Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints

Finite element methods for a model elliptic distributed optimal control problem with pointwise state constraints are considered from the perspective of fourth order boundary value problems

Adaptive interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

In this paper we conduct a priori and a posteriori error analysis of the C0 interior penalty method for Hamilton–Jacobi–Bellman equations, with coefficients that satisfy the Cordes condition. These estimates show the quasi-optimality of the method, and provide one with an adaptive finite element method. In accordance with the proven regularity theory, we only assume that the solution of the Hamilton–Jacobi–Bellman equation belongs to H2

Multispace and Multilevel BDDC

BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory.Comment: 26 pages, 3 figures, 2 tables, 20 references. Formal changes onl

A weighted reduced basis method for parabolic PDEs with random data

This work considers a weighted POD-greedy method to estimate statistical outputs parabolic PDE problems with parametrized random data. The key idea of weighted reduced basis methods is to weight the parameter-dependent error estimate according to a probability measure in the set-up of the reduced space. The error of stochastic finite element solutions is usually measured in a root mean square sense regarding their dependence on the stochastic input parameters. An orthogonal projection of a snapshot set onto a corresponding POD basis defines an optimum reduced approximation in terms of a Monte Carlo discretization of the root mean square error. The errors of a weighted POD-greedy Galerkin solution are compared against an orthogonal projection of the underlying snapshots onto a POD basis for a numerical example involving thermal conduction. In particular, it is assessed whether a weighted POD-greedy solutions is able to come significantly closer to the optimum than a non-weighted equivalent. Additionally, the performance of a weighted POD-greedy Galerkin solution is considered with respect to the mean absolute error of an adjoint-corrected functional of the reduced solution.Comment: 15 pages, 4 figure

Paleomagnetic evidence for a long-lived, potentially reversing martian dynamo at ~3.9 Ga

The 4.1-billion-year-old meteorite Allan Hills 84001 (ALH 84001) may preserve a magnetic record of the extinct martian dynamo. However, previous paleomagnetic studies have reported heterogeneous, nonunidirectional magnetization in the meteorite at submillimeter scales, calling into question whether it records a dynamo field. We use the quantum diamond microscope to analyze igneous Fe-sulfides in ALH 84001 that may carry remanence as old as 4.1 billion years (Ga). We find that individual, 100-μm-scale ferromagnetic mineral assemblages are strongly magnetized in two nearly antipodal directions. This suggests that the meteorite recorded strong fields following impact heating at 4.1 to 3.95 Ga, after which at least one further impact heterogeneously remagnetized the meteorite in a nearly antipodal local field. These observations are most simply explained by a reversing martian dynamo that was active until 3.9 Ga, thereby implying a late cessation for the martian dynamo and potentially documenting reversing behavior in a nonterrestrial planetary dynamo

Analysis of discontinuous Galerkin methods using mesh-dependent norms and applications to problems with rough data

We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine some problems with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control

Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide

We consider an inverse problem of reconstructing the conductivity function in a hyperbolic equation using single space-time domain noisy observations of the solution on the backscattering boundary of the computational domain. We formulate our inverse problem as an optimization problem and use Lagrangian approach to minimize the corresponding Tikhonov functional. We present a theorem of a local strong convexity of our functional and derive error estimates between computed and regularized as well as exact solutions of this functional, correspondingly. In numerical simulations we apply domain decomposition finite element-finite difference method for minimization of the Lagrangian. Our computational study shows efficiency of the proposed method in the reconstruction of the conductivity function in three dimensions

Haptic search with finger movements: using more fingers does not necessarily reduce search times

Two haptic serial search tasks were used to investigate how the separations between items, and the number of fingers used to scan them, influence the search time and search strategy. In both tasks participants had to search for a target (cross) between a fixed number of non-targets (circles). The items were placed in a straight line. The target’s position was varied within blocks, and inter-item separation was varied between blocks. In the first experiment participants used their index finger to scan the display. As expected, search time depended on target position as well as on item separation. For larger separations participants’ movements were jerky, resembling ‘saccades’ and ‘fixations’, while for the shortest separation the movements were smooth. When only considering time in contact with an item, search times were the same for all separation conditions. Furthermore, participants never continued their movement after they encountered the target. These results suggest that participants did not use the time during which they were moving between the items to process information about the items. The search times were a little shorter than those in a static search experiment (Overvliet et al. in Percept Psychophys, 2007a), where multiple items were presented to the fingertips simultaneously. To investigate whether this is because the finger was moving or because only one finger was stimulated, we conducted a second experiment in which we asked participants to put three fingers in line and use them together to scan the items. Doing so increased the time in contact with the items for all separations, so search times were presumably longer in the static search experiment because multiple fingers were involved. This may be caused by the time that it takes to switch from one finger to the other