28,057 research outputs found
Dimension-Independent MCMC Sampling for Inverse Problems with Non-Gaussian Priors
The computational complexity of MCMC methods for the exploration of complex
probability measures is a challenging and important problem. A challenge of
particular importance arises in Bayesian inverse problems where the target
distribution may be supported on an infinite dimensional space. In practice
this involves the approximation of measures defined on sequences of spaces of
increasing dimension. Motivated by an elliptic inverse problem with
non-Gaussian prior, we study the design of proposal chains for the
Metropolis-Hastings algorithm with dimension independent performance.
Dimension-independent bounds on the Monte-Carlo error of MCMC sampling for
Gaussian prior measures have already been established. In this paper we provide
a simple recipe to obtain these bounds for non-Gaussian prior measures. To
illustrate the theory we consider an elliptic inverse problem arising in
groundwater flow. We explicitly construct an efficient Metropolis-Hastings
proposal based on local proposals, and we provide numerical evidence which
supports the theory.Comment: 26 pages, 7 figure
Backreaction for Einstein-Rosen waves coupled to a massless scalar field
We present a one-parameter family of exact solutions to Einstein equations
that may be used to study the nature of the Green-Wald backreaction framework.
Our explicit example is a family of Einstein-Rosen waves coupled to a massless
scalar field. This solution may be reinterpreted as a generalized three-torus
polarized Gowdy cosmology with scalar and gravitational waves. We use it to
illustrate essential properties of the Green-Wald approach. Among other things
we show that within our model the Green-Wald framework uniquely determines
backreaction for finite size inhomogeneities on a predefined background. The
results agree with those calculated in the Charach-Malin approach. In the
vacuum limit, the Green-Wald, the Charach-Malin and the Isaacson method imply
identical backreaction as expected.Comment: 26 pages; minor changes to match published versio
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Robust permanence for ecological equations with internal and external feedbacks.
Species experience both internal feedbacks with endogenous factors such as trait evolution and external feedbacks with exogenous factors such as weather. These feedbacks can play an important role in determining whether populations persist or communities of species coexist. To provide a general mathematical framework for studying these effects, we develop a theorem for coexistence for ecological models accounting for internal and external feedbacks. Specifically, we use average Lyapunov functions and Morse decompositions to develop sufficient and necessary conditions for robust permanence, a form of coexistence robust to large perturbations of the population densities and small structural perturbations of the models. We illustrate how our results can be applied to verify permanence in non-autonomous models, structured population models, including those with frequency-dependent feedbacks, and models of eco-evolutionary dynamics. In these applications, we discuss how our results relate to previous results for models with particular types of feedbacks
Robust permanence for interacting structured populations
The dynamics of interacting structured populations can be modeled by
where , , and
are matrices with non-negative off-diagonal entries. These models are
permanent if there exists a positive global attractor and are robustly
permanent if they remain permanent following perturbations of .
Necessary and sufficient conditions for robust permanence are derived using
dominant Lyapunov exponents of the with respect to
invariant measures . The necessary condition requires for all ergodic measures with support in the boundary of the
non-negative cone. The sufficient condition requires that the boundary admits a
Morse decomposition such that for all invariant
measures supported by a component of the Morse decomposition. When the
Morse components are Axiom A, uniquely ergodic, or support all but one
population, the necessary and sufficient conditions are equivalent.
Applications to spatial ecology, epidemiology, and gene networks are given
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