Indefinite System Research Articles

Symmetric Quasidefinite Matrices

It is stated here that a symmetric matrix K is quasidefinite if it has the form \[ K = \begin{bmatrix} - E & A^T A & F \end{bmatrix}, \] where E and F are symmetric positive definite matrices. Although such matrices are indefinite, it is shown that any symmetric permutation of a quasidefinite matrix yields a factorization $LDL^T $.This result is applied to obtain a new approach for solving the symmetric indefinite systems arising in interior-point methods for linear and quadratic programming. These systems are typically solved either by reducing to a positive definite system or by performing a Bunch–Parlett factorization of the full indefinite system at every iteration. This is an intermediate approach based on reducing to a quasidefinite system. This approach entails less fill-in than further reducing to a positive definite system, but is based on a static ordering and is therefore more efficient than performing Bunch–Parlett factorizations of the original indefinite system.

A gauged A, V-A-/spl Psi/ formulation without A/spl middot/n=0 on conductor boundaries

In the present paper a new A, V-A-/spl Psi/ formulation for transient quasi-static eddy current problems using V=0, instead of A/spl middot/n=0, on conductor boundaries is proposed. This choice appears to advantageous for finite element solution problems without any symmetry plane or involving curved boundaries, even though unsuitable if the fields change very slowly. The differential formulation is carefully defined; the ways in which the Coulomb gauge is imposed and the existence and the uniqueness of the solution are discussed. A new method to obtain the symmetry of the final algebraic system is introduced. This symmetrisation technique leads to an indefinite system of algebraic equations with peculiar structure, that led to the development of a special solver. The results are extremely encouraging, the formulation is based on thorough theoretical basis, and it is fairly easy to impose the conditions on conductor boundaries.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the convergence of multigrid iteration scheme for mixed finite element equation

We present a multigrid iteration algorithm for mixed finite element equations and a proof of its convergence where the rate of convergence is independent of the mesh size. Whereas techniques such as the “multigrid preconditioning” and “non-nested multigrid schemes” have been recently proposed to solve the indefinite linear system arising from the mixed finite element approximation, our analysis here clearly indicates that the multigrid idea for solving linear systems could be applied directly to mixed finite element subspace. We hope that this fact encourages one to consider multigrid solution for other kinds of non-conforming finite element equations for the elliptic PDEs. To establish our convergence theorem, we derive and apply an energy inequality that is weaker than some norm estimates used by others; in fact, such norm estimates are not applicable to establish our convergence result.

Quasi-newtonian methods for the solution of hierarchical finite element equations

This paper exploits the solution of hierarchical finite element equations via quasi-Newtonian (QN) methods, and discovers that QN methods can compete with the preconditioned conjugate gradient (PCG) method in the rate of convergence. The currently available version of the PCG method is unable to solve the indefinite system of equations, because the preconditioning matrices involve the arithmetic operations on the square roots of diagonal pivots of the coefficient matrix. On the contrary, the QN methods investigated in this paper do not have this limitation. Thus, they have the potential to become popular solution strategies for nonlinear hierarchical finite element analysis in the future. Details on how the QN algorithms are implemented for solving the linear hierarchical system of equations are presented. The QN algorithms as well as the PCG method are tested by using an example in which the coefficient matrices are symmetric positive definite. Comparisons on computational efficiency and convergence rate for the positive definite case are then discussed. Also, the capability of QN algorithms for solving the indefinite system of equations is also presented via a numerical experiment using another example.

Symmetric indefinite systems for interior point methods

We present a unified framework for solving linear and convex quadratic programs via interior point methods. At each iteration, this method solves an indefinite system whose matrix is\(\left[ {\begin{array}{*{20}c} { - D^{ - 2} } & {A^T } \\ A & 0 \\ \end{array} } \right]\) instead of reducing to obtain the usualAD 2 A T system. This methodology affords two advantages: (1) it avoids the fill created by explicitly forming the productAD 2 A T whenA has dense columns; and (2) it can easily be used to solve nonseparable quadratic programs since it requires only thatD be symmetric. We also present a procedure for converting nonseparable quadratic programs to separable ones which yields computational savings when the matrix of quadratic coefficients is dense.

The solution of large-scale least-squares problems on supercomputers

We discuss methods for solving medium to large-scale sparse least-squares problems on supercomputers, illustrating our remarks by experiments on the CRAY-2 supercomputer at Harwell. The method we are primarily concerned with is an augmented system approach which has the merit of both robustness and accuracy, in addition to a kernel operation that is just the solution of a symmetric indefinite system. We consider extensions to handle weighted and constrained problems, and include experiments on systems similar to those arising in the Karmarkar algorithm for linear programming. We indicate how recent improvements to the kernel software could greatly improve the performance of the least-squares code.

