A multi-level solver for Gaussian constrained CMB realizations

Updated to match published version (no major changes)Updated to match published version (no major changes)Updated to match published version (no major changes)We present a multi-level solver for drawing constrained Gaussian realizations or finding the maximum likelihood estimate of the CMB sky, given noisy sky maps with partial sky coverage. The method converges substantially faster than existing Conjugate Gradient (CG) methods for the same problem. For instance, for the 143 GHz Planck frequency channel, only 3 multi-level W-cycles result in an absolute error smaller than 1 microKelvin in any pixel. Using 16 CPU cores, this translates to a computational expense of 6 minutes wall time per realization, plus 8 minutes wall time for a power spectrum-dependent precomputation. Each additional W-cycle reduces the error by more than an order of magnitude, at an additional computational cost of 2 minutes. For comparison, we have never been able to achieve similar absolute convergence with conventional CG methods for this high signal-to-noise data set, even after thousands of CG iterations and employing expensive preconditioners. The solver is part of the Commander 2 code, which is available with an open source license at http://commander.bitbucket.org/

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

Accurate discretization of poroelasticity without Darcy stability --- Stokes-Biot stability revisited

In this manuscript we focus on the question: what is the correct notion of Stokes-Biot stability? Stokes-Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot’s equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes-Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes-Biot stable Euler-Galerkin discretization schemes

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