Approximation of Bayesian inverse problems for PDEs

Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier–Stokes equation from Lagrangian and Eulerian observations, respectively

Variational data assimilation using targetted random walks

The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis (offline hindcasting). In either of these scenarios it can be important to assess uncertainties in the assimilated state. Ideally it would be desirable to have complete information concerning the Bayesian posterior distribution for unknown state, given data. The purpose of this paper is to show that complete computational probing of this posterior distribution is now within reach in the offline situation. In this paper we will introduce an MCMC method which enables us to directly sample from the Bayesian\ud posterior distribution on the unknown functions of interest, given observations. Since we are aware that these\ud methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walk-type MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however more sophisticated MCMC methods are available\ud which do exploit derivative information. For simplicity of exposition we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number (Stokes flow) scenario in a two dimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces

MCMC methods for functions modifying old algorithms to make\ud them faster

Many problems arising in applications result in the need\ud to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the unknown function. We describe an approach to modifying a whole range of MCMC methods which ensures that their speed of convergence is robust under mesh refinement. In the applications of interest the data is often sparse and the prior specification is an essential part of the overall modeling strategy. The algorithmic approach that we describe is applicable whenever the desired probability measure has density with respect to a Gaussian process or Gaussian random field prior, and to some useful non-Gaussian priors constructed through random truncation. Applications are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration. The key design principle is to formulate the MCMC method for functions. This leads to algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems

The Redshift Distribution of FIRST Radio Sources at 1 mJy

We present spectra for a sample of radio sources from the FIRST survey, and use them to define the form of the redshift distribution of radio sources at mJy levels.We targeted 365 sources and obtained 46 redshifts (13 per cent of the sample). We find that our sample is complete in redshift measurement to R $\sim 18.6$, corresponding to $z\sim 0.2$. Early-type galaxies represent the largest subset (45 per cent) of the sample and have redshifts 0.15\la z \la 0.5 ; late-type galaxies make up 15 per cent of the sample and have redshifts 0.05\la z \la 0.2; starbursting galaxies are a small fraction ($\sim 6$ per cent), and are very nearby (z\la 0.05). Some 9 per cent of the population have Seyfert1/quasar-type spectra, all at z\ga 0.8, and there are 4 per cent are Seyfert2 type galaxies at intermediate redshifts ($z\sim 0.2$). Using our measurements and data from the Phoenix survey, we obtain an estimate for $N(z)$ at $S_{1.4 \rm {GHz}}\ge 1$ mJy and compare this with model predictions. At variance with previous conclusions, we find that the population of starbursting objects makes up \la 5 per cent of the radio population at S $\sim 1$ mJy.Comment: 20 pages, sumbitted to MNRA

Deep spectroscopy of z~1 6C radio galaxies - II. Breaking the redshift-radio power degeneracy

The results of a spectroscopic analysis of 3CR and 6C radio galaxies at redshift z~1 are contrasted with the properties of lower redshift radio galaxies, chosen to be matched in radio luminosity to the 6C sources studied at z~1, thus enabling the P-z degeneracy to be broken. Partial rank correlations and principal component analysis have been used to determine which of z and P are the critical parameters underlying the observed variation of the ionization state andd kinematics of the emission line gas. [OII]/H-beta is shown to be a useful ionization mechanism diagnostic. Statistical analysis of the data shows that the ionization state of the emission line gas is strongly correlated with radio power, once the effects of other parameters are removed. No dependence of ionization state on z is observed, implying that the ionization state of the emission line gas is solely a function of the AGN properties rather than the hostt galaxy and/or environment. Statistical analysis of the kinematic properties of the emission line gas shows that these are strongly correlated independently withh both P and z. The correlation with redshift is the stronger of the two, suggesting that host galaxy composition or environment may play a role in producing the less extreme gas kinematics observed in the emission line regions of low redshift galaxies. For both the ionization and kinematic properties of thee galaxies, the independent correlations observed with radio size are strongest. Radio source age is a determining factor for the extended emission line regions.Comment: 10 pages, 5 figures, accepted for publication in MNRA

Rapid communications M u m p s in Ir e l a n d, 2004-2008

Following a national mumps outbreak that began in November 2004 and continued into 2005, the number of mumps notifications in Ireland waned in the latter half of 2006 and during 2007 (Figure 1). However, mumps notifications have started to increase again in 2008 (Figure 1). The number of mumps notifications annuall

Listeriolysin S, a Novel Peptide Haemolysin Associated with a Subset of Lineage I Listeria monocytogenes

peer-reviewedStreptolysin S (SLS) is a bacteriocin-like haemolytic and cytotoxic virulence factor that plays a key role in the virulence of Group A Streptococcus (GAS), the causative agent of pharyngitis, impetigo, necrotizing fasciitis and streptococcal toxic shock syndrome. Although it has long been thought that SLS and related peptides are produced by GAS and related streptococci only, there is evidence to suggest that a number of the most notorious Gram-positive pathogenic bacteria, including Listeria monocytogenes, Clostridium botulinum and Staphylococcus aureus, produce related peptides. The distribution of the L. monocytogenes cluster is particularly noteworthy in that it is found exclusively among a subset of lineage I strains; i.e., those responsible for the majority of outbreaks of listeriosis. Expression of these genes results in the production of a haemolytic and cytotoxic factor, designated Listeriolysin S, which contributes to virulence of the pathogen as assessed by murine- and human polymorphonuclear neutrophil–based studies. Thus, in the process of establishing the existence of an extended family of SLS-like modified virulence peptides (MVPs), the genetic basis for the enhanced virulence of a proportion of lineage I L. monocytogenes may have been revealed.Work is funded by the Irish Government under the National Development Plan, through a Science Foundation Ireland Investigator award to CH, PR and PC (06/IN.1/B98)