710 research outputs found
Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces
Optimal linear prediction (also known as kriging) of a random field
indexed by a compact metric space
can be obtained if the mean value function
and the covariance function
of are known. We
consider the problem of predicting the value of at some location
based on observations at locations which
accumulate at as (or, more generally, predicting
based on for linear functionals ).
Our main result characterizes the asymptotic performance of linear predictors
(as increases) based on an incorrect second order structure
, without any restrictive assumptions on such as stationarity. We, for the first time, provide
necessary and sufficient conditions on for
asymptotic optimality of the corresponding linear predictor holding uniformly
with respect to . These general results are illustrated by an example on the
sphere for the case of two isotropic covariance functions.Comment: 36 page
The rational SPDE approach for Gaussian random fields with general smoothness
A popular approach for modeling and inference in spatial statistics is to
represent Gaussian random fields as solutions to stochastic partial
differential equations (SPDEs) of the form , where
is Gaussian white noise, is a second-order differential
operator, and is a parameter that determines the smoothness of .
However, this approach has been limited to the case ,
which excludes several important models and makes it necessary to keep
fixed during inference.
We propose a new method, the rational SPDE approach, which in spatial
dimension is applicable for any , and thus remedies
the mentioned limitation. The presented scheme combines a finite element
discretization with a rational approximation of the function to
approximate . For the resulting approximation, an explicit rate of
convergence to in mean-square sense is derived. Furthermore, we show that
our method has the same computational benefits as in the restricted case
. Several numerical experiments and a statistical
application are used to illustrate the accuracy of the method, and to show that
it facilitates likelihood-based inference for all model parameters including
.Comment: 28 pages, 4 figure
Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
The numerical approximation of the solution to a stochastic partial
differential equation with additive spatial white noise on a bounded domain is
considered. The differential operator is assumed to be a fractional power of an
integer order elliptic differential operator. The solution is approximated by
means of a finite element discretization in space and a quadrature
approximation of an integral representation of the fractional inverse from the
Dunford-Taylor calculus.
For the resulting approximation, a concise analysis of the weak error is
performed. Specifically, for the class of twice continuously Fr\'echet
differentiable functionals with second derivatives of polynomial growth, an
explicit rate of weak convergence is derived, and it is shown that the
component of the convergence rate stemming from the stochasticity is doubled
compared to the corresponding strong rate. Numerical experiments for different
functionals validate the theoretical results.Comment: 22 pages, 1 figur
Numerical solution of fractional elliptic stochastic PDEs with spatial white noise
The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in R-d is considered. The differential operator is given by the fractional power L-beta, beta is an element of (0, 1) of an integer-order elliptic differential operator L and is therefore nonlocal. Its inverse L-beta is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation the inverse fractional-order operator L-beta is approximated by a weighted sum of nonfractional resolvents (I + exp(2yl)L)(-1) at certain quadrature nodes t(j) > 0. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for L = kappa(2) - Delta, kappa > 0 with homogeneous Dirichlet boundary conditions on the unit cube (0, 1)(d) in d = 1, 2, 3 spatial dimensions for varying beta is an element of (0, 1) attest to the theoretical results
Spreading Static Analysis with Frama-C in Industrial Contexts
International audienceThis article deals with the usage of Frama-C to detect runtime-errors. As static analysis for runtime-error detection is not a novelty, we will present significant new usages in industrial contexts, which represent a change in the ways this kind of tool is employed. The main goal is to have a scalable methodology for using static analysis through the development process and by a development team. This goal is achieved by performing analysis on partial pieces of code, by using the ACSL language for interface definitions, by choosing a bottom-up strategy to process the code, and by enabling a well-balanced definition of actors and skills. The methodology, designed during the research project U3CAT, has been applied in industrial contexts with good results as for the quality of verifications and for the performance in the industrial process
Toward a Broadband Astro-comb: Effects of Nonlinear Spectral Broadening in Optical Fibers
We propose and analyze a new approach to generate a broadband astro-comb by
spectral broadening of a narrowband astro-comb inside a highly nonlinear
optical fiber. Numerical modeling shows that cascaded four-wave-mixing
dramatically degrades the input comb's side-mode suppression and causes
side-mode amplitude asymmetry. These two detrimental effects can systematically
shift the center-of-gravity of astro-comb spectral lines as measured by an
astrophysical spectrograph with resolution \approx100,000; and thus lead to
wavelength calibration inaccuracy and instability. Our simulations indicate
that this performance penalty, as a result of nonlinear spectral broadening,
can be compensated by using a filtering cavity configured for double-pass. As
an explicit example, we present a design based on an Yb-fiber source comb (with
1 GHz repetition rate) that is filtered by double-passing through a low finesse
cavity (finesse = 208), and subsequent spectrally broadened in a 2-cm,
SF6-glass photonic crystal fiber. Spanning more than 300 nm with 16 GHz line
spacing, the resulting astro-comb is predicted to provide 1 cm/s (~10 kHz)
radial velocity calibration accuracy for an astrophysical spectrograph. Such
extreme performance will be necessary for the search for and characterization
of Earth-like extra-solar planets, and in direct measurements of the change of
the rate of cosmological expansion.Comment: 9 pages, 6 figure
Microsatellite instability, KRAS mutations and cellular distribution of TRAIL-receptors in early stage colorectal cancer.
Thus, we evaluated the immunofluorescence pattern of TRAIL-receptors and E-cadherin to assess the fraction of membrane-bound TRAIL-receptors in 231 selected patients with early-stage CRC undergoing surgical treatment only. Moreover, we investigated whether membrane staining for TRAIL-receptors as well as the presence of KRAS mutations or of microsatellite instability (MSI) had an effect on survival and thus a prognostic effect.
The fact that the receptors for the TNF-related apoptosis inducing ligand (TRAIL) are almost invariably expressed in colorectal cancer (CRC) represents the rationale for the employment of TRAIL-receptors targeting compounds for the therapy of patients affected by this tumor. Yet, first reports on the use of these bioactive agents provided disappointing results. We therefore hypothesized that loss of membrane-bound TRAIL-R might be a feature of some CRC and that the evaluation of membrane staining rather than that of the overall expression of TRAIL-R might predict the response to TRAIL-R targeting compounds in this tumor. As expected, almost all CRC samples stained positive for TRAIL-R1 and 2. Instead, membrane staining for these receptors was positive in only 71% and 16% of samples respectively. No correlation between KRAS mutation status or MSI-phenotype and prognosis could be detected. TRAIL-R1 staining intensity correlated with survival in univariate analysis, but only membranous staining of TRAIL-R1 and TRAIL-R2 on cell membranes was an independent predictor of survival (cox multivariate analysis: TRAIL-R1: p = 0.019, RR 2.06[1.12-3.77]; TRAIL-R2: p = 0.033, RR 3.63[1.11-11.84]). In contrast to the current assumptions, loss of membrane staining for TRAIL-receptors is a common feature of early stage CRC which supersedes the prognostic significance of their staining intensity. Failure to achieve therapeutic effects in recent clinical trials using TRAIL-receptors targeting compounds might be due to insufficient selection of patients bearing tumors with membrane-bound TRAIL-receptors
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