Corrector theory for MsFEM and HMM in random media

We analyze the random fluctuations of several multi-scale algorithms such as the multi-scale finite element method (MsFEM) and the finite element heterogeneous multiscale method (HMM), that have been developed to solve partial differential equations with highly heterogeneous coefficients. Such multi-scale algorithms are often shown to correctly capture the homogenization limit when the highly oscillatory random medium is stationary and ergodic. This paper is concerned with the random fluctuations of the solution about the deterministic homogenization limit. We consider the simplified setting of the one dimensional elliptic equation, where the theory of random fluctuations is well understood. We develop a fluctuation theory for the multi-scale algorithms in the presence of random environments with short-range and long-range correlations. What we find is that the computationally more expensive method MsFEM captures the random fluctuations both for short-range and long-range oscillations in the medium. The less expensive method HMM correctly captures the fluctuations for long-range oscillations and strongly amplifies their size in media with short-range oscillations. We present a modified scheme with an intermediate computational cost that captures the random fluctuations in all cases.Comment: 41 page

Calcul des doses d'irrigation pour les sols salés : application d'une méthode géochimique de calcul à un sol tunisien

La méthode de prévision du volume de lessivage recommandée par la FAO (méthode classique) néglige la précipitation des minéraux à partir de la solution du sol. La méthode dite géochimique prend en compte ces phénomènes par le biais d'un modèle thermodynamique. L'application de ces deux méthodes à la station de Cherfech en Tunisie permet de comparer leur validit

Two-scale convergence for locally-periodic microstructures and homogenization of plywood structures

The introduced notion of locally-periodic two-scale convergence allows to average a wider range of microstructures, compared to the periodic one. The compactness theorem for the locally-periodic two-scale convergence and the characterisation of the limit for a sequence bounded in $H^1(\Omega)$ are proven. The underlying analysis comprises the approximation of functions, which periodicity with respect to the fast variable depends on the slow variable, by locally-periodic functions, periodic in subdomains smaller than the considered domain, but larger than the size of microscopic structures. The developed theory is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.Comment: 22 pages, 4 figure

Convergence of the Generalized Volume Averaging Method on a Convection-Diffusion Problem: A Spectral Perspective

A mixed formulation is proposed and analyzed mathematically for coupled convection-diffusion in heterogeneous medias. Transfer in solid parts driven by pure diffusion is coupled with convection-diffusion transfer in fluid parts. This study is carried out for translation-invariant geometries (general infinite cylinders) and unidirectional flows. This formulation brings to the fore a new convection-diffusion operator, the properties of which are mathematically studied: its symmetry is first shown using a suitable scalar product. It is proved to be self-adjoint with compact resolvent on a simple Hilbert space. Its spectrum is characterized as being composed of a double set of eigenvalues: one converging towards −∞ and the other towards +∞, thus resulting in a nonsectorial operator. The decomposition of the convection-diffusion problem into a generalized eigenvalue problem permits the reduction of the original three-dimensional problem into a two-dimensional one. Despite the operator being nonsectorial, a complete solution on the infinite cylinder, associated to a step change of the wall temperature at the origin, is exhibited with the help of the operator’s two sets of eigenvalues/eigenfunctions. On the computational point of view, a mixed variational formulation is naturally associated to the eigenvalue problem. Numerical illustrations are provided for axisymmetrical situations, the convergence of which is found to be consistent with the numerical discretization

Evaluation of novel data-driven metrics of amyloid β deposition for longitudinal PET studies

