Singular Blocks Research Articles

Two block triangular preconditioners for asymmetric saddle point problems

In this paper, two block triangular preconditioners for the asymmetric saddle point problems with singular (1,1) block are presented. The spectral characteristics of the preconditioned matrices are discussed in detail. Theoretical analysis shows that all the eigenvalues of the preconditioned matrices are strongly clustered. Numerical experiments are reported to the efficiency of the proposed preconditioners.

Some homological properties of category [formula omitted]. III

We prove that thick category O associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick category O and the associated categories of Harish–Chandra bimodules, but disprove it for singular blocks.

Spectrum analysis of a more general augmentation block preconditioner for generalized saddle point matrices

In this paper, some errors are pointed out and the correct results are given for the paper Axelsson et al. (J Comput Appl Math 280:141–157, 2015). Then, a class of general augmentation block preconditioners for solving generalized saddle point systems with singular (1, 1) blocks are considered. Results concerning the eigenvalue distribution and forms of the eigenvectors of the preconditioned generalized saddle point matrix are presented. These results extend previous one in the literature.

Categorification of a parabolic Hecke module via sheaves on moment graphs

We investigate certain categories, associated by Fiebig with the geometric representation of a Coxeter system, via sheaves on Bruhat graphs. We modify Fiebig's definition of translation functors in order to extend it to the singular setting and use it to categorify a parabolic Hecke module. As an application we obtain a combinatorial description of indecomposable projective objects of (truncated) non-critical singular blocks of (a deformed version of) category $\mathcal{O}$, using indecomposable special modules over the structure algebra of the corresponding Bruhat graph.

On Augmented Lagrangian Methods for Saddle-Point Linear Systems with Singular or Semidefinite (1, 1) Blocks

An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite (1, 1) blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems. Mathematics subject classification: 65F10, 65F50.

A new preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

In this paper, based on the preconditioners presented by Cao [A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices, Journal of Computational and Applied Mathematics 238(15) (2013), pp. 109–115], we introduce and study a new augmentation block preconditioners for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and its minimal polynomial. Finally, numerical examples show that the eigenvalue distribution with presented preconditioner has the same spectral clustering with preconditioners in the literature when choosing the optimal parameters and the preconditioner in this paper and in the literature improve the convergence of BICGSTAB and GMRES iteration efficiently when they are applied to the preconditioned BICGSTAB and GMRES to solve the Stokes equation and two-dimensional time-harmonic Maxwell equations by choosing different parameters.

New Preconditioners for Nonsymmetric Saddle Point Systems with Singular (1,1) Block

We investigate the solution of large linear systems of saddle point type with singular (1,1) block by preconditioned iterative methods and consider two parameterized block triangular preconditioners used with Krylov subspace methods which have the attractive property of improved eigenvalue clustering with increased ill-conditioning of the (1,1) block of the saddle point matrix, including the choice of the parameter. Meanwhile, we analyze the spectral characteristics of two preconditioners and give the optimal parameter in practice. Numerical experiments that validate the analysis are presented.

A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices

In this note we consider a set of augmentation block preconditioners for solving generalized saddle point systems whose coefficient matrices have singular (1,1) blocks. Results concerning the eigenvalue distribution and forms of the eigenvectors of the augmentation block preconditioned generalized saddle point matrix and its minimal polynomial are given, and an optimal augmentation block preconditioner of the set is derived. These results extend previous ones in the literature. Numerical experiments that show the very effective performance of the optimal augmentation block preconditioner are reported.

A note on spectrum analysis of augmentation block Schur complement preconditioners

In this note, as a generalization of the preconditioner presented by Greif et al. (SIAM J Matrix Anal Appl 27:779---792, 2006), we consider a set of augmentation block Schur complement preconditioners for solving saddle point systems whose coefficient matrices have singular (1,1) blocks. The spectral properties of the preconditioned matrices are analyzed and an optimal preconditioner is derived.

