The use of orbitals and full spectra to identify misalignment

In this paper, a SpectraQuest demonstrator is used to introduce misalignment in a rotating set-up. The vibrations caused by misalignment is measured with both accelerometers on the bearings and eddy current probes on the shaft itself. A comparison is made between the classical spectral analysis, orbitals and full spectra. Orbitals are used to explain the physical interpretation of the vibration caused by misalignment. Full spectra allow to distinguish unbalance from misalignment by looking at the forward and reversed phenomena. This analysis is done for different kinds of misalignment, couplings, excitation forces and combined machinery faults

Schur complement reduction in the mixed-hybrid approximation of Darcy's law: rounding error analysis

AbstractMixed-hybrid finite element approximation of the potential fluid flow problem leads to the solution of a large symmetric indefinite system for the velocity and potential head vector components. Such discretization gives rise to a very accurate approximation of the continuity equation in every element, and for low-order discretizations, the structural properties of the discrete matrix blocks allow cheap block elimination of the positive-definite diagonal block and subsequent reduction to the Schur complement system for the pressure and Lagrangian vector components. This system is then frequently solved by the iterative conjugate gradient-type method. Whereas this approach is well known, considerably less attention has been paid to the numerical stability aspects of such transformation. It was shown in [5] that block LU factorization can be unstable even when the system matrix is symmetric positive definite. In this paper we examine this type of conditional stability for a particular application in the underground water flow modelling. We show that the actual error of the computed approximate solution depends not only on the user-defined tolerance in the conjugate gradient process but also on the spectral properties of the corresponding matrix blocks eliminated during the Schur complement reduction. It is often observed that although the backward error of the approximate solution in the iterative part is reduced to the level of machine accuracy, the total residual norm after the back-substitution process remains at certain accuracy level. We give a bound for this maximal attainable accuracy and illustrate our theoretical results on a model example

Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras

The assignment (nonstable K_0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that VF is equivalent to the identity. (3) There is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0. By using categorical tools from an earlier paper (larders, lifters, CLL), we deduce that there exists a unital exchange ring of cardinality aleph three (resp., an aleph three-separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(R) is the positive cone of a dimension group and V(R) is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea

A Schur complement approach to preconditioning sparse linear least-squares problems with some dense rows

The effectiveness of sparse matrix techniques for directly solving large-scale linear least-squares problems is severely limited if the system matrix A has one or more nearly dense rows. In this paper, we partition the rows of A into sparse rows and dense rows (A s and A d ) and apply the Schur complement approach. A potential difficulty is that the reduced normal matrix AsTA s is often rank-deficient, even if A is of full rank. To overcome this, we propose explicitly removing null columns of A s and then employing a regularization parameter and using the resulting Cholesky factors as a preconditioner for an iterative solver applied to the symmetric indefinite reduced augmented system. We consider complete factorizations as well as incomplete Cholesky factorizations of the shifted reduced normal matrix. Numerical experiments are performed on a range of large least-squares problems arising from practical applications. These demonstrate the effectiveness of the proposed approach when combined with either a sparse parallel direct solver or a robust incomplete Cholesky factorization algorithm