PURPOSE: Positron emission tomography (PET) provides in vivo quantification of amyloid-β (Aβ) pathology. Established methods for assessing Aβ burden can be affected by physiological and technical factors. Novel, data-driven metrics have been developed to account for these sources of variability. We aimed to evaluate the performance of four data-driven amyloid PET metrics against conventional techniques, using a common set of criteria. METHODS: Three cohorts were used for evaluation: Insight 46 (N=464, [18F]florbetapir), AIBL (N=277, [18F]flutemetamol), and an independent test-retest data (N=10, [18F]flutemetamol). Established metrics of amyloid tracer uptake included the Centiloid (CL) and where dynamic data was available, the non-displaceable binding potential (BPND). The four data driven metrics computed were the amyloid load (Aβ load), the Aβ PET pathology accumulation index (Aβ index), the Centiloid derived from non-negative matrix factorisation (CLNMF), and the amyloid pattern similarity score (AMPSS). These metrics were evaluated using reliability and repeatability in test-retest data, associations with BPND and CL, and sample size estimates to detect a 25% slowing in Aβ accumulation. RESULTS: All metrics showed good reliability. Aβ load, Aβ index and CLNMF were strong associated with the BPND. The associations with CL suggests that cross-sectional measures of CLNMF, Aβ index and Aβ load are robust across studies. Sample size estimates for secondary prevention trial scenarios were the lowest for CLNMF and Aβ load compared to the CL. CONCLUSION: Among the novel data-driven metrics evaluated, the Aβ load, the Aβ index and the CLNMF can provide comparable performance to more established quantification methods of Aβ PET tracer uptake. The CLNMF and Aβ load could offer a more precise alternative to CL, although further studies in larger cohorts should be conducted

Association of pelvic fracture patterns, pelvic binder use and arterial angio-embolization with transfusion requirements and mortality rates; a 7-year retrospective cohort study.

Pelvic fractures are severe injuries with frequently associated multi-system trauma and a high mortality rate. The value of the pelvic fracture pattern for predicting transfusion requirements and mortality is not entirely clear. To address hemorrhage from pelvic injuries, the early application of pelvic binders is now recommended and arterial angio-embolization is widely used for controlling arterial bleeding. Our aim was to assess the association of the pelvic fracture pattern according to the Tile classification system with transfusion requirements and mortality rates, and to evaluate the correlation between the use of pelvic binders and arterial angio-embolization and the mortality of patients with pelvic fractures. Single-center retrospective cohort study including all consecutive patients with a pelvic fracture from January 2008 to June 2015. All radiological fracture patterns were independently reviewed and grouped according to the Tile classification system. Data on patient demographics, use of pelvic binders and arterial angio-embolization, transfusion requirements and mortality were extracted from the institutional trauma registry and analyzed. The present study included 228 patients. Median patient age was 43.5 years and 68.9% were male. The two independent observers identified 105 Tile C (46.1%), 71 Tile B (31.1%) and 52 Tile A (22.8%) fractures, with substantial to almost perfect interobserver agreement (Kappa 0.70-0.83). Tile C fractures were associated with a higher mortality rate (p = 0.001) and higher transfusion requirements (p < 0.0001) than Tile A or B fractures. Arterial angio-embolization for pelvic bleeding (p = 0.05) and prehospital pelvic binder placement (p = 0.5) were not associated with differences in mortality rates. Tile C pelvic fractures are associated with higher transfusion requirements and a higher mortality rate than Tile A or B fractures. No association between the use of pelvic binders or arterial angio-embolization and survival was observed in this cohort of patients with pelvic fractures

Assessment of a polygenic hazard score for the onset of pre-clinical Alzheimer’s disease

Background: With a growing number of loci associated with late-onset (sporadic) Alzheimer’s disease (AD), the polygenic contribution to AD is now well established. The development of polygenic risk score approaches have shown promising results for identifying individuals at higher risk of developing AD, thereby facilitating the development of preventative and therapeutic strategies. A polygenic hazard score (PHS) has been proposed to quantify age-specific genetic risk for AD. In this study, we assessed the predictive power and transferability of this PHS in an independent cohort, to support its clinical utility. Results: Using genotype and imaging data from 780 individuals enrolled in the Australian Imaging, Biomarkers and Lifestyle (AIBL) study, we investigated associations between the PHS and several AD-related traits, including 1) cross-sectional Aβ-amyloid (Aβ) deposition, 2) longitudinal brain atrophy, 3) longitudinal cognitive decline, 4) age of onset. Except in the cognitive domain, we obtained results that were consistent with previously published findings. The PHS was associated with increased Aβ burden, faster regional brain atrophy and an earlier age of onset. Conclusion: Overall, the results support the predictive power of a PHS, however, with only marginal improvement compared to apolipoprotein E alone