In a variety of modern applications there arises a need to tessellate the domain into representative regions,called Voronoi cells.A particular type of such tessellations,called centroidal Voronoi tessellations or CVTs...In a variety of modern applications there arises a need to tessellate the domain into representative regions,called Voronoi cells.A particular type of such tessellations,called centroidal Voronoi tessellations or CVTs,are in big demand due to their optimality properties important for many applications.The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings.This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems.Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems,and O(k)complexity estimation is provided for a problem with k generators.展开更多
Orbital-free density functional theory(OFDFT)is a quantum mechanical method in which the energy of a material depends only on the electron density and ionic positions.We examine some popular algorithms for optimizing ...Orbital-free density functional theory(OFDFT)is a quantum mechanical method in which the energy of a material depends only on the electron density and ionic positions.We examine some popular algorithms for optimizing the electron density distribution in OFDFT,explaining their suitability,benchmarking their performance,and suggesting some improvements.We start by describing the constrained optimization problem that encompasses electron density optimization.Next,we discuss the line search(including Wolfe conditions)and the nonlinear conjugate gradient and truncated Newton algorithms,as implemented in our open source OFDFT code.We finally focus on preconditioners derived from OFDFT energy functionals.Newlyderived preconditioners are successful for simulation cells of all sizes without regions of low electron-density and for small simulation cells with such regions.展开更多
We consider the Thomas-Fermi-von Weizsacker energy functional,with the Wang-Teter correction,and present an efficient real space method for Orbital-Free Density Functional Theory.It is proved that the energy minimizer...We consider the Thomas-Fermi-von Weizsacker energy functional,with the Wang-Teter correction,and present an efficient real space method for Orbital-Free Density Functional Theory.It is proved that the energy minimizer satisfies a second order quasilinear elliptic equation,even at the points where the electron density vanishes.This information is used to construct an efficient energy minimization method for the resulting constrained problem,based on the truncated Newton method for unconstrained optimization.The Wang-Teter kernel is analyzed,and its behavior in real space at short and far distances is determined.A second order accurate discretization of the energy is obtained using finite differences.The efficiency and accuracy of the method is illustrated with numerical simulations in an Aluminium FCC lattice.展开更多
基金supported by the U.S.Department of Energy under Award DE-SC-0001691support from the ORAU Ralph E.Powe Junior Faculty Enhancement Award and from the National Science Foundation under the grants DMS-1056821 and DMS-0915013.
文摘In a variety of modern applications there arises a need to tessellate the domain into representative regions,called Voronoi cells.A particular type of such tessellations,called centroidal Voronoi tessellations or CVTs,are in big demand due to their optimality properties important for many applications.The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings.This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems.Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems,and O(k)complexity estimation is provided for a problem with k generators.
基金We would like to thank the National Defense Science and Engineering Graduate Fellowship program(L.H.)and the National Science Foundation(E.A.C.)for funding.
文摘Orbital-free density functional theory(OFDFT)is a quantum mechanical method in which the energy of a material depends only on the electron density and ionic positions.We examine some popular algorithms for optimizing the electron density distribution in OFDFT,explaining their suitability,benchmarking their performance,and suggesting some improvements.We start by describing the constrained optimization problem that encompasses electron density optimization.Next,we discuss the line search(including Wolfe conditions)and the nonlinear conjugate gradient and truncated Newton algorithms,as implemented in our open source OFDFT code.We finally focus on preconditioners derived from OFDFT energy functionals.Newlyderived preconditioners are successful for simulation cells of all sizes without regions of low electron-density and for small simulation cells with such regions.
基金I would like to thank Prof.Weinan E for suggesting this problem to me.I would also like to thank Prof.Emily A.Carter,and the members of her group,in particular Gregory Ho and Vincent Ligneres,for explaining to me some of the intricacies of OFDFT.The simulations presented in this article were carried out in an HP workstation purchased with funds provided by NSF grant DMS-0411504in a Beowulf cluster purchased by the Mathematics department at UCSB with funds provided by an NSF SCREMS grant DMS-0112388and in an SGI Altix system located at the CNSI Computer Facilities at UCSB,purchased with funds provided by NSF grant CHE-0321368.
文摘We consider the Thomas-Fermi-von Weizsacker energy functional,with the Wang-Teter correction,and present an efficient real space method for Orbital-Free Density Functional Theory.It is proved that the energy minimizer satisfies a second order quasilinear elliptic equation,even at the points where the electron density vanishes.This information is used to construct an efficient energy minimization method for the resulting constrained problem,based on the truncated Newton method for unconstrained optimization.The Wang-Teter kernel is analyzed,and its behavior in real space at short and far distances is determined.A second order accurate discretization of the energy is obtained using finite differences.The efficiency and accuracy of the method is illustrated with numerical simulations in an Aluminium FCC lattice.
