307 research outputs found
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An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons
In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Trees with Convex Faces and Optimal Angles
We consider drawings of trees in which all edges incident to leaves can be
extended to infinite rays without crossing, partitioning the plane into
infinite convex polygons. Among all such drawings we seek the one maximizing
the angular resolution of the drawing. We find linear time algorithms for
solving this problem, both for plane trees and for trees without a fixed
embedding. In any such drawing, the edge lengths may be set independently of
the angles, without crossing; we describe multiple strategies for setting these
lengths.Comment: 12 pages, 10 figures. To appear at 14th Int. Symp. Graph Drawing,
200
Packing identical simple polygons is NP-hard
Given a small polygon S, a big simple polygon B and a positive integer k, it
is shown to be NP-hard to determine whether k copies of the small polygon
(allowing translation and rotation) can be placed in the big polygon without
overlap. Previous NP-hardness results were only known in the case where the big
polygon is allowed to be non-simple. A novel reduction from Planar-Circuit-SAT
is presented where a small polygon is constructed to encode the entire circuit
Discrete-Continuous ADMM for Transductive Inference in Higher-Order MRFs
This paper introduces a novel algorithm for transductive inference in
higher-order MRFs, where the unary energies are parameterized by a variable
classifier. The considered task is posed as a joint optimization problem in the
continuous classifier parameters and the discrete label variables. In contrast
to prior approaches such as convex relaxations, we propose an advantageous
decoupling of the objective function into discrete and continuous subproblems
and a novel, efficient optimization method related to ADMM. This approach
preserves integrality of the discrete label variables and guarantees global
convergence to a critical point. We demonstrate the advantages of our approach
in several experiments including video object segmentation on the DAVIS data
set and interactive image segmentation
Linear Complexity Hexahedral Mesh Generation
We show that any polyhedron forming a topological ball with an even number of
quadrilateral sides can be partitioned into O(n) topological cubes, meeting
face to face. The result generalizes to non-simply-connected polyhedra
satisfying an additional bipartiteness condition. The same techniques can also
be used to reduce the geometric version of the hexahedral mesh generation
problem to a finite case analysis amenable to machine solution.Comment: 12 pages, 17 figures. A preliminary version of this paper appeared at
the 12th ACM Symp. on Computational Geometry. This is the final version, and
will appear in a special issue of Computational Geometry: Theory and
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