9,229 research outputs found
Non-uniqueness in conformal formulations of the Einstein constraints
Standard methods in non-linear analysis are used to show that there exists a
parabolic branching of solutions of the Lichnerowicz-York equation with an
unscaled source. We also apply these methods to the extended conformal thin
sandwich formulation and show that if the linearised system develops a kernel
solution for sufficiently large initial data then we obtain parabolic solution
curves for the conformal factor, lapse and shift identical to those found
numerically by Pfeiffer and York. The implications of these results for
constrained evolutions are discussed.Comment: Arguments clarified and typos corrected. Matches published versio
Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples
We examine some differential geometric approaches to finding approximate
solutions to the continuous time nonlinear filtering problem. Our primary focus
is a new projection method for the optimal filter infinite dimensional
Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric
and on a family of normal mixtures. We compare this method to earlier
projection methods based on the Hellinger distance/Fisher metric and
exponential families, and we compare the L2 mixture projection filter with a
particle method with the same number of parameters, using the Levy metric. We
prove that for a simple choice of the mixture manifold the L2 mixture
projection filter coincides with a Galerkin method, whereas for more general
mixture manifolds the equivalence does not hold and the L2 mixture filter is
more general. We study particular systems that may illustrate the advantages of
this new filter over other algorithms when comparing outputs with the optimal
filter. We finally consider a specific software design that is suited for a
numerically efficient implementation of this filter and provide numerical
examples.Comment: Updated and expanded version published in the Journal reference
below. Preprint updates: January 2016 (v3) added projection of Zakai Equation
and difference with projection of Kushner-Stratonovich (section 4.1). August
2014 (v2) added Galerkin equivalence proof (Section 5) to the March 2013 (v1)
versio
Iterative Estimation of Solutions to Noisy Nonlinear Operator Equations in Nonparametric Instrumental Regression
This paper discusses the solution of nonlinear integral equations with noisy
integral kernels as they appear in nonparametric instrumental regression. We
propose a regularized Newton-type iteration and establish convergence and
convergence rate results. A particular emphasis is on instrumental regression
models where the usual conditional mean assumption is replaced by a stronger
independence assumption. We demonstrate for the case of a binary instrument
that our approach allows the correct estimation of regression functions which
are not identifiable with the standard model. This is illustrated in computed
examples with simulated data
A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation
We linearize the Einstein-scalar field equations, expressed relative to
constant mean curvature (CMC)-transported spatial coordinates gauge, around
members of the well-known family of Kasner solutions on . The Kasner solutions model a spatially uniform scalar field
evolving in a (typically) spatially anisotropic spacetime that expands towards
the future and that has a "Big Bang" singularity at . We
place initial data for the linearized system along and study the linear solution's behavior in the collapsing
direction . Our first main result is the proof of an
approximate monotonicity identity for the linear solutions. Using it, we
prove a linear stability result that holds when the background Kasner solution
is sufficiently close to the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW)
solution. In particular, we show that as , various
time-rescaled components of the linear solution converge to regular functions
defined along . In addition, we motivate the preferred
direction of the approximate monotonicity by showing that the CMC-transported
spatial coordinates gauge can be viewed as a limiting version of a family of
parabolic gauges for the lapse variable; an approximate monotonicity identity
and corresponding linear stability results also hold in the parabolic gauges,
but the corresponding parabolic PDEs are locally well-posed only in the
direction . Finally, based on the linear stability results, we
outline a proof of the following result, whose complete proof will appear
elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing
direction under small perturbations of its data at .Comment: 73 page
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