chains. The application of cumulant expansion has been generalized to the ferrimagnetic chain in the presence of an external magnetic field. Using cumulants, we have obtained the field dependent effective Hamiltonian in terms of the classical variables up to the second order of quantum corrections. Thermodynamic functions, the internal energy, the specific heat and the magnetic susceptibility are obtained from the effective Hamiltonian. We have also examined the modified spin wave theory to derive the same physical properties. Finally, we have studied our model using quantum Monte Carlo simulation to obtain accurate results. The comparison of the above results and also the high temperature series expansion shows that cumulant expansion gives good results for moderate and high temperature regions while the modified spin wave theory is good for low temperatures. Moreover, the convergence regions of the cumulant expansion and the modified spin wave theory overlap each other which propose these two as a set of complement methods to get the thermodynamic properties of spin models.