13,269 research outputs found
Convergence and Optimality of Adaptive Mixed Methods on Surfaces
In a 1988 article, Dziuk introduced a nodal finite element method for the
Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear
triangulation, initiating a line of research into surface finite element
methods (SFEM). Demlow and Dziuk built on the original results, introducing an
adaptive method for problems on 2-surfaces, and Demlow later extended the a
priori theory to 3-surfaces and higher order elements. In a separate line of
research, the Finite Element Exterior Calculus (FEEC) framework has been
developed over the last decade by Arnold, Falk and Winther and others as a way
to exploit the observation that mixed variational problems can be posed on a
Hilbert complex, and Galerkin-type mixed methods can be obtained by solving
finite dimensional subproblems. In 2011, Holst and Stern merged these two lines
of research by developing a framework for variational crimes in abstract
Hilbert complexes, allowing for application of the FEEC framework to problems
that violate the subcomplex assumption of Arnold, Falk and Winther. When
applied to Euclidean hypersurfaces, this new framework recovers the original a
priori results and extends the theory to problems posed on surfaces of
arbitrary dimensions. In yet another seemingly distinct line of research,
Holst, Mihalik and Szypowski developed a convergence theory for a specific
class of adaptive problems in the FEEC framework. Here, we bring these ideas
together, showing convergence and optimality of an adaptive finite element
method for the mixed formulation of the Hodge Laplacian on hypersurfaces.Comment: 22 pages, no figures. arXiv admin note: substantial text overlap with
arXiv:1306.188
Subsumption Algorithms for Three-Valued Geometric Resolution
In our implementation of geometric resolution, the most costly operation is
subsumption testing (or matching): One has to decide for a three-valued,
geometric formula, if this formula is false in a given interpretation. The
formula contains only atoms with variables, equality, and existential
quantifiers. The interpretation contains only atoms with constants. Because the
atoms have no term structure, matching for geometric resolution is hard. We
translate the matching problem into a generalized constraint satisfaction
problem, and discuss several approaches for solving it efficiently, one direct
algorithm and two translations to propositional SAT. After that, we study
filtering techniques based on local consistency checking. Such filtering
techniques can a priori refute a large percentage of generalized constraint
satisfaction problems. Finally, we adapt the matching algorithms in such a way
that they find solutions that use a minimal subset of the interpretation. The
adaptation can be combined with every matching algorithm. The techniques
presented in this paper may have applications in constraint solving independent
of geometric resolution.Comment: This version was revised on 18.05.201
Completeness of Randomized Kinodynamic Planners with State-based Steering
Probabilistic completeness is an important property in motion planning.
Although it has been established with clear assumptions for geometric planners,
the panorama of completeness results for kinodynamic planners is still
incomplete, as most existing proofs rely on strong assumptions that are
difficult, if not impossible, to verify on practical systems. In this paper, we
focus on an important class of kinodynamic planners, namely those that
interpolate trajectories in the state space. We provide a proof of
probabilistic completeness for these planners under assumptions that can be
readily verified from the system's equations of motion and the user-defined
interpolation function. Our proof relies crucially on a property of
interpolated trajectories, termed second-order continuity (SOC), which we show
is tightly related to the ability of a planner to benefit from denser sampling.
We analyze the impact of this property in simulations on a low-torque pendulum.
Our results show that a simple RRT using a second-order continuous
interpolation swiftly finds solution, while it is impossible for the same
planner using standard Bezier curves (which are not SOC) to find any solution.Comment: 21 pages, 5 figure
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