Two augmentation preconditioners for nonsymmetric and indefinite saddle point linear systems with singular (1, 1) blocks

In this paper, two preconditioners based on augmentation are introduced for the solution of large saddle point-type systems with singular (1, 1) blocks. We study the spectral characteristics of the preconditioners, show that all eigenvalues of preconditioned matrices are strongly clustered. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.

Singular blocks of parabolic category [formula omitted] and finite W-algebras

We show that each integral infinitesimal block of parabolic category O (including singular ones) for a semi-simple Lie algebra can be realized as a full subcategory of a modified category O over a finite W-algebra for the same Lie algebra. The nilpotent used to construct this finite W-algebra is determined by the central character of the block, and the subcategory taken is that killed by a particular two-sided ideal depending on the original parabolic. The equivalences in question are induced by those defined by Miličić–Soergel and Losev. We also give a proof of a result of some independent interest: the singular blocks of parabolic category O can be geometrically realized as “partial Whittaker sheaves” on partial flag varieties.

A New Preconditioner with Two Variable Relaxation Parameters for Saddle Point Linear Systems with Highly Singular(1,1) Blocks

In this paper, we provide new preconditioner for saddle point linear systems with (1,1) blocks that have a high nullity. The preconditioner is block triangular diagonal with two variable relaxation paremeters and it is extension of results in [1] and [2]. Theoretical analysis shows that all eigenvalues of preconditioned matrix is strongly clustered. Finally, numerical tests confirm our analysis.

Nonzero Infinitesimal Blocks of Category $\boldsymbol{\mathcal{O}}_{\textbf{\emph{S}}}$

Category \(\mathcal{O}\) is a nice category of modules for a finite dimensional semisimple Lie algebra \(\mathfrak{g}\) It was first introduced by Bernstein–Gelfand–Gelfand in 1976. In the early 1980’s, Rocha–Caridi introduced parabolic category \(\mathcal{O}_S\), where S is a subset of simple roots. Category \(\mathcal{O}_S\) is a generalization of ordinary category O. These are highest weight categories that decompose into certain subcategories, called infinitesimal blocks. An infinitesimal block contains at most finitely many simple modules, and some contain only the zero module. The representation type of the infinitesimal blocks of category \(\mathcal{O}_S\) has been studied by Futorny–Nakano–Pollack, Brustle–Konig–Mazorchuk, and Boe–Nakano. The representation type of the singular blocks of category \(\mathcal{O}_S\) is still generally unknown, though Boe–Nakano classified certain of these. Understanding when a singular block is zero is an important step to understanding the singular blocks in general. In this work, we will answer this question. It is given in terms of nilpotent orbits of \(\mathfrak{g}\).

A combinatorial approach to functorial quantum sl k knot invariants

This paper contains a categorification of the ${\frak sl}(k)$ link invariant using parabolic singular blocks of category ${\cal{O}}$. Our approach is intended to be as elementary as possible, providing essentially combinatorial arguments for the main results of Sussan. The justification that our combinatorial arguments and steps are correct uses non-combinatorial geometric and representation theoretic results (e.g., the Kazhdan-Lusztig and Soergel's theorems). We take these results as granted and use them like axioms (called {\it Facts\/} in the text). We first construct an exact functor valued invariant of webs or ``special'' trivalent graphs labelled with $1, 2, k-1, k$ satisfying the MOY relations. Afterwards we extend it to the ${\frak sl}(k)$-invariant of links by passing to the derived categories. The approach of Khovanov using foams appears naturally in this context. More generally, we expect that our approach provides a representation theoretic interpretation of the ${\frak sl}(k)$-homology, based on foams and the Kapustin-Li formula, from Mackaay, Stosic, and Vaz. Conjecturally this implies that the Khovanov-Rozansky link homology is obtained from our invariant by restriction.

New Block Triangular Preconditioners for Saddle Point Linear Systems with Highly Singular (1,1) Blocks

We establish two types of block triangular preconditioners applied to the linear saddle point problems with the singular (1,1) block. These preconditioners are based on the results presented in the paper of Rees and Greif (2007). We study the spectral characteristics of the preconditioners and show that all eigenvalues of the preconditioned matrices are strongly clustered. The choice of the parameter is involved. Furthermore, we give the optimal parameter in practical. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.