A new implementation of the force method and the optimum design of large trusses

AbstractAn implementation of the force method is proposed in which the forces and the displacements are simultaneously obtained by the solution of a sparse symmetric indefinite system. The matrix of coefficients is formed by just the concatenation of the element flexibility and equilibrium matrices. No computational procedure is required to generate the compatibility conditions (or the self‐stress matrix) and no partitioning of the force vector is made into a basic set and a redundant set, unlike the conventional force method. A slightly modified sparse unsymmetric system can be written in which the stresses and the displacements are the unknowns. This is used as constraints in the formulation of the minimum weight design problem for large structures under static loading conditions. A sparse generalized reduced gradient package is used as the optimizer. A class of test problems involving large truss structures is solved. The results indicate that the present implementation of the force method is better than the displacement method for the optimum design of large structures.

A conjugate-residual-FEM for incompressible viscous flow analysis

The application of the conjugate residual method to the indefinite system resulting from the finite element approximation of the incompressible Navier-Stokes equations is studied. The possibility to use an element-by-element preconditioner for the acceleration of the convergence is also discussed. With this scheme, one can achieve significant savings in storage without ever forming the large total matrix. Furthermore, the hybrid finite element mesh, a mixture of the structured and unstructured mesh subdivisions, is employed to minimize the storage requirement. The numerical simulations of the Karman vortex street, a driven cavity flow and the flow in a safety valve are presented to demonstrate the applicability of the proposed strategies.

Societies and Academies

LONDON Linnean Society, November 6.—Prof. Allman, president, in the chair.—Mr. W. H. Twelvetrees (of Orenburg, Russia) was elected a Fellow of the Society.—The President, in opening the session, briefly alluded to the demise of Mr. W. Wilson Saunders and Mr. John Miers, whose scientific and official labours in connection with the Society have been well appreciated.—Mr. W. T. Thiselton Dyer exhibited and made remarks on some photographs of vegetation, including Cinchona Ledgeriana, in the Botanic Garden of Buitenzorg, Java.—Mr. D. Morris, recently returned from investigating the coffee leaf disease of Ceylon and South India, read a paper on the structure and habit of Hemileia vastatrix. He supports the Rev. R. Abbay's statements as to the destructive character of the fungus and its evident gradual extension over the coffee-producing regions of the East; he even expresses fears of its ultimately being carried to the West Indies and Brazil;—2,000,000l., the estimated annual deficiency in Ceylon alone, is no mean sum to be debited from the revenue and interests of the planters. Mr. Abbay has described the spores as attached to the inner surface of the orange yellow sporanges (a notion opposed to received ideas respecting free cell formation); but Mr. Morris's observations are opposed to those of the former. The author explains the hitherto puzzling dark brown bodies beneath the sporanges as composed of closely interwoven threads of mycelium. During February, March, and April, both bark and leaves are everywhere covered exteriorly by mycelial filamentous threads which reproduce by germinating spores. In the wet weather these do not enter the stomata. It is in this stage that conidial growth supervenes according to Abbay (secondary spores of Thwaites), but the author has failed to substantiate this phase, though starved plants on glass slides raised conidia. It is during the filamentous stage before penetration that remedial agents-dusting with sulphur and lime, &c.—have a chance of being effective; but a serious disturbing element offers in the large area of abandoned crop still continuing to propagate the fungus.—Dr. F. Day read a paper on the instincts and emotions of fish. Biologists of late have been less attracted by the faculties of fish than of other animals, and even Cuvier's estimate of their total want of intelligence has been quite recently quoted asauthentic. Theauthor combats this notion, and, from his own experience and data afforded by other writers, claims evidence of emotions and affections. He shows they construct nests, transport their eggs, protect and defend their young, exhibit affection for each other, recognise human beings, can be tamed, manifest fear, anger, hatred, and revenge, utter sounds, hide from danger, betake themselves for protection to the bodies of other animals, and have other peculiar modes of defence, leave the water for food, and even different families combine for attack and defence. Their faculties, notwithstanding, are greatly subordinated and modified compared with those of higher races of the vertebrata.—The Rev. G. Henslow read a paper on the origin of the (socalled) scorpioid cyme. He pointed out some errors in deducing this from the dichotomous cyme: I. Opposite pairs of bracts, being successively in planes at right angles, the resulting sympode would be a volute, and not a helix. 2. The position of the bracts (when present, as in Borago) are not opposite the flowers. 3. There are always two rows of flowers, not a single one. 4. The appearance of a flower in the fork between the two branches of the inflorescence (as in Myosotis) is not usual, and is due to the adhesion between the terminal and the highest axillary raceme. This has given rise to a false impression of dichotomy. 5. Authors have hitherto confounded the “true scorpioid raceme” (Henslow) with spicate degradations of sympodial inflorescence. He refers it to the indefinite system, and explains its origin by a new principle of phyllotaxis, which he first discovered in Lagerstromia, viz., in resolving opposite and decussate leaves into alternate, instead of their lying on a continuous spiral line, the line oscillates through three-fourths of a circle, and if a line be drawn from flower to bract, it will represent the so-called scorpioid cyme of Boragineæ.