288 research outputs found
Multilevel quasiseparable matrices in PDE-constrained optimization
Optimization problems with constraints in the form of a partial differential
equation arise frequently in the process of engineering design. The
discretization of PDE-constrained optimization problems results in large-scale
linear systems of saddle-point type. In this paper we propose and develop a
novel approach to solving such systems by exploiting so-called quasiseparable
matrices. One may think of a usual quasiseparable matrix as of a discrete
analog of the Green's function of a one-dimensional differential operator. Nice
feature of such matrices is that almost every algorithm which employs them has
linear complexity. We extend the application of quasiseparable matrices to
problems in higher dimensions. Namely, we construct a class of preconditioners
which can be computed and applied at a linear computational cost. Their use
with appropriate Krylov methods leads to algorithms of nearly linear
complexity
Secants of minuscule and cominuscule minimal orbits
We study the geometry of the secant and tangential variety of a cominuscule
and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods
inspired by statistics we provide an explicit local isomorphism with a product
of an affine space with a variety which is the Zariski closure of the image of
a map defined by generalized determinants. In particular, equations of the
secant or tangential variety correspond to relations among generalized
determinants. We also provide a representation theoretic decomposition of
cubics in the ideal of the secant variety of any Grassmannian
Symmetric polynomials and divided differences in formulas of intersection theory
The goal of the paper is two-fold. At first, we attempt to give a survey of
some recent applications of symmetric polynomials and divided differences to
intersection theory. We discuss: polynomials universally supported on
degeneracy loci; some explicit formulas for the Chern and Segre classes of
Schur bundles with applications to enumerative geometry; flag degeneracy loci;
fundamental classes, diagonals and Gysin maps; intersection rings of G/P and
formulas for isotropic degeneracy loci; numerically positive polynomials for
ample vector bundles.
Apart of surveyed results, the paper contains also some new results as well
as some new proofs of earlier ones: how to compute the fundamental class of a
subvariety from the class of the diagonal of the ambient space; how to compute
the class of the relative diagonal using Gysin maps; a new formula for pushing
forward Schur's Q- polynomials in Grassmannian bundles; a new formula for the
total Chern class of a Schur bundle; another proof of Schubert's and
Giambelli's enumeration of complete quadrics; an operator proof of the
Jacobi-Trudi formula; a Schur complex proof of the Giambelli-Thom-Porteous
formula.Comment: 58 pages; to appear in the volume "Parameter Spaces", Banach Center
Publications vol 36 (1996) AMSTE
Hua\u27s Matrix Equality and Schur Complements
The purpose of this paper is to revisit Hua\u27s matrix equality (and inequality) through the Schur complement. We present Hua\u27s original proof and two new proofs with some extensions of Hua\u27s matrix equality and inequalities. The new proofs use a result concerning Shur complements and a generalization of Sylvester\u27s law of inertia, each of which is useful in its own right
Fast computation of the matrix exponential for a Toeplitz matrix
The computation of the matrix exponential is a ubiquitous operation in
numerical mathematics, and for a general, unstructured matrix it
can be computed in operations. An interesting problem arises
if the input matrix is a Toeplitz matrix, for example as the result of
discretizing integral equations with a time invariant kernel. In this case it
is not obvious how to take advantage of the Toeplitz structure, as the
exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself.
The main contribution of this work are fast algorithms for the computation of
the Toeplitz matrix exponential. The algorithms have provable quadratic
complexity if the spectrum is real, or sectorial, or more generally, if the
imaginary parts of the rightmost eigenvalues do not vary too much. They may be
efficient even outside these spectral constraints. They are based on the
scaling and squaring framework, and their analysis connects classical results
from rational approximation theory to matrices of low displacement rank. As an
example, the developed methods are applied to Merton's jump-diffusion model for
option pricing
Rank Equalities Related to Generalized Inverses of Matrices and Their Applications
This paper is divided into two parts. In the first part, we develop a general
method for expressing ranks of matrix expressions that involve Moore-Penrose
inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose
inverses of matrices. Through this method we establish a variety of valuable
rank equalities related to generalized inverses of matrices mentioned above.
Using them, we characterize many matrix equalities in the theory of generalized
inverses of matrices and their applications. In the second part, we consider
maximal and minimal possible ranks of matrix expressions that involve variant
matrices, the fundamental work is concerning extreme ranks of the two linear
matrix expressions and . As applications,
we present a wide range of their consequences and applications in matrix
theory.Comment: 245 pages, LaTe
A combinatorial characterization of tight fusion frames
In this paper we give a combinatorial characterization of tight fusion frame
(TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case
has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our
characterization does not have this limitation. We also develop some methods
for generating TFF sequences. The basic technique is a majorization principle
for TFF sequences combined with spatial and Naimark dualities. We use these
methods and our characterization to give necessary and sufficient conditions
which are satisfied by the first three highest ranks. We also give a
combinatorial interpretation of spatial and Naimark dualities in terms of
Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences
which have unique maximal elements with respect to majorization partial order.
Finally, we give several examples illustrating our techniques including an
example of tight fusion frame which can not be constructed by the existing
spectral tetris techniques. We end the paper by giving a complete list of
maximal TFF sequences in dimensions less than ten.Comment: 31 page
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