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 $\{Z(x)\}_{x\in\mathcal{X}}$ indexed by a compact metric space $(\mathcal{X},d_{\mathcal{X}})$ can be obtained if the mean value function $m\colon\mathcal{X}\to\mathbb{R}$ and the covariance function $\varrho\colon\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ of $Z$ are known. We consider the problem of predicting the value of $Z(x^*)$ at some location $x^*\in\mathcal{X}$ based on observations at locations $\{x_j\}_{j=1}^n$ which accumulate at $x^*$ as $n\to\infty$ (or, more generally, predicting $f(Z)$ based on $\{ f_j(Z) \}_{j=1}^n$ for linear functionals $f, f_1, \ldots, f_n$). Our main result characterizes the asymptotic performance of linear predictors (as $n$ increases) based on an incorrect second order structure $(\tilde{m},\tilde{\varrho})$, without any restrictive assumptions on $\varrho, \tilde{\varrho}$ such as stationarity. We, for the first time, provide necessary and sufficient conditions on $(\tilde{m},\tilde{\varrho})$ for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to $f$. These general results are illustrated by an example on the sphere $\mathbb{S}^2$ 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 $L^{\beta}u = \mathcal{W}$, where $\mathcal{W}$ is Gaussian white noise, $L$ is a second-order differential operator, and $\beta>0$ is a parameter that determines the smoothness of $u$. However, this approach has been limited to the case $2\beta\in\mathbb{N}$, which excludes several important models and makes it necessary to keep $\beta$ fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension $d\in\mathbb{N}$ is applicable for any $\beta>d/4$, and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function $x^{-\beta}$ to approximate $u$. For the resulting approximation, an explicit rate of convergence to $u$ in mean-square sense is derived. Furthermore, we show that our method has the same computational benefits as in the restricted case $2\beta\in\mathbb{N}$. 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 $\beta$.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