Linear Form of Canonical Gravity

Recent work in the literature has shown that general relativity can be formulated in terms of a jet bundle which, in local coordinates, has five entries: local coordinates on Lorentzian space-time, tetrads, connection one-forms, multivelocities corresponding to the tetrads and multivelocities corresponding to the connection one-forms. The derivatives of the Lagrangian with respect to the latter class of multivelocities give rise to a set of multimomenta which naturally occur in the constraint equations. Interestingly, all the constraint equations of general relativity are linear in terms of this class of multimomenta. This construction has been then extended to complex general relativity, where Lorentzian space-time is replaced by a four-complex-dimensional complex-Riemannian manifold. One then finds a holomorphic theory where the familiar constraint equations are replaced by a set of equations linear in the holomorphic multimomenta, providing such multimomenta vanish on a family of two-complex-dimensional surfaces. In quantum gravity, the problem arises to quantize a real or a holomorphic theory on the extended space where the multimomenta can be defined.Comment: 5 pages, plain-te

Quasi-linear SPDEs in divergence-form

We develop a solution theory in Hölder spaces for a quasi-linear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE lemmas which establish a priori Hölder bounds for a parabolic equation in divergence form with irregular right-hand-side term. We apply these bounds to the case of a right-hand-side noise term which is white in time and trace class in space, to obtain stretched exponential bounds for the Hölder semi-norms of the solution

Formal Adjoints and a Canonical Form for Linear Operators

We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.Comment: 3 page

Augmented resolution of linear hyperbolic systems under nonconservative form

Hyperbolic systems under nonconservative form arise in numerous applications modeling physical processes, for example from the relaxation of more general equations (e.g. with dissipative terms). This paper reviews an existing class of augmented Roe schemes and discusses their application to linear nonconservative hyperbolic systems with source terms. We extend existing augmented methods by redefining them within a common framework which uses a geometric reinterpretation of source terms. This results in intrinsically well-balanced numerical discretizations. We discuss two equivalent formulations: (1) a nonconservative approach and (2) a conservative reformulation of the problem. The equilibrium properties of the schemes are examined and the conditions for the preservation of the well-balanced property are provided. Transient and steady state test cases for linear acoustics and hyperbolic heat equations are presented. A complete set of benchmark problems with analytical solution, including transient and steady situations with discontinuities in the medium properties, are presented and used to assess the equilibrium properties of the schemes. It is shown that the proposed schemes satisfy the expected equilibrium and convergence properties

Fixed-Form Variational Posterior Approximation through Stochastic Linear Regression

We propose a general algorithm for approximating nonstandard Bayesian posterior distributions. The algorithm minimizes the Kullback-Leibler divergence of an approximating distribution to the intractable posterior distribution. Our method can be used to approximate any posterior distribution, provided that it is given in closed form up to the proportionality constant. The approximation can be any distribution in the exponential family or any mixture of such distributions, which means that it can be made arbitrarily precise. Several examples illustrate the speed and accuracy of our approximation method in practice