Domain Decomposition for Wavelet Single Layer on Geometries with Patches

We focus on the single layer formulation which provides an integral equation of the first kind that is very badly conditioned. The condition number of the unpreconditioned system increases exponentially with the multiscale levels. A remedy utilizing overlapping domain decompositions applied to the Boundary Element Method by means of wavelets is examined. The width of the overlapping of the subdomains plays an important role in the estimation of the eigenvalues as well as the condition number of the additive domain decomposition operator. We examine the convergence analysis of the domain decomposition method which depends on the wavelet levels and on the size of the subdomain overlaps. Our theoretical results related to the additive Schwarz method are corroborated by numerical outputs.

Analytical Polarizable Continuum Model for Wavelets on NURBS Patches

This article concerns the application of wavelet techniques on molecular surfaces constituted of four-sided patches. The Polarizable Continuum Model, which is governed by the Poisson-Boltzmann equation, is treated by means of boundary integral equations. The media inside and outside the molecular surface consist respectively of the solute and the solvent. For a given electrically charged molecule, the principal unknown is the electrostatic solvation energy when the permittivity is specified. The wavelet basis functions are constructed on the unit square which are subsequently mapped onto the patches that are assumed to be isotropically shaped and to admit similar surface areas. The initial transmission problem is recast as an integral equation in term of both the single and the double layers. Domain decomposition preconditioner serves as acceleration of the linear solver of the single layer which is badly conditioned.

Multiwavelet Boundary Element Method for Cavities Admitting Many NURBS Patches

We consider the modeling and simulation by means of multiwavelets on many patches. Our focus is on molecular surfaces which are represented in the form of Solvent Excluded Surfaces that are featured by smooth blendings between the constituting atoms. The wavelet bases are constructed on the unit square which maps bijectively onto the patches embedded in the space. The cavity which designates the surface bounding a molecular model is acquired from the nuclei coordinates and the Van-der-Waals radii. We use multi-wavelets for which the wavelet basis functions are organized hierarchically on several levels. Our assembly of the linear system is accomplished by using a hierarchical tree which enables the treatment of large molecules admitting thousands of patches. Along with the patch construction, some wavelet simulation outcomes which are applied to realistic patches are reported.

Error Estimator Using Higher Order FEM for an Interface Problem

A higher order finite element method is considered to treat an interface problem. The polynomial degree is allowed to be arbitrary but it is fixed for the FEM-variational formulation. We propose an error estimator which turns out to be efficient and reliable. We demonstrate upper and lower bounds of the error estimator with respect to the exact accuracy. For the transmission problem, the coefficients for the internal and external domains can be highly dissimilar. One major difficulty is the characteristic of the estimator at the interface. The a-posteriori error estimates can be computed very efficiently element by element. To corroborate the theoretical analysis, we report on a few numerical results.

Erratum to “Multilevel B-Spline Repulsive Energy in Nanomodeling of Graphenes” [Journal of Surface Engineered Materials and Advanced Technology Vol. 4 No. 2 (April 2014) 75-86]

Quantum energies which are used in applications are usually composed of repulsive and attractive terms. The objective of this study is to use an accurate and efficient fitting of the repulsive energy instead of using standard parametrizations. The investigation is based on Density Functional Theory and Tight Binding simulations. Our objective is not only to capture the values of the repulsive terms but also to efficiently reproduce the elastic properties and the forces. The elasticity values determine the rigidity of a material when some traction or load is applied on it. The pair-potential is based on an exponential term corrected by B-spline terms. In order to accelerate the computations, one uses a hierarchical optimization for the B-splines on different levels. Carbon graphenes constitute the configurations used in the simulations. We report on some results to show the efficiency of the B-splines on different levels.

Multilevel B-Spline Repulsive Energy in Nanomodeling of Graphenes

Quantum energies which are used in applications are usually composed of repulsive and attractive terms. The objective of this study is to use an accurate and efficient fitting of the repulsive energy instead of using standard parametrizations. The investigation is based on Density Functional Theory and Tight Binding simulations. Our objective is not only to capture the values of the repulsive terms but also to efficiently reproduce the elastic properties and the forces. The elasticity values determine the rigidity of a material when some traction or load is applied on it. The pair-potential is based on an exponential term corrected by B-spline terms. In order to accelerate the computations, one uses a hierarchical optimization for the B-splines on different levels. Carbon graphenes constitute the configurations used in the simulations. We report on some results to show the efficiency of the B-splines on different levels.

Erratum to “On DFT Molecular Simulation for Non-Adaptive Kernel Approximation” [Advances in Materials Physics and Chemistry Vol. 4 No. 6 (June 2014) 105-115]

The original online version of this article (Randrianarivony, M. (June 2014) On DFT Molecular Simulation for Non-Adaptive Kernel Approximation. Advances in Materials Physics and Chemistry, Vol. 4 No. 6, 105-115. http://dx.doi.org/10.4236/ampc.2014.46013) did not contain any acknowledgment. The author wishes to add the following acknowledgements: Acknowledgements: This work was partially supported by Eurostars Project E!6935 funded by German Federal Ministry of Education and Research.

On DFT Molecular Simulation for Non-Adaptive Kernel Approximation

Using accurate quantum energy computations in nanotechnologic applications is usually very computationally intensive. That makes it difficult to apply in subsequent quantum simulation. In this paper, we present some preliminary results pertaining to stochastic methods for alleviating the numerical expense of quantum estimations. The initial information about the quantum energy originates from the Density Functional Theory. The determination of the parameters is performed by using methods stemming from machine learning. We survey the covariance method using marginal likelihood for the statistical simulation. More emphasis is put at the position of equilibrium where the total atomic energy attains its minimum. The originally intensive data can be reproduced efficiently without losing accuracy. A significant acceleration gain is perceived by using the proposed method.