Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov-Galerkin, and Monotone Mixed Methods

This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residual minimization in dual norms, and its inexact version using discrete dual norms. It is shown that this development, in the case of strictly-convex reflexive Banach spaces with strictly-convex dual, gives rise to a class of nonlinear Petrov-Galerkin methods and, equivalently, abstract mixed methods with monotone nonlinearity. Crucial in the formulation of these methods is the (nonlinear) bijective duality map. Under the Fortin condition, we prove discrete stability of the abstract inexact method, and subsequently carry out a complete error analysis. As part of our analysis, we prove new bounds for best-approximation projectors, which involve constants depending on the geometry of the underlying Banach space. The theory generalizes and extends the classical Petrov-Galerkin method as well as existing residual-minimization approaches, such as the discontinuous Petrov-Galerkin method.Comment: 43 pages, 2 figure

Discontinuities without discontinuity: The Weakly-enforced Slip Method

Tectonic faults are commonly modelled as Volterra or Somigliana dislocations in an elastic medium. Various solution methods exist for this problem. However, the methods used in practice are often limiting, motivated by reasons of computational efficiency rather than geophysical accuracy. A typical geophysical application involves inverse problems for which many different fault configurations need to be examined, each adding to the computational load. In practice, this precludes conventional finite-element methods, which suffer a large computational overhead on account of geometric changes. This paper presents a new non-conforming finite-element method based on weak imposition of the displacement discontinuity. The weak imposition of the discontinuity enables the application of approximation spaces that are independent of the dislocation geometry, thus enabling optimal reuse of computational components. Such reuse of computational components renders finite-element modeling a viable option for inverse problems in geophysical applications. A detailed analysis of the approximation properties of the new formulation is provided. The analysis is supported by numerical experiments in 2D and 3D.Comment: Submitted for publication in CMAM

Highlights from the SoilCAM project: Soil Contamination, Advanced integrated characterisation and time-lapse Monitoring

The SoilCAM project (Soil Contamination, Advanced integrated characterisation and time-lapse Monitoring 2008-2012, EU-FP7-212663) is aimed at improving current methods for monitoring contaminant distribution and biodegradation in the subsurfac

Effects of episodic gas infall on the chemical abundances in galaxies

The chemical evolution of galaxies that undergo an episode of massive and rapid accretion of metal-poor gas is investigated with models using both simplified and detailed nucleosynthesis recipes. The rapid decrease of the oxygen abundance during infall is followed by a slower evolution which leads back to the closed-box relation, thus forming a loop in the N/O-O/H diagram. For large excursions from the closed-box relation, the mass of the infalling material needs to be substantially larger than the gas remaining in the galaxy, and the accretion rate should be larger than the star formation rate. We apply this concept to the encounter of high velocity clouds with galaxies of various masses, finding that the observed properties of these clouds are indeed able to cause substantial effects not only in low mass galaxies, but also in the partial volumes in large massive galaxies that would be affected by the collision. Numerical models with detailed nucleosynthesis prescriptions are constructed. We assume star formation timescales and scaled yields that depend on the galactic mass, and which are adjusted to reproduce the average relations of gas fraction, oxygen abundance, and effective oxygen yield observed in irregular and spiral galaxies. The resulting excursions in the N/O-O/H diagram due to a single accretion event involving a high velocity cloud are found to be appreciable, which could thus provide a contribution to the large scatter in the N/O ratio found among irregular galaxies. Nonetheless, the N/O-O/H diagram remains an important indicator for stellar nucleosynthesis.Comment: 13 pages, 23 postscript figures, Latex, to appear in Astronomy & Astrophysic

Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations

We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed for an abstract nonlinear parabolic PDE and subsequently applied to linear heat and nonlinear Cahn-Hilliard equations. The error due to finite element approximations is estimated with a residual weighted approximate-dual solution which is computed with two primal approximations at nested levels. We prove that the exact error is estimated by our estimator up to higher-order remainder terms. Numerical experiments confirm the theory regarding consistency of the dual-based two-level estimator. We also present a novel space-time adaptive strategy to control errors based on the new estimator

Gibbs Phenomena for LqL^q-Best Approximation in Finite Element Spaces -- Some Examples

Recent developments in the context of minimum residual finite element methods are paving the way for designing finite element methods in non-standard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties. One of the biggest challenges in designing finite element methods are non-physical oscillations near thin layers and jump discontinuities. In this article we investigate Gibbs phenomena in the context of $L^q$-best approximation of discontinuities in finite element spaces with $1\leq q<\infty$. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as $q$ tends to $1$. The aim here is to show the potential of $L^1$ as a solution space in connection with suitably designed meshes

Genome-wide methylome analysis using MethylCap-seq uncovers 4 hypermethylated markers with high sensitivity for both adeno- and squamous-cell cervical carcinoma

Background: Cytology-based screening methods for cervical adenocarcinoma (ADC) and to a lesser extent squamous-cell carcinoma (SCC) suffer from low sensitivity. DNA hypermethylation analysis in cervical scrapings may improve detection of SCC, but few methylation markers have been described for ADC. We aimed to identify novel methylation markers for the early detection of both ADC and SCC. Results: Genome-wide methylation profiling for 20 normal cervices, 6 ADC and 6 SCC using MethylCap-seq yielded 53 candidate regions hypermethylated in both ADC and SCC. Verification and independent validation of the 15 most significant regions revealed 5 markers with differential methylation between 17 normals and 13 cancers. Quantitative methylation-specific PCR on cervical cancer scrapings resulted in detection rates ranging between 80% and 92% while between 94% and 99% of control scrapings tested negative. Four markers (SLC6A5, SOX1, SOX14 and TBX20) detected ADC and SCC with similar sensitivity. In scrapings from women referred with an abnormal smear (n = 229), CIN3+ sensitivity was between 36% and 71%, while between 71% and 93% of adenocarcinoma in situ (AdCIS) were detected; and CIN0/1 specificity was between 88% and 98%. Compared to hrHPV, the combination SOX1/SOX14 showed a similar CIN3+ sensitivity (80% vs. 75%, respectively, P>0.2), while specificity improved (42% vs. 84%, respectively, P < 10(-5)). Conclusion: SOX1 and SOX14 are methylation biomarkers applicable for screening of all cervical cancer types

Restructuring the Tridiagonal and Bidiagonal QR Algorithms for Performance

We show how both the tridiagonal and bidiagonal QR algorithms can be restructured so that they be- come rich in operations that can achieve near-peak performance on a modern processor. The key is a novel, cache-friendly algorithm for applying multiple sets of Givens rotations to the eigenvector/singular vector matrix. This algorithm is then implemented with optimizations that (1) leverage vector instruction units to increase floating-point throughput, and (2) fuse multiple rotations to decrease the total number of memory operations. We demonstrate the merits of these new QR algorithms for computing the Hermitian eigenvalue decomposition (EVD) and singular value decomposition (SVD) of dense matrices when all eigen- vectors/singular vectors are computed. The approach yields vastly improved performance relative to the traditional QR algorithms for these problems and is competitive with two commonly used alternatives— Cuppen’s Divide and Conquer algorithm and the Method of Multiple Relatively Robust Representations— while inheriting the more modest O(n) workspace requirements of the original QR algorithms. Since the computations performed by the restructured algorithms remain essentially identical to those performed by the original methods, robust numerical properties are preserved