209 research outputs found
Reaching the superlinear convergence phase of the CG method
The rate of convergence of the conjugate gradient method takes place in essen-
tially three phases, with respectively a sublinear, a linear and a superlinear rate.
The paper examines when the superlinear phase is reached. To do this, two methods
are used. One is based on the K-condition number, thereby separating the eigenval-
ues in three sets: small and large outliers and intermediate eigenvalues. The other
is based on annihilating polynomials for the eigenvalues and, assuming various an-
alytical distributions of them, thereby using certain refined estimates. The results
are illustrated for some typical distributions of eigenvalues and with some numerical
tests
Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy
We study an inexact inner-outer generalized Golub-Kahan algorithm for the
solution of saddle-point problems with a two-times-two block structure. In each
outer iteration, an inner system has to be solved which in theory has to be
done exactly. Whenever the system is getting large, an inner exact solver is,
however, no longer efficient or even feasible and iterative methods must be
used. We focus this article on a numerical study showing the influence of the
accuracy of an inner iterative solution on the accuracy of the solution of the
block system. Emphasis is further given on reducing the computational cost,
which is defined as the total number of inner iterations. We develop relaxation
techniques intended to dynamically change the inner tolerance for each outer
iteration to further minimize the total number of inner iterations. We
illustrate our findings on a Stokes problem and validate them on a mixed
formulation of the Poisson problem.Comment: 25 pages, 9 figure
Hessian Averaging in Stochastic Newton Methods Achieves Superlinear Convergence
We consider minimizing a smooth and strongly convex objective function using
a stochastic Newton method. At each iteration, the algorithm is given an oracle
access to a stochastic estimate of the Hessian matrix. The oracle model
includes popular algorithms such as the Subsampled Newton and Newton Sketch,
which can efficiently construct stochastic Hessian estimates for many tasks.
Despite using second-order information, these existing methods do not exhibit
superlinear convergence, unless the stochastic noise is gradually reduced to
zero during the iteration, which would lead to a computational blow-up in the
per-iteration cost. We address this limitation with Hessian averaging: instead
of using the most recent Hessian estimate, our algorithm maintains an average
of all past estimates. This reduces the stochastic noise while avoiding the
computational blow-up. We show that this scheme enjoys local -superlinear
convergence with a non-asymptotic rate of ,
where is proportional to the level of stochastic noise in the
Hessian oracle. A potential drawback of this (uniform averaging) approach is
that the averaged estimates contain Hessian information from the global phase
of the iteration, i.e., before the iterates converge to a local neighborhood.
This leads to a distortion that may substantially delay the superlinear
convergence until long after the local neighborhood is reached. To address this
drawback, we study a number of weighted averaging schemes that assign larger
weights to recent Hessians, so that the superlinear convergence arises sooner,
albeit with a slightly slower rate. Remarkably, we show that there exists a
universal weighted averaging scheme that transitions to local convergence at an
optimal stage, and still enjoys a superlinear convergence~rate nearly (up to a
logarithmic factor) matching that of uniform Hessian averaging.Comment: 40 pages, 16 figure
Weak KAM for commuting Hamiltonians
For two commuting Tonelli Hamiltonians, we recover the commutation of the
Lax-Oleinik semi-groups, a result of Barles and Tourin ([BT01]), using a direct
geometrical method (Stoke's theorem). We also obtain a "generalization" of a
theorem of Maderna ([Mad02]). More precisely, we prove that if the phase space
is the cotangent of a compact manifold then the weak KAM solutions (or
viscosity solutions of the critical stationary Hamilton-Jacobi equation) for G
and for H are the same. As a corrolary we obtain the equality of the Aubry
sets, of the Peierls barrier and of flat parts of Mather's functions.
This is also related to works of Sorrentino ([Sor09]) and Bernard ([Ber07b]).Comment: 23 pages, accepted for publication in NonLinearity (january 29th
2010). Minor corrections, fifth part added on Mather's function (or
effective Hamiltonian
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