7,376 research outputs found
Stability of de Sitter spacetime under isotropic perturbations in semiclassical gravity
A spatially flat Robertson-Walker spacetime driven by a cosmological constant
is non-conformally coupled to a massless scalar field. The equations of
semiclassical gravity are explicitly solved for this case, and a
self-consistent de Sitter solution associated with the Bunch-Davies vacuum
state is found (the effect of the quantum field is to shift slightly the
effective cosmological constant). Furthermore, it is shown that the corrected
de Sitter spacetime is stable under spatially-isotropic perturbations of the
metric and the quantum state. These results are independent of the free
renormalization parameters.Comment: 19 pages, REVTeX
Stochastic ordinary differential equations in applied and computational mathematics
Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation
A path-integral approach to Bayesian inference for inverse problems using the semiclassical approximation
We demonstrate how path integrals often used in problems of theoretical
physics can be adapted to provide a machinery for performing Bayesian inference
in function spaces. Such inference comes about naturally in the study of
inverse problems of recovering continuous (infinite dimensional) coefficient
functions from ordinary or partial differential equations (ODE, PDE), a problem
which is typically ill-posed. Regularization of these problems using
function spaces (Tikhonov regularization) is equivalent to Bayesian
probabilistic inference, using a Gaussian prior. The Bayesian interpretation of
inverse problem regularization is useful since it allows one to quantify and
characterize error and degree of precision in the solution of inverse problems,
as well as examine assumptions made in solving the problem -- namely whether
the subjective choice of regularization is compatible with prior knowledge.
Using path-integral formalism, Bayesian inference can be explored through
various perturbative techniques, such as the semiclassical approximation, which
we use in this manuscript. Perturbative path-integral approaches, while
offering alternatives to computational approaches like Markov-Chain-Monte-Carlo
(MCMC), also provide natural starting points for MCMC methods that can be used
to refine approximations.
In this manuscript, we illustrate a path-integral formulation for inverse
problems and demonstrate it on an inverse problem in membrane biophysics as
well as inverse problems in potential theories involving the Poisson equation.Comment: Fixed some spelling errors and the author affiliations. This is the
version accepted for publication by J Stat Phy
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