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Anderson Acceleration of the Alternating Projections Method for Computing the Nearest Correlation Matrix
Get PDFIn a wide range of applications it is required to compute the nearest correlation matrix in the Frobenius norm to a given symmetric but indefinite matrix. Of the available methods with guaranteed convergence to the unique solution of this problem the easiest to implement, and perhaps the most widely used, is the alternating projections method. However, the rate of convergence of this method is at best linear, and it can require a large number of iterations to converge to within a given tolerance. We show that Anderson acceleration, a technique for accelerating the convergence of fixed-point iterations, can be applied to the alternating projections method and that in practice it brings a significant reduction in both the number of iterations and the computation time. We also show that Anderson acceleration remains effective, and indeed can provide even greater improvements, when it is applied to the variants of the nearest correlation matrix problem in which specified elements are fixed or a lower bound is imposed on the smallest eigenvalue. Alternating projections is a general method for finding a point in the intersection of several sets and ours appears to be the first demonstration that this class of methods can benefit from Anderson acceleration
Finite groups and Lie rings with an automorphism of order
Get PDFSuppose that a finite group admits an automorphism \f of
order such that the fixed-point subgroup C_G(\f
^{2^{n-1}}) of the involution \f ^{2^{n-1}}
is nilpotent of class . Let m=|C_G(\f )| be the number of fixed points of \f. It is proved that has a soluble subgroup of derived length bounded in terms of whose index is bounded in terms of . A similar result is also proved for Lie rings
An optimal iterative solver for linear systems arising from SFEM approximation of diffusion equations with random coefficients
Get PDFThis paper discusses the design and implementation of efficient solution algorithms for symmetric linear systems associated with stochastic Galerkin approximation of elliptic PDE problems with correlated random data. The novel feature of our iterative solver is the incorporation of error control in the natural "energy" norm in combination with an effective a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error
Null-space preconditioners for saddle point problems
Get PDFThe null-space method is a technique that has been used for many years to reduce a saddle point system to a smaller, easier to solve, symmetric positive-definite system. This method can be understood as a block factorization of the system. Here we explore the use of preconditioners based on incomplete versions of a particular null-space factorization, and compare their performance with the equivalent Schur-complement based preconditioners. We also describe how to apply the non-symmetric preconditioners proposed using the conjugate gradient method (CG) with a non-standard inner product. This requires an exact solve with the (1,1) block, and the resulting algorithm is applicable in other cases where Bramble-Pasciak CG is used. We verify the efficiency of the newly proposed preconditioners on a number of test cases from a range of applications
Topics in Finite Groups: Homology Groups, Pi-product Graphs, Wreath Products and Cuspidal Characters
Get PDFIn this thesis we investigate four separate topics in finite group theory:
1) Homology groups of presheaves of abelian groups;
2) Pi-product graphs for the symmetric groups;
3) Wreath products of cyclic groups of order p; and
4) p-Cuspidal characters.
In each case we use combinatorial and computational arguments to ascertain properties of the groups/graphs/characters in question
The Hill and Eshelby tensors for ellipsoidal inhomogeneities in the Newtonian potential problem and linear elastostatics.
Get PDFOne of the most cited papers in Applied Mechanics is the work of Eshelby from 1957 who showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in a homogeneous isotropic host would feel uniform strains and stresses when uniform strains or stresses are applied in the far-field. Of specific importance is the uniformity of \textit{Eshelby's tensor} . Following this paper a vast literature has been generated using and developing Eshelby's result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor arises, where is the host medium compliance tensor. The tensor is associated with \textit{Hill} and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc.\ and in the vector problems of elastostatics. Knowledge of the Hill and Eshelby tensors permit a number of interesting aspects to be studied associated with inhomogeneity problems and more generally for inhomogeneous media. Micromechanical methods established mainly over the last half-century have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases.
Of specific interest is that a number of important limits of the ellipsoidal inhomogeneity can be taken in order to be employed in predictions of the effective properties of e.g.\ layered media, fibre reinforced composites, voids and cracks to name but a few. In the main, results for the Hill and Eshelby tensors associated with these problems are distributed over a wide range of articles and books, using different notation and terminology and so it is often difficult to extract the necessary information for the tensor that one requires. The case of an anisotropic host phase is also frequently non-trivial due to the requirement of the associated Green's tensor. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled. A scaling argument leads to the derivation of the Eshelby tensor for potential problems where the host phase is at most orthotropic, without the requirement of using the anisotropic Green's function. Concentration tensors are derived for a wide variety of problems that can be used directly in the various micromechanical schemes. Both tensor and matrix formulations are considered and contrasted
A new strain energy function for modelling ligaments and tendons whose fascicles have a helical arrangement of fibrils
Get PDFA new strain energy function for the hyperelastic modelling of ligaments and tendons whose fascicles have a helical arrangement of fibrils is derived. The stress-strain response of a single fascicle whose fibrils exhibit varying levels of crimp throughout its radius is calculated and used to determine the form of the strain energy function. The new constitutive law is used to model uniaxial extension test data for human patellar tendon and is shown to provide an excellent fit, with the average relative error being 9.8%. It is then used to model shear and predicts that the stresses required to shear a tendon are much smaller than those required to uniaxially stretch it to the same strain level. Finally, the strain energy function is used to model ligaments and tendons whose fascicles are helical, and the relative effects of the fibril helix angle, the fascicle helix angle and the fibril crimp variable are compared. It is shown that they all have a significant effect; the fibril crimp variable governs the non-linearity of the stress-strain curve, whereas the helix angles primarily affect its stiffness. Smaller values of the helix angles lead to stiffer tendons; therefore, the model predicts that one would expect to see fewer helical sub-structures in stiff positional tendons, and more in those that are required to be more flexible
Max-Balanced Hungarian Scalings
Get PDFA Hungarian scaling is a diagonal scaling of a matrix that is typically applied
along with a permutation to a sparse linear system before calling a direct or
iterative solver. A matrix that has been Hungarian scaled and reordered has all
entries of modulus less than or equal to 1 and entries of modulus 1 on the
diagonal.
An important fact that has been overlooked by the previous research into
Hungarian scaling of linear systems is that a given matrix typically has a range
of possible Hungarian scalings and direct or iterative solvers may behave quite
differently under each of these scalings.
Since standard algorithms for computing Hungarian scalings return only one
scaling, it is natural to ask whether a superior performing scaling can be
obtained by searching within the set of all possible Hungarian scalings. To this
end we propose a method for computing a Hungarian scaling that is optimal from
the point of view of diagonal dominance.
Our method uses max-balancing, which minimizes the largest off-diagonal entries
in the matrix.
Numerical experiments illustrate the increased diagonal dominance produced by
max-balanced Hungarian scaling as well as the reduced need for row interchanges
in Gaussian elimination with partial pivoting and the improved stability of LU
factorizations without pivoting.
We additionally find that applying the max-balancing scaling before computing
incomplete LU preconditioners improves the convergence rate of certain
iterative methods