hp-dGFEM for Second-Order Elliptic Problems in Plyhedra II: Exponential Convergence

The goal of this paper is to establish exponential convergence of $hp$-version interior penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and piecewise analytic data in three-dimensional polyhedral domains. More precisely, we shall analyze the convergence of the $hp$-IP dG methods considered in [D. Schötzau, C. Schwab, T. P. Wihler, SIAM J. Numer. Anal., 51 (2013), pp. 1610--1633] based on axiparallel $\sigma$-geometric anisotropic meshes and $\bm{s}$-linear anisotropic polynomial degree distributions.

Are great Cascadia earthquakes recorded in the sedimentary records from small forearc lakes?

Here we investigate sedimentary records from four small inland lakes located in the southern Cascadia forearc region for evidence of earthquakes. Three of these lakes are in the Klamath Mountains near the Oregon–California border, and one is in the central Oregon Coast range. The sedimentary sequences recovered from these lakes are composed of normal lake sediment interbedded with disturbance event layers. The thickest of these layers are graded, and appear to be turbidites or linked debrites (turbidites with a basal debris-flow deposit), suggesting rapid deposition. Variations in particle size and organic content of these layers are reflected in the density and magnetic susceptibility data. The frequency and timing of these events, based on radiocarbon ages from detrital organics, is similar to the offshore seismogenic turbidite record from trench and slope basin cores along the Cascadia margin. Stratigraphic correlation of these anomalous deposits based on radiocarbon ages, down-core density, and magnetic susceptibility data between lake and offshore records suggest synchronous triggering. The areal extent and multiple depositional environments over which these events appear to correlate suggest that these deposits were most likely caused by shaking during great Cascadia earthquakes.

Tracking the Flow of Information through the Hippocampal Formation in the Rat

The hippocampus receives input from upper levels of the association cortex and is implicated in many mnemonic processes, but the exact mechanisms by which it codes and stores information is an unresolved topic. This work examines the flow of information through the hippocampal formation while attempting to determine the computations that each of the hippocampal subfields performs in learning and memory. The formation, storage, and recall of hippocampal-dependent memories theoretically utilize an autoassociative attractor network that functions by implementing two competitive, yet complementary, processes. Pattern separation, hypothesized to occur in the dentate gyrus (DG), refers to the ability to decrease the similarity among incoming information by producing output patterns that overlap less than the inputs. In contrast, pattern completion, hypothesized to occur in the CA3 region, refers to the ability to reproduce a previously stored output pattern from a partial or degraded input pattern. Prior to addressing the functional role of the DG and CA3 subfields, the spatial firing properties of neurons in the dentate gyrus were examined. The principal cell of the dentate gyrus, the granule cell, has spatially selective place fields; however, the behavioral correlates of another excitatory cell, the mossy cell of the dentate polymorphic layer, are unknown. This report shows that putative mossy cells have spatially selective firing that consists of multiple fields similar to previously reported properties of granule cells. Other cells recorded from the DG had single place fields. Compared to cells with multiple fields, cells with single fields fired at a lower rate during sleep, were less likely to burst, and were more likely to be recorded simultaneously with a large population of neurons that were active during sleep and silent during behavior. These data suggest that single-field and multiple-field cells constitute at least two distinct cell classes in the DG. Based on these characteristics, we propose that putative mossy cells tend to fire in multiple, distinct locations in an environment, whereas putative granule cells tend to fire in single locations, similar to place fields of the CA1 and CA3 regions. Experimental evidence supporting the theories of pattern separation and pattern completion comes from both behavioral and electrophysiological tests. These studies specifically focused on the function of each subregion and made implicit assumptions about how environmental manipulations changed the representations encoded by the hippocampal inputs. However, the cell populations that provided these inputs were in most cases not directly examined. We conducted a series of studies to investigate the neural activity in the entorhinal cortex, dentate gyrus, and CA3 in the same experimental conditions, which allowed a direct comparison between the input and output representations. The results show that the dentate gyrus representation changes between the familiar and cue altered environments more than its input representations, whereas the CA3 representation changes less than its input representations. These findings are consistent with longstanding computational models proposing that (1) CA3 is an associative memory system performing pattern completion in order to recall previous memories from partial inputs, and (2) the dentate gyrus performs pattern separation to help store different memories in ways that reduce interference when the memories are subsequently recalled.

Transition metal ligands as novel DNA-base substitutes

The base modified nucleoside dBP, carrying a non-hydrogen-bonding non-shape complementary base was incorporated into oligonucleotides (Brotschi, C.; Haberli, A.; Leumann C.J. Angew. Chem. Int. Ed. 2001, 40, 3012-3014). This base was designed to coordinate transition metal ions into well defined positions within a DNA double helix. Melting experiments revealed that the stability of a dBP:dBP base couple in a DNA duplex is similar to a dG:dC base pair even in the absence of transition metal ions. In the presence of transition metal ions, melting experiments revealed a decrease in duplex stability which is on a similar order for all metal ions (Mn2+, Cu2+, Zn2+, Ni2+) tested

hp-DGFEM for second-order mixed elliptic problems polyhedra

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming dG interpolant given by local L 2 -projections on elements away from corners and edges, and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting on Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of [11, 12] for the pure Dirichlet case.

A total variation discontinuous Galerkin approach for image restoration

The focal point of this paper is to propose and analyze a P 0 discontinuous Galerkin (DG) formulation for image denoising. The scheme is based on a total variation approach which has been applied successfully in previous papers on image processing. The main idea of the new scheme is to model the restoration process in terms of a discrete energy minimization problem and to derive a corresponding DG variational formulation. Furthermore, we will prove that the method exhibits a unique solution and that a natural maximum principle holds. In addition, a number of examples illustrate the effectiveness of the method.

Die Lösung des Antoninusrätsels

von Rudolf Leszynsky