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Approximation of Bayesian Inverse Problems for PDEs
Get PDFInverse 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
Space variant PSF parameterization in image space using printed point source arrays on the HiRez PET/CT
Get PDFA new practical and computationally efficient method for deriving and applying the space-variant blurring component of the system matrix is proposed and applied to the HiRez PET/CT scanner. The point spread function (PSF) was sampled at 14400 locations within the field-of-view (FOV) using an array of 120 18-F printed point sources. An HP printer was modified to print the sources on a sheet of A4 paper. To provide enough annihilating material for the positrons and move the array accurately within the FOV a Perspex phantom was designed. The reconstructed PSFs were parameterized in image space and modeled with a pair of multidimensional (3-D) Gaussian distributions. Through the fitting of appropriate functions, model parameters were interpolated and extrapolated for the remaining positions in the FOV. Image reconstruction with resolution modeling was implemented using the expectation maximization algorithm (OP-OSEM) and space variant image based convolution operations. Initial analysis shows significant improvements in the resolution using space variant kernels (reduction of FWHM from 5.5mm down to 2mm at 20cm radially). The improvements are more pronounced at the edge of the FOV when compared to the space invariant method where the discrepancy between the measured space variant blurring kernel and the invariant kernel are larger. Using the printer for producing radioactive point sources, the PSF was sampled at 14400 positions in less than 24h. Parameterizing the kernels in image space also provides a computation efficient alternative to projection space PSF parameterization with similar resolution improvements (a uniform resolution of 2mm throughout the FOV) and minimal increase in the reconstruction time
Further developments towards a genome-scale metabolic model of yeast
Get PDFBackground
To date, several genome-scale network reconstructions have been used to describe the metabolism of the yeast Saccharomyces cerevisiae, each differing in scope and content. The recent community-driven reconstruction, while rigorously evidenced and well annotated, under-represented metabolite transport, lipid metabolism and other pathways, and was not amenable to constraint-based analyses because of lack of pathway connectivity.
Results
We have expanded the yeast network reconstruction to incorporate many new reactions from the literature and represented these in a well-annotated and standards-compliant manner. The new reconstruction comprises 1102 unique metabolic reactions involving 924 unique metabolites - significantly larger in scope than any previous reconstruction. The representation of lipid metabolism in particular has improved, with 234 out of 268 enzymes linked to lipid metabolism now present in at least one reaction. Connectivity is emphatically improved, with more than 90% of metabolites now reachable from the growth medium constituents. The present updates allow constraint-based analyses to be performed; viability predictions of single knockouts are comparable to results from in vivo experiments and to those of previous reconstructions.
Conclusions
We report the development of the most complete reconstruction of yeast metabolism to date that is based upon reliable literature evidence and richly annotated according to MIRIAM standards. The reconstruction is available in the Systems Biology Markup Language (SBML) and via a publicly accessible database
http://www.comp-sys-bio.org/yeastnet
Collective Choice as Information Theory: Towards a Theory of Gravitas
Get PDFThe present paper introduces a new approach to the the theory of voting in
the context of binary collective choice, which seeks to define a dynamic optimal voting rule by using insights derived from the mathematical theory of information. In order to de¯ne such a voting rule, a method of defining a real-valued measure of the weight of independent opinion of an arbitrary set of voters is suggested, which is value free to the extent that it depends only on probabilistic information extracted from previous patterns of voting, but does not require for its definition any direct information concerning either the correctness or incorrectness of previous voting decisions, or the content of those decisions. The approach to the definition of such a measure, which the author calls gravitas, is axiomatic. The voting rule is then defined by comparing the gravitas of the set of those voters who vote for a given motion with the gravitas of the set of those
who vote against that motion
An inhomogeneous stochastic rate process for evolution from states in an information geometric neighbourhood of uniform fitness
Get PDFThis study elaborates some examples of a simple evolutionary stochastic rate process
where the population rate of change depends on the distribution of properties---so
different cohorts change at different rates. We investigate
the effect on the evolution arising from parametrized perturbations of
uniformity for the initial inhomogeneity. The information geometric
neighbourhood system yields also solutions
for a wide range of other initial inhomogeneity distributions,
including approximations to truncated Gaussians of arbitrarily small variance
and distributions with pronounced extreme values.
It is found that, under quite
considerable alterations in the shape and variance of the initial distribution of inhomogeneity
in unfitness, the decline of the mean does change markedly with the variation in starting conditions,
but the net population evolution seems surprisingly stable
Noncommutative Fourier Analysis
Get PDFThis is a high-level, hopefully readable, expository account of the Plancherel Theorem for liminal groups, with special emphasis on reductive groups over local fields
Towards a genome-scale kinetic model of cellular metabolism
Get PDFBackground
Advances in bioinformatic techniques and analyses have led to the availability of genome-scale metabolic reconstructions. The size and complexity of such networks often means that their potential behaviour can only be analysed with constraint-based methods. Whilst requiring minimal experimental data, such methods are unable to give insight into cellular substrate concentrations. Instead, the long-term goal of systems biology is to use kinetic modelling to characterize fully the mechanics of each enzymatic reaction, and to combine such knowledge to predict system behaviour.
Results
We describe a method for building a parameterized genome-scale kinetic model of a metabolic network. Simplified linlog kinetics are used and the parameters are extracted from a kinetic model repository. We demonstrate our methodology by applying it to yeast metabolism. The resultant model has 956 metabolic reactions involving 820 metabolites, and, whilst approximative, has considerably broader remit than any existing models of its type. Control analysis is used to identify key steps within the system.
Conclusions
Our modelling framework may be considered a stepping-stone toward the long-term goal of a fully-parameterized model of yeast metabolism. The model is available in SBML format from the BioModels database (BioModels ID: MODEL1001200000) and at http://www.mcisb.org/resources/genomescale/
Fast iterative solvers for buoyancy driven flow problems
Get PDFWe outline a new class of robust and efficient methods for
solving the Navier-Stokes equations with a
Boussinesq model for buoyancy driven flow. We describe
a general solution strategy that has two
basic building blocks: an implicit time integrator using
a stabilized trapezoid rule with an explicit
Adams-Bashforth method for error control, and a
robust Krylov subspace solver for the spatially discretized system.
We present numerical experiments illustrating the efficiency of
the chosen preconditioning schemes with respect to the
discretization parameters
Normalizers of 2-subgroups in black-box groups
Get PDFIn this paper we rene and extend the applicability of the algorithms in Bates and Rowley (Arch. Math. 92 (2009) 7-13) for computing part of the normalizer of a 2-subgroup in a black-box group
Geometric structure in the tempered dual of SL(N)
Get PDFThe Aubert-Baum-Plymen conjecture says that there is a simple geometric structure within the representation theory of p-adic groups such as SL(N).
This PhD thesis will focus on SL(2), SL(3) and SL(4). We prove the conjecture for SL(2); part (3) of the conjecture for SL(3); and the principal series case for SL(4, Q_p) with p >2.
The constructions are, for the most part, very explicit.
One case is especially interesting: we reveal a tetrahedron of reducibility in the tempered dual of SL(4, Q_2)