Numerical Computations with H(div)-Finite Elements for the Brinkman Problem

The H(div)-conforming approach for the Brinkman equation is studied numerically, verifying the theoretical a priori and a posteriori analysis in previous work of the authors. Furthermore, the results are extended to cover a non-constant permeability. A hybridization technique for the problem is presented, complete with a convergence analysis and numerical verification. Finally, the numerical convergence studies are complemented with numerical examples of applications to domain decomposition and adaptive mesh refinement.Comment: Minor clarifications, added references. Reordering of some figures. To appear in Computational Geosciences, final article available at http://www.springerlink.co

Robustness of common hemodynamic indicators with respect to numerical resolution in 38 middle cerebral artery aneurysms

Background: Using computational fluid dynamics (CFD) to compute the hemodynamics in cerebral aneurysms has received much attention in the last decade. The usability of these methods depends on the quality of the computations, highlighted in recent discussions. The purpose of this study is to investigate the convergence of common hemodynamic indicators with respect to numerical resolution. Methods: 38 middle cerebral artery bifurcation aneurysms were studied at two different resolutions (one comparable to most studies, and one finer). Relevant hemodynamic indicators were collected from two of the most cited studies, and were compared at the two refinements. In addition, correlation to rupture was investigated. Results: Most of the hemodynamic indicators were very well resolved at the coarser resolutions, correlating with the finest resolution with a correlation coefficient >0.95. The oscillatory shear index (OSI) had the lowest correlation coefficient of 0.83. A logarithmic Bland-Altman plot revealed noticeable variations in the proportion of the aneurysm under low shear, as well as in spatial and temporal gradients not captured by the correlation alone. Conclusion: Statistically, hemodynamic indicators agree well across the different resolutions studied here. However, there are clear outliers visible in several of the hemodynamic indicators, which suggests that special care should be taken when considering individual assessment

An observation on the uniform preconditioners for the mixed Darcy problem

When solving a multiphysics problem one often decomposes a monolithic system into simpler, frequently single‐physics, subproblems. A comprehensive solution strategy may commonly be attempted, then, by means of combining strategies devised for the constituent subproblems. When decomposing the monolithic problem, however, it may be that requiring a particular scaling for one subproblem enforces an undesired scaling on another. In this manuscript we consider the H(div)‐based mixed formulation of the Darcy problem as a single‐physics subproblem; the hydraulic conductivity, K, is considered intrinsic and not subject to any rescaling. Preconditioners for such porous media flow problems in mixed form are frequently based on H(div) preconditioners rather than the pressure Schur complement. We show that when the hydraulic conductivity, K, is small the pressure Schur complement can also be utilized for H(div)‐based preconditioners. The proposed approach employs an operator preconditioning framework to establish a robust, K‐uniform block preconditioner. The mapping property of the continuous operator is a key component in applying the theoretical framework point of view. As such, a main challenge addressed here is establishing a K‐uniform inf‐sup condition with respect to appropriately weighted Hilbert intersection‐ and sum spaces

Parameter robust preconditioning by congruence for multiple-network poroelasticity

The mechanical behavior of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network poroelasticity (MPET) equations or multiporosity/multipermeability systems. These equations generalize Biot's equations, which describe the mechanics of the one network case. The efficient numerical solution of the MPET equations is challenging, in part due to the complexity of the system and in part due to the presence of interacting parameter regimes. In this paper, we present a new strategy for efficiently and robustly solving the MPET equations numerically. In particular, we discuss an approach to formulating finite element methods and associated preconditioners for the MPET equations based on simultaneous diagonalization of the element matrices. We demonstrate the technique for the multicompartment Darcy equations, with large exchange variability, and the MPET equations for a nearly incompressible medium with large exchange variability. The approach is based on designing transformations of variables that simultaneously diagonalize (by congruence) the equations' key operators and subsequently constructing parameter-robust block diagonal preconditioners for the transformed system. The proposed approach is supported by theoretical considerations as well as by numerical results

Multi-resolution Bayesian CMB component separation through Wiener filtering with a pseudo-inverse preconditioner

We present a Bayesian model for multi-resolution component separation for cosmic microwave background (CMB) applications based on Wiener filtering and/or computation of constrained realizations, extending a previously developed framework. We also develop an efficient solver for the corresponding linear system for the associated signal amplitudes. The core of this new solver is an efficient preconditioner based on the pseudo-inverse of the coefficient matrix of the linear system. In the full sky coverage case, the method gives an increased speed of the preconditioner, and it is easier to implement in terms of practical computer code. In the case where a mask is applied and prior-driven constrained realization is sought within the mask, this is the first time full convergence has been achieved at the full resolution of the Planck data set

Sub-voxel Perfusion Modeling in Terms of Coupled 3d-1d Problem

We study perfusion by a multiscale model coupling diffusion in the tissue and diffusion along the one-dimensional segments representing the vasculature. We propose a block-diagonal preconditioner for the model equations and demonstrate its robustness by numerical experiments. We compare our model to a macroscale model by Tofts [Modelling in DCE MRI, 2012]

Can Shunt Response in Patients with Idiopathic Normal Pressure Hydrocephalus Be Predicted from Preoperative Brain Imaging? A Retrospective Study of the Diagnostic Use of the Normal Pressure Hydrocephalus Radscale in 119 Patients

BACKGROUND AND PURPOSE: The Normal Pressure Hydrocephalus Radscale is a combined scoring of 7 different structural imaging markers on preoperative brain CT or MR imaging in patients with idiopathic normal pressure hydrocephalus: callosal angle, Evans Index, Sylvian fissure dilation, apical sulcal narrowing, mean temporal horn diameter, periventricular WM lesions, and focal sulcal dilation. The purpose of this retrospective study was to assess the performance of the Normal Pressure Hydrocephalus Radscale in distinguishing idiopathic normal pressure hydrocephalus shunt responders from nonresponders. MATERIALS AND METHODS: The preoperative MR imaging and CT scans of 119 patients with idiopathic normal pressure hydrocephalus were scored using the Normal Pressure Hydrocephalus Radscale. A summary shunt-response score assessed within 6 months from ventriculoperitoneal shunt surgery, combining the effect on cognition, gait, and urinary incontinence, was used as a reference. The difference between the mean Normal Pressure Hydrocephalus Radscale for responders and nonresponders was tested using the Student t test. The area under the curve was calculated for the Normal Pressure Hydrocephalus Radscale to assess shunt response. To ascertain reproducibility, we assessed the interobserver agreement between the 2 independent observers as intraclass correlation coefficients for the Normal Pressure Hydrocephalus Radscale for 74 MR imaging scans and 19 CT scans. RESULTS: Ninety-four (79%) of 119 patients were shunt responders. The mean Normal Pressure Hydrocephalus Radscale score for shunt responders was 8.35 (SD, 1.53), and for nonresponders, 7.48 (SD, 1.53) (P = .02). The area under the curve for the Normal Pressure Hydrocephalus Radscale was 0.66 (range, 0.54–0.78). The intraclass correlation coefficient for the Normal Pressure Hydrocephalus Radscale was 0.86 for MR imaging and 0.82 for CT. CONCLUSIONS: The Normal Pressure Hydrocephalus Radscale showed moderate discrimination for shunt response but cannot, on its own, be used for selecting patients with idiopathic normal pressure hydrocephalus for shunt surgery