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On a Construction of a Hierarchy of Best Linear Spline Approximations Using a Finite Element Approach
September/October 2004 (vol. 10 no. 5)
pp. 548-563
ASCII Text
x 
 
David F. Wiley, Martin Bertram, Bernd Hamann, "On a Construction of a Hierarchy of Best Linear Spline Approximations Using a Finite Element Approach," IEEE Transactions on Visualization and Computer Graphics, vol. 10, no. 5, pp. 548-563, September/October, 2004.
 
 
BibTex
x
 
@article{ 10.1109/TVCG.2004.29,
author = {David F. Wiley and Martin Bertram and Bernd Hamann},
title = {On a Construction of a Hierarchy of Best Linear Spline Approximations Using a Finite Element Approach},
journal ={IEEE Transactions on Visualization and Computer Graphics},
volume = {10},
number = {5},
issn = {1077-2626},
year = {2004},
pages = {548-563},
doi = {http://doi.ieeecomputersociety.org/10.1109/TVCG.2004.29},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Visualization and Computer Graphics
TI - On a Construction of a Hierarchy of Best Linear Spline Approximations Using a Finite Element Approach
IS - 5
SN - 1077-2626
SP548
EP563
EPD - 548-563
A1 - David F. Wiley,
A1 - Martin Bertram,
A1 - Bernd Hamann,
PY - 2004
KW - Approximation
KW - finite element method
KW - grid generation
KW - multiresolution method optimization
KW - Ritz approximation
KW - scattered data
KW - spline
KW - triangulation
KW - unstructured grid
KW - visualization.
VL - 10
JA - IEEE Transactions on Visualization and Computer Graphics
ER -
 
We present a method for the hierarchical approximation of functions in one, two, or three variables based on the finite element method (Ritz approximation). Starting with a set of data sites with associated function, we first determine a smooth (scattered-data) interpolant. Next, we construct an initial triangulation by triangulating the region bounded by the minimal subset of data sites defining the convex hull of all sites. We insert only original data sites, thus reducing storage requirements. For each triangulation, we solve a minimization problem: computing the best linear spline approximation of the interpolant of all data, based on a functional involving function values and first derivatives. The error of a best linear spline approximation is computed in a Sobolev-like norm, leading to element-specific error values. We use these interval/triangle/tetrahedron-specific values to identify the element to subdivide next. The subdivision of an element with largest error value requires the recomputation of all spline coefficients due to the global nature of the problem. We improve efficiency by 1) subdividing multiple elements simultaneously and 2) by using a sparse-matrix representation and system solver.

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Index Terms:
Approximation, finite element method, grid generation, multiresolution method optimization, Ritz approximation, scattered data, spline, triangulation, unstructured grid, visualization.
Citation:
David F. Wiley, Martin Bertram, Bernd Hamann, "On a Construction of a Hierarchy of Best Linear Spline Approximations Using a Finite Element Approach," IEEE Transactions on Visualization and Computer Graphics, vol. 10, no. 5, pp. 548-563, Sept.-Oct. 2004, doi:10.1109/TVCG.2004.29
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