One main issue, when numerically integrating autonomous Hamiltonian systems,
is the long-term conservation of some of its invariants, among which the
Hamiltonian function itself. Recently, a new class of methods, named
Hamiltonian Boundary Value Methods (HBVMs) has been introduced and analysed,
which are able to exactly preserve polynomial Hamiltonians of arbitrarily high
degree. We here study a further property of such methods, namely that of
having, when cast as a Runge-Kutta method, a matrix of the Butcher tableau with
the same spectrum (apart from the zero eigenvalues) as that of the
corresponding Gauss-Legendre method, independently of the considered abscissae.
Consequently, HBVMs are always perfectly A-stable methods. This, in turn,
allows to elucidate the existing connections with classical Runge-Kutta
collocation methods.
miércoles, 24 de febrero de 2010
Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their connections with Runge-Kutta collocation methods. (arXiv:1002.4394v1 [math.NA])
FE-BE coupling for a transmission problem involving microstructure. (arXiv:1002.4385v1 [math.NA])
We analyze a finite element/boundary element procedure to solve a non-convex
contact problem for the double-well potential. After relaxing the associated
functional, the degenerate minimization problem is reduced to a boundary/domain
variational inequality, a discretized saddle point formulation of which may
then be solved numerically. The convergence of the Galerkin approximations to
certain macroscopic quantities and a corresponding a posteriori estimate for
the approximation error are discussed.
Computing the R of the QR factorization of tall and skinny matrices using MPI_Reduce. (arXiv:1002.4250v1 [math.NA])
A QR factorization of a tall and skinny matrix with n columns can be
represented as a reduction. The operation used along the reduction tree has in
input two n-by-n upper triangular matrices and in output an n-by-n upper
triangular matrix which is defined as the R factor of the two input matrices
stacked the one on top of the other. This operation is binary, associative, and
commutative. We can therefore leverage the MPI library capabilities by using
user-defined MPI operations and MPI_Reduce to perform this reduction. The
resulting code is compact and portable. In this context, the user relies on the
MPI library to select a reduction tree appropriate for the underlying
architecture.
Sparse Channel Separation using Random Probes. (arXiv:1002.4222v1 [math.NA])
This paper considers the problem of estimating the channel response (or
Green's function) between multiple source-receiver pairs. Typically, the
channel responses are estimated one-at-a-time: a single source sends out a
known probe signal, the receiver measures the probe signal convolved with the
channel response, and the responses are recovered using deconvolution. In this
paper, we show that if the channel responses are sparse and the probe signals
are random, then we can significantly reduce the total amount of time required
to probe the channels by activating all of the sources simultaneously. With all
sources activated simultaneously, the receiver measures a superposition of all
the channel responses convolved with the respective probe signals. Separating
this cumulative response into individual channel responses can be posed as a
linear inverse problem.
We show that channel response separation is possible (and stable) even when
the probing signals are relatively short in spite of the corresponding linear
system of equations becoming severely underdetermined. We derive a theoretical
lower bound on the length of the source signals that guarantees that this
separation is possible with high probability. The bound is derived by putting
the problem in the context of finding a sparse solution to an underdetermined
system of equations, and then using mathematical tools from the theory of
compressive sensing. Finally, we discuss some practical applications of these
results, which include forward modeling for seismic imaging, channel
equalization in multiple-input multiple-output communication, and increasing
the field-of-view in an imaging system by using coded apertures.
Finite sections of band-dominated operators on discrete groups. (arXiv:1002.4258v1 [math.NA])
Let $\Gamma$ be a finitely generated discrete exact group. We consider
operators on $l^2(\Gamma)$ which are composed by operators of multiplication by
a function in $l^\infty (\Gamma)$ and by the operators of left-shift by
elements of $\Gamma$. These operators generate a $C^*$-subalgebra of
$L(l^2(\Gamma))$ the elements of which we call band-dominated operators on
$\Gamma$. We study the stability of the finite sections method for
band-dominated operators with respect to a given generating system of $\Gamma$.
Our approach is based on the equivalence of the stability of a sequence and the
Fredholmness of an associated operator, and on Roe's criterion for the
Fredholmness of a band-dominated operator on a exact discrete group, which we
formulate in terms of limit operators. Special emphasis is paid to the
quasicommutator ideal of the algebra generated by the finite sections sequences
and to the stability of sequences in that algebra. For both problems, the
sequence of the discrete boundaries plays an essential role.
A Finite Element Method for Density Estimation with Gaussian Process Priors
A variational problem characterizing the density estimator defined by the maximum a posteriori method with Gaussian process priors is derived. It is shown that this problem is well posed and can be solved with Newton's method. Numerically, the solution is approximated by a Galerkin/finite element m ... [SIAM J. Numer. Anal. 47, 4759 (2010)] published Wed Feb 24, 2010."