9,134 research outputs found
Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods
Use of the stochastic Galerkin finite element methods leads to large systems
of linear equations obtained by the discretization of tensor product solution
spaces along their spatial and stochastic dimensions. These systems are
typically solved iteratively by a Krylov subspace method. We propose a
preconditioner which takes an advantage of the recursive hierarchy in the
structure of the global matrices. In particular, the matrices posses a
recursive hierarchical two-by-two structure, with one of the submatrices block
diagonal. Each one of the diagonal blocks in this submatrix is closely related
to the deterministic mean-value problem, and the action of its inverse is in
the implementation approximated by inner loops of Krylov iterations. Thus our
hierarchical Schur complement preconditioner combines, on each level in the
approximation of the hierarchical structure of the global matrix, the idea of
Schur complement with loops for a number of mutually independent inner Krylov
iterations, and several matrix-vector multiplications for the off-diagonal
blocks. Neither the global matrix, nor the matrix of the preconditioner need to
be formed explicitly. The ingredients include only the number of stiffness
matrices from the truncated Karhunen-Lo\`{e}ve expansion and a good
preconditioned for the mean-value deterministic problem. We provide a condition
number bound for a model elliptic problem and the performance of the method is
illustrated by numerical experiments.Comment: 15 pages, 2 figures, 9 tables, (updated numerical experiments
Resolvent estimates for operators belonging to exponential classes
For let be the set of all compact operators on
a separable Hilbert space such that , where
denotes the -th singular number of . We provide upper bounds for
the norm of the resolvent of in terms of a quantity
describing the departure from normality of and the distance of to the
spectrum of . As a consequence we obtain upper bounds for the Hausdorff
distance of the spectra of two operators in .Comment: AMS-LaTeX, 20 page
Holographic three-point functions of giant gravitons
Working within the AdS/CFT correspondence we calculate the three-point
function of two giant gravitons and one pointlike graviton using methods of
semiclassical string theory and considering both the case where the giant
gravitons wrap an S^3 in S^5 and the case where the giant gravitons wrap an S^3
in AdS_5. We likewise calculate the correlation function in N=4 SYM using two
Schur polynomials and a single trace chiral primary. We find that the gauge and
string theory results have structural similarities but do not match perfectly,
and interpret this in terms of the Schur polynomials' inability to interpolate
between dual giant and pointlike gravitons.Comment: 21 page
An Interesting Class of Operators with unusual Schatten-von Neumann behavior
We consider the class of integral operators Q_\f on of the form
(Q_\f f)(x)=\int_0^\be\f (\max\{x,y\})f(y)dy. We discuss necessary and
sufficient conditions on to insure that is bounded, compact,
or in the Schatten-von Neumann class \bS_p, . We also give
necessary and sufficient conditions for to be a finite rank
operator. However, there is a kind of cut-off at , and for membership in
\bS_{p}, , the situation is more complicated. Although we give
various necessary conditions and sufficient conditions relating to
Q_{\phi}\in\bS_{p} in that range, we do not have necessary and sufficient
conditions. In the most important case , we have a necessary condition and
a sufficient condition, using and modulus of continuity,
respectively, with a rather small gap in between. A second cut-off occurs at
: if \f is sufficiently smooth and decays reasonably fast, then \qf
belongs to the weak Schatten-von Neumann class \wS{1/2}, but never to
\bS_{1/2} unless \f=0.
We also obtain results for related families of operators acting on
and .
We further study operations acting on bounded linear operators on
related to the class of operators Q_\f. In particular we
study Schur multipliers given by functions of the form and
we study properties of the averaging projection (Hilbert-Schmidt projection)
onto the operators of the form Q_\f.Comment: 87 page
- …