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Scalable atomistic simulation algorithms for materials research
 in Supercomputing ’01: Proceedings of the 2001 ACM/IEEE conference on Supercomputing (CDROM
"... Abstract. A suite of scalable atomistic simulation programs has been developed for materials research based on spacetime multiresolution algorithms. Design and analysis of parallel algorithms are presented for molecular dynamics (MD) simulations and quantummechanical (QM) calculations based on th