Symmetric functions, parabolic category O, and the Springer fiber

We prove that the center of a regular block of parabolic category O for the general linear Lie algebra is isomorphic to the cohomology algebra of a corresponding Springer fiber. This was conjectured by Khovanov [K]. We also find presentations for the centers of singular blocks, which are cohomology algebras of Spaltenstein varieties

Augmentation block preconditioners for saddle point‐type matrices with singular (1, 1) blocks

AbstractWe consider the use of block preconditioners for the application of the preconditioned Krylov subspace iterative methods to the solution of large saddle point‐type systems with singular (1, 1) blocks. Two block triangular preconditioners are introduced and the block diagonal preconditioner in Greif and Schötzau (Electron. Trans. Numer. Anal. 2006; 22:114–121) is extended to nonsymmetric saddle point systems. All these preconditioners are based on augmentation, using nonsingular weight matrices. If the nullity of the (1, 1) block takes its highest possible value, the preconditioned matrix with either block triangular preconditioner has precisely three distinct eigenvalues, and the preconditioned matrix with the block diagonal preconditioner has precisely two distinct eigenvalues, giving rise to immediate convergence of preconditioned GMRES. Finally, numerical experiments that validate the analysis are reported. Copyright © 2008 John Wiley & Sons, Ltd.

Complexity of an algorithm for solving saddle-point systems with singular blocks arising in wavelet-Galerkin discretizations

The paper deals with fast solving of large saddle-point systems arising in wavelet-Galerkin discretizations of separable elliptic PDEs. The periodized orthonormal compactly supported wavelets of the tensor product type together with the fictitious domain method are used. A special structure of matrices makes it possible to utilize the fast Fourier transform that determines the complexity of the algorithm. Numerical experiments confirm theoretical results.

Category 𝒪: Quivers and endomorphism rings of projectives

We describe an algorithm for computing quivers of category O \mathcal O of a finite dimensional semisimple Lie algebra. The main tool for this is Soergel’s description of the endomorphism ring of the antidominant indecomposable projective module of a regular block as an algebra of coinvariants. We give explicit calculations for root systems of rank 1 and 2 for regular and singular blocks and also quivers for regular blocks for type A 3 A_3 . The main result in this paper is a necessary and sufficient condition for an endomorphism ring of an indecomposable projective object of O \mathcal O to be commutative. We give also an explicit formula for the socle of a projective object with a short proof using Soergel’s functor V \mathbb V and finish with a generalization of this functor to Harish-Chandra bimodules and parabolic versions of category O \mathcal O .

The sedimentary record and rates of Quaternary vertical tectonic movements in NW Poland

Analysis based on drilling, lithostratigraphic, geophysical and geodetic data showed that the spatial differentiation pattern of the Pleistocene cover displays a direct connection with the structural pattern of the Permo-Mesozoic complex. The main features of the Pleistocene cover, such as the thickness, stratigraphic section, facies variability and the shape of the sub-Quaternary surface, are adapted to the distribution of local Permo-Mesozoic tectonic elements, salt tectonic folds, grabens and singular blocks. This is expressed by a reduction in the thickness and stratigraphic section of Pleistocene sediments overlying uplifted tectonic elements, in relation to those which were lowered. Facies variability within the Pleistocene complex is in many places connected with the location of salt tectonic folds, and this connection is manifested by the gradual disappearance of water-laid series over salt anticlines. The most intensive, isostatically driven, vertical displacements occurred during the Cromerian, Holsteinian and Eemian interglacial periods. Transformation of repeated levelling data allowed the velocity of recent vertical displacements between examined tectonic elements to be estimated at 0.3–2.0 mm/yr, and this is comparable to the velocity of Pleistocene movements. The Pleistocene displacements between tectonic elements exerted a decisive influence on the development of their glaciogenic sedimentary cover.