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

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

Homogenization of linear transport equations in a stationary ergodic setting

We study the homogenization of a linear kinetic equation which models the evolution of the density of charged particles submitted to a highly oscillating electric field. The electric field and the initial density are assumed to be random and stationary. We identify the asymptotic microscopic and macroscopic profiles of the density, and we derive formulas for these profiles when the space dimension is equal to one.Comment: 24 page

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

Cortical thickness measurement from magnetic resonance images using partial volume estimation

Measurement of the cortical thickness from 3D Magnetic Resonance Imaging (MRI) can aid diagnosis and longitudinal studies of a wide range of neurodegenerative diseases. We estimate the cortical thickness using a Laplacian approach whereby equipotentials analogous to layers of tissue are computed. The thickness is then obtained using an Eulerian approach where partial differential equations (PDE) are solved, avoiding the explicit tracing of trajectories along the streamlines gradient. This method has the advantage of being relatively fast and insure unique correspondence points between the inner and outer boundaries of the cortex. The original method is challenged when the thickness of the cortex is of the same order of magnitude as the image resolution since partial volume (PV) effect is not taken into account at the gray matter (GM) boundaries. We propose a novel way to take into account PV which improves substantially accuracy and robustness. We model PV by computing a mixture of pure Gaussian probability distributions and use this estimate to initialize the cortical thickness estimation. On synthetic phantoms experiments, the errors were divided by three while reproducibility was improved when the same patients was scanned three consecutive times

Supervised method to build an atlas database for multi-atlas segmentation-propagation

Multi-atlas based segmentation-propagation approaches have been shown to obtain accurate parcelation of brain structures. However, this approach requires a large number of manually delineated atlases, which are often not available. We propose a supervised method to build a population specific atlas database, using the publicly available Internet Brain Segmentation Repository (IBSR). The set of atlases grows iteratively as new atlases are added, so that its segmentation capability may be enhanced in the multi-atlas based approach. Using a dataset of 210 MR images of elderly subjects (170 elderly controls, 40 Alzheimer's disease) from the Australian Imaging, Biomarkers and Lifestyle (AIBL) study, 40 MR images were segmented to build a population specific atlas database for the purpose of multi-atlas segmentation-propagation. The population specific atlases were used to segment the elderly population of 210 MR images, and were evaluated in terms of the agreement among the propagated labels. The agreement was measured by using the entropy H of the probability image produced when fused by voting rule and the partial moment mu(2) of the histogram. Compared with using IBSR atlases, the population specific atlases obtained a higher agreement when dealing with images of elderly subjects

Cross-sectional and Longitudinal Analysis of the Relationship Between A beta Deposition, Cortical Thickness, and Memory in Cognitively Unimpaired Individuals and in Alzheimer Disease

IMPORTANCE beta-amyloid (A beta) deposition is one of the hallmarks of Alzheimer disease. A beta deposition accelerates gray matter atrophy at early stages of the disease even before objective cognitive impairment is manifested. Identification of at-risk individuals at the presymptomatic stage has become a major research interest because it will allow early therapeutic interventions before irreversible synaptic and neuronal loss occur. We aimed to further characterize the cross-sectional and longitudinal relationship between A beta deposition, gray matter atrophy, and cognitive impairment

Systemic perturbations of the kynurenine pathway precede progression to dementia independently of amyloid-β

Increasing evidence suggests that kynurenine pathway (KP) dyshomeostasis may promote disease progression in dementia. Studies in Alzheimer's disease (AD) patients confirm KP dyshomeostasis in plasma and cerebrospinal fluid (CSF) which correlates with amyloid-β and tau pathology. Herein, we performed the first comprehensive study assessing baseline levels of KP metabolites in participants enrolling in the Australian Imaging Biomarkers Flagship Study of Aging. Our purpose was to test the hypothesis that changes in KP metabolites may be biomarkers of dementia processes that are largely silent. We used a cross-sectional analytical approach to assess non-progressors (N = 73); cognitively normal (CN) or mild cognitive impairment (MCI) participants at baseline and throughout the study, and progressors (N = 166); CN or MCI at baseline but progressing to either MCI or AD during the study. Significant KP changes in progressors included increased 3-hydroxyanthranilic acid (3-HAA) and 3-hydroxyanthranilic acid/anthranilic acid (3-HAA/AA) ratio, the latter having the largest effect on the odds of an individual being a progressor (OR 35.3; 95% CI between 14 and 104). 3-HAA levels were hence surprisingly bi-phasic, high in progressors but low in non-progressors or participants who had already transitioned to MCI or dementia. This is a new, unexpected and interesting result, as most studies of the KP in neurodegenerative disease show reduced 3-HAA/AA ratio after diagnosis. The neuroprotective metabolite picolinic acid was also significantly decreased while the neurotoxic metabolite 3-hydroxykynurenine increased in progressors. These results were significant even after adjustment for confounders. Considering the magnitude of the OR to predict change in cognition, it is important that these findings are replicated in other populations. Independent validation of our findings may confirm the utility of 3-HAA/AA ratio to predict change in cognition leading to dementia in clinical settings

Effect of BDNF Val66Met on memory decline and hippocampal atrophy in prodromal alzheimer\u27s disease: A preliminary study

Objective: Cross-sectional genetic association studies have reported equivocal results on the relationship between the brain-derived neurotrophic factor (BDNF) Val66Met and risk of Alzheimer\u27s disease (AD). As AD is a neurodegenerative disease, genetic influences may become clearer from prospective study. We aimed to determine whether BDNF Val66Met polymorphism influences changes in memory performance, hippocampal volume, and Aβ accumulation in adults with amnestic mild cognitive impairment (aMCI) and high Aβ. Methods: Thirty-four adults with aMCI were recruited from the Australian, Imaging, Biomarkers and Lifestyle (AIBL) Study. Participants underwent PiB-PET and structural MRI neuroimaging, neuropsychological assessments and BDNF genotyping at baseline, 18 month, and 36 month assessments. Results: In individuals with aMCI and high Aβ, Met carriers showed significant and large decline in episodic memory (d = 0.90, p = .020) and hippocampal volume (d = 0.98, p = .035). BDNF Val66Met was unrelated to the rate of Aβ accumulation (d = -0.35, p = .401). Conclusions: Although preliminary due to the small sample size, results of this study suggest that high Aβ levels and Met carriage may be useful prognostic markers of accelerated decline in episodic memory, and reductions in hippocampal volume in individuals in the prodromal or MCI stage of AD