A posteriori error estimates for elliptic problems in two and three space dimensions

Let $u \in H$ be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some $\tilde u \in \mathcal{S}$, $\mathcal{S}$ being a suitable finite-element space. Efficient and reliable a posteriors estimates of the error $\| {u - \tilde u} \|$, measuring the (local) quality of $\tilde u$, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well-known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques. In this paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations

Hierarchical error estimates for the energy functional in obstacle problems

We present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems. The main result is that the error in the energy minimum is, up to oscillation terms, equivalent to an appropriate hierarchical estimator. The proof does not invoke any saturation assumption. We even show that small oscillation implies a related saturation assumption. In addition, we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators. Finally, we illustrate our theoretical results by numerical computations

Validation of the German Revised Addenbrooke's Cognitive Examination for Detecting Mild Cognitive Impairment, Mild Dementia in Alzheimer's Disease and Frontotemporal Lobar Degeneration

Background/Aims: The diagnostic accuracy of the German version of the revised Addenbrooke's Cognitive Examination (ACE-R) in identifying mild cognitive impairment (MCI), mild dementia in Alzheimer's disease (AD) and mild dementia in frontotemporal lobar degeneration (FTLD) in comparison with the conventional Mini Mental State Examination (MMSE) was assessed. Methods: The study encompasses 76 cognitively healthy elderly individuals, 75 patients with MCI, 56 with AD and 22 with FTLD. ACE-R and MMSE were validated against an expert diagnosis based on a comprehensive diagnostic procedure. Statistical analysis was performed using the receiver operating characteristic method and regression analyses. Results: The optimal cut-off score for the ACE-R for detecting MCI, AD, and FTLD was 86/87, 82/83 and 83/84, respectively. ACE-R was superior to MMSE only in the detection of patients with FTLD {[}area under the curve (AUC): 0.97 vs. 0.92], whilst the accuracy of the two instruments did not differ in identifying MCI and AD. The ratio of the scores of the memory ACE-R subtest to verbal fluency subtest contributed significantly to the discrimination between AD and FTLD (optimal cut-off score: 2.30/2.31, AUC: 0.77), whereas the MMSE and ACE-R total scores did not. Conclusion: The German ACE-R is superior to the most commonly employed MMSE in detecting mild dementia in FTLD and in the differential diagnosis between AD and FTLD. Thus it might serve as a valuable instrument as part of a comprehensive diagnostic workup in specialist centres/clinics contributing to the diagnosis and differential diagnosis of the cause of dementia. Copyright (C) 2010 S. Karger AG, Base

Fredholm integral equations for function approximation and the training of neural networks

We present a novel and mathematically transparent approach to function approximation and the training of large, high-dimensional neural networks, based on the approximate least-squares solution of associated Fredholm integral equations of the first kind by Ritz-Galerkin discretization, Tikhonov regularization and tensor-train methods. Practical application to supervised learning problems of regression and classification type confirm that the resulting algorithms are competitive with state-of-the-art neural network-based methods. Patrick , Aizhan , Ral

Anatomy, morphology and evolution of the patella in squamate lizards and tuatara (Sphenodon punctatus)

The patella (kneecap) is the largest and best-known of the sesamoid bones, postulated to confer biomechanical advantages including increasing joint leverage and reinforcing the tendon against compression. It has evolved several times independently in amniotes, but despite apparently widespread occurrence in lizards, the patella remains poorly characterised in this group and is, as yet, completely undescribed in their nearest extant relative Sphenodon (Rhynchocephalia). Through radiography, osteological and fossil studies we examined patellar presence in diverse lizard and lepidosauromorph taxa, and using computed tomography, dissection and histology we investigated in greater depth the anatomy and morphology of the patella in 16 lizard species and 19 Sphenodon specimens. We have found the first unambiguous evidence of a mineralised patella in Sphenodon, which appears similar to the patella of lizards and shares several gross and microscopic anatomical features. Although there may be a common mature morphology, the squamate patella exhibits a great deal of variability in development (whether from a cartilage anlage or not, and in the number of mineralised centres) and composition (bone, mineralised cartilage or fibrotendinous tissue). Unlike in mammals and birds, the patella in certain lizards and Sphenodon appears to be a polymorphic trait. We have also explored the evolution of the patella through ancestral state reconstruction, finding that the patella is ancestral for lizards and possibly Lepidosauria as a whole. Clear evidence of the patella in rhynchocephalian or stem lepidosaurian fossil taxa would clarify the evolutionary origin(s) of the patella, but due to the small size of this bone and the opportunity for degradation or loss we could not definitively conclude presence or absence in the fossils examined. The pattern of evolution in lepidosaurs is unclear but our data suggest that the emergence of this sesamoid may be related to the evolution of secondary ossification centres and/or changes in knee joint conformation, where enhancement of extensor muscle leverage would be more beneficial.Sophie Regnault, Marc E. H. Jones, Andrew A. Pitsillides, John R. Hutchinso

Evolving surface finite element methods for random advection-diffusion equations

In this paper, we introduce and analyse a surface finite element discretization of advection-diffusion equations with uncertain coefficients on evolving hypersurfaces. After stating unique solvability of the resulting semi-discrete problem, we prove optimal error bounds for the semi-discrete solution and Monte Carlo samplings of its expectation in appropriate Bochner spaces. Our theoretical findings are illustrated by numerical experiments in two and three space dimensions

A monotone multigrid solver for two body contact problems in biomechanics

The purpose of the paper is to apply monotone multigrid methods to static and dynamic biomechanical contact problems. In space, a finite element method involving a mortar discretization of the contact conditions is used. In time, a new contact-stabilized Newmark scheme is presented. Numerical experiments for a two body Hertzian contact problem and a biomechanical application are reported