27,773 research outputs found
Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors
This article extends the framework of Bayesian inverse problems in
infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer.
19:451--559, 2010) and others, to the case of a heavy-tailed prior measure in
the family of stable distributions, such as an infinite-dimensional Cauchy
distribution, for which polynomial moments are infinite or undefined. It is
shown that analogues of the Karhunen--Lo\`eve expansion for square-integrable
random variables can be used to sample such measures on quasi-Banach spaces.
Furthermore, under weaker regularity assumptions than those used to date, the
Bayesian posterior measure is shown to depend Lipschitz continuously in the
Hellinger metric upon perturbations of the misfit function and observed data.Comment: To appear in Inverse Problems and Imaging. This preprint differs from
the final published version in layout and typographical detail
Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors
The Bayesian perspective on inverse problems has attracted much mathematical
attention in recent years. Particular attention has been paid to Bayesian
inverse problems (BIPs) in which the parameter to be inferred lies in an
infinite-dimensional space, a typical example being a scalar or tensor field
coupled to some observed data via an ODE or PDE. This article gives an
introduction to the framework of well-posed BIPs in infinite-dimensional
parameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) and
others. This framework has the advantage of ensuring uniformly well-posed
inference problems independently of the finite-dimensional discretisation used
for numerical solution. Recently, this framework has been extended to the case
of a heavy-tailed prior measure in the family of stable distributions, such as
an infinite-dimensional Cauchy distribution, for which polynomial moments are
infinite or undefined. It is shown that analogues of the Karhunen--Lo\`eve
expansion for square-integrable random variables can be used to sample such
measures on quasi-Banach spaces. Furthermore, under weaker regularity
assumptions than those used to date, the Bayesian posterior measure is shown to
depend Lipschitz continuously in the Hellinger and total variation metrics upon
perturbations of the misfit function and observed data.Comment: To appear in the proceedings of the 88th Annual Meeting of the
International Association of Applied Mathematics and Mechanics (GAMM), Weimar
2017. This preprint differs from the final published version in pagination
and typographical detai
Evaluation of a Stirling engine heater bypass with the NASA Lewis nodal-analysis performance code
In support of the U.S. Department of Energy's Stirling Engine Highway Vehicle Systems program, the NASA Lewis Research Center investigated whether bypassing the P-40 Stirling engine heater during regenerative cooling would improve engine performance. The Lewis nodal-analysis Stirling engine computer simulation was used for this investigation. Results for the heater-bypass concept showed no significant improvement in the indicated thermal efficiency for the P-40 Stirling engine operating at full-power and part-power conditions. Optimizing the heater tube length produced a small increase in the indicated thermal efficiency with the heater-bypass concept
Quasi-invariance of countable products of Cauchy measures under non-unitary dilations
Consider an infinite sequence (Un)n∈N of independent Cauchy random variables, defined by a sequence (δn)n∈N of location parameters and a sequence (γn)n∈N of scale parameters. Let (Wn)n∈N be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence (σnγn)n∈N of scale parameters, with σn≠0 for all n∈N. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of (Un)n∈N and (Wn)n∈N are equivalent if and only if the sequence (|σn|−1)n∈N is square-summable
Equivalence of weak and strong modes of measures on topological vector spaces
A strong mode of a probability measure on a normed space can be defined
as a point such that the mass of the ball centred at uniformly
dominates the mass of all other balls in the small-radius limit. Helin and
Burger weakened this definition by considering only pairwise comparisons with
balls whose centres differ by vectors in a dense, proper linear subspace of
, and posed the question of when these two types of modes coincide. We show
that, in a more general setting of metrisable vector spaces equipped with
measures that are finite on bounded sets, the density of and a uniformity
condition suffice for the equivalence of these two types of modes. We
accomplish this by introducing a new, intermediate type of mode. We also show
that these modes can be inequivalent if the uniformity condition fails. Our
results shed light on the relationships between among various notions of
maximum a posteriori estimator in non-parametric Bayesian inference.Comment: 22 pages, 3 figure
Convergence Rates of Gaussian ODE Filters
A recently-introduced class of probabilistic (uncertainty-aware) solvers for
ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to
initial value problems. These methods model the true solution and its first
derivatives \emph{a priori} as a Gauss--Markov process ,
which is then iteratively conditioned on information about . This
article establishes worst-case local convergence rates of order for a
wide range of versions of this Gaussian ODE filter, as well as global
convergence rates of order in the case of and an integrated Brownian
motion prior, and analyses how inaccurate information on coming from
approximate evaluations of affects these rates. Moreover, we show that, in
the globally convergent case, the posterior credible intervals are well
calibrated in the sense that they globally contract at the same rate as the
truncation error. We illustrate these theoretical results by numerical
experiments which might indicate their generalizability to .Comment: 26 pages, 5 figure
Exact active subspace Metropolis-Hastings, with applications to the Lorenz-96 system
We consider the application of active subspaces to inform a
Metropolis-Hastings algorithm, thereby aggressively reducing the computational
dimension of the sampling problem. We show that the original formulation, as
proposed by Constantine, Kent, and Bui-Thanh (SIAM J. Sci. Comput.,
38(5):A2779-A2805, 2016), possesses asymptotic bias. Using pseudo-marginal
arguments, we develop an asymptotically unbiased variant. Our algorithm is
applied to a synthetic multimodal target distribution as well as a Bayesian
formulation of a parameter inference problem for a Lorenz-96 system
Strong convergence rates of probabilistic integrators for ordinary differential equations
Probabilistic integration of a continuous dynamical system is a way of
systematically introducing model error, at scales no larger than errors
introduced by standard numerical discretisation, in order to enable thorough
exploration of possible responses of the system to inputs. It is thus a
potentially useful approach in a number of applications such as forward
uncertainty quantification, inverse problems, and data assimilation. We extend
the convergence analysis of probabilistic integrators for deterministic
ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\
Comput.}, 2017), to establish mean-square convergence in the uniform norm on
discrete- or continuous-time solutions under relaxed regularity assumptions on
the driving vector fields and their induced flows. Specifically, we show that
randomised high-order integrators for globally Lipschitz flows and randomised
Euler integrators for dissipative vector fields with polynomially-bounded local
Lipschitz constants all have the same mean-square convergence rate as their
deterministic counterparts, provided that the variance of the integration noise
is not of higher order than the corresponding deterministic integrator. These
and similar results are proven for probabilistic integrators where the random
perturbations may be state-dependent, non-Gaussian, or non-centred random
variables.Comment: 25 page
Single stage, low noise, advanced technology fan. Volume 1: Aerodynamic design
The aerodynamic design for a half-scale fan vehicle, which would have application on an advanced transport aircraft, is described. The single stage advanced technology fan was designed to a pressure ratio of 1.8 at a tip speed of 503 m/sec 11,650 ft/sec). The fan and booster components are designed in a scale model flow size convenient for testing with existing facility and vehicle hardware. The design corrected flow per unit annulus area at the fan face is 215 kg/sec sq m (44.0 lb m/sec sq ft) with a hub-tip ratio of 0.38 at the leading edge of the fan rotor. This results in an inlet corrected airflow of 117.9 kg/sec (259.9 lb m/sec) for the selected rotor tip diameter if 90.37 cm (35.58 in.). The variable geometry inlet is designed utilizing a combination of high throat Mach number and acoustic treatment in the inlet diffuser for noise suppression (hybrid inlet). A variable fan exhaust nozzle was assumed in conjunction with the variable inlet throat area to limit the required area change of the inlet throat at approach and hence limit the overall diffusion and inlet length. The fan exit duct design was primarily influenced by acoustic requirements, including length of suppressor wall treatment; length, thickness and position on a duct splitter for additional suppressor treatment; and duct surface Mach numbers
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