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Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems

100%
Axelsson O., Karátson J.
Open Mathematics
|
2013
|
tom 11
|
nr 8
1441-1457
EN
A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.
2
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Some variants of the method of fundamental solutions: regularization using radial and nearly radial basis functions

100%
Gáspár C.
Open Mathematics
|
2013
|
tom 11
|
nr 8
1429-1440
EN
The method of fundamental solutions and some versions applied to mixed boundary value problems are considered. Several strategies are outlined to avoid the problems due to the singularity of the fundamental solutions: the use of higher order fundamental solutions, and the use of nearly fundamental solutions and special fundamental solutions concentrated on lines instead of points. The errors of the approximations as well as the problem of ill-conditioned matrices are illustrated via numerical examples.
3
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Analysis of fast boundary-integral approximations for modeling electrostatic contributions of molecular binding

64%
Kreienkamp A. B., Liu L. Y., Minkara M. S., Knepley M. G., Bardhan J. P., Radhakrishnan M. L.
Molecular Based Mathematical Biology
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2013
|
tom 1
124-150
EN
We analyze and suggest improvements to a recently developed approximate continuum-electrostatic model for proteins. The model, called BIBEE/I (boundary-integral based electrostatics estimation with interpolation), was able to estimate electrostatic solvation free energies to within a mean unsigned error of 4% on a test set of more than 600 proteins¶a significant improvement over previous BIBEE models. In this work, we tested the BIBEE/I model for its capability to predict residue-by-residue interactions in protein–protein binding, using the widely studied model system of trypsin and bovine pancreatic trypsin inhibitor (BPTI). Finding that the BIBEE/I model performs surprisingly less well in this task than simpler BIBEE models, we seek to explain this behavior in terms of the models’ differing spectral approximations of the exact boundary-integral operator. Calculations of analytically solvable systems (spheres and tri-axial ellipsoids) suggest two possibilities for improvement. The first is a modified BIBEE/I approach that captures the asymptotic eigenvalue limit correctly, and the second involves the dipole and quadrupole modes for ellipsoidal approximations of protein geometries. Our analysis suggests that fast, rigorous approximate models derived from reduced-basis approximation of boundaryintegral equations might reach unprecedented accuracy, if the dipole and quadrupole modes can be captured quickly for general shapes.
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