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1
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Graphical Processing Unit accelerated Poisson equation solver and its application for calculation of single ion potential in ion-channels

100%
Simakov N. A., Kurnikova M. G.
Molecular Based Mathematical Biology
|
2013
|
tom 1
151-163
EN
Poisson and Poisson-Boltzmann equations (PE and PBE) are widely used in molecular modeling to estimate the electrostatic contribution to the free energy of a system. In such applications, PE often needs to be solved multiple times for a large number of system configurations. This can rapidly become a highly demanding computational task. To accelerate such calculations we implemented a graphical processing unit (GPU) PE solver described in this work. The GPU solver performance is compared to that of our central processing unit (CPU) implementation of the solver. During the performance analysis the following three characteristics were studied: (1) precision associated with the modeled system discretization on the grid, (2) numeric precision associated with the floating point representation of real numbers (this is done via comparison of calculations with single precision (SP) and double precision (DP)), and (3) execution time. Two types of example calculations were carried out to evaluate the solver performance: (1) solvation energy of a single ion and a small protein (lysozyme), and (2) a single ion potential in a large ion-channel (α-hemolysin). In addition, influence of various boundary condition (BC) choices was analyzed, to determine the most appropriate BC for the systems that include a membrane, typically represented by a slab with the dielectric constant of low value. The implemented GPU PE solver is overall about 7 times faster than the CPU-based version (including all four cores). Therefore, a single computer equipped with multiple GPUs can offer a computational power comparable to that of a small cluster. Our calculations showed that DP versions of CPU and GPU solvers provide nearly identical results. SP versions of the solvers have very similar behavior: in the grid scale range of 1-4 grids/Å the difference between SP and DP versions is less than the difference stemming from the system discretization. We found that for the membrane protein, the use of a focusing technique with periodic boundary conditions in rough grid provides significantly better results than using a focusing technique with the electric potential set to zero at the boundaries.
2
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Ground states for asymptotically periodic Schrödinger-Poisson systems with critical growth

61%
Zhang H., Xu J., Zhang F., Du M.
Open Mathematics
|
2014
|
tom 12
|
nr 10
1484-1499
EN
For a class of asymptotically periodic Schrödinger-Poisson systems with critical growth, the existence of ground states is established. The proof is based on the method of Nehari manifold and concentration compactness principle.
3
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A brief review of some application driven fast algorithms for elliptic partial differential equations

52%
Daripa P.
Open Mathematics
|
2012
|
tom 10
|
nr 1
204-216
EN
Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and parallel implementation of these algorithms and their applications to some pure and applied problems are also briefly reviewed.
4
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Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients

52%
Melnikov Y.
Open Mathematics
|
2010
|
tom 8
|
nr 1
53-72
EN
Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable for equations with piecewise-constant coefficients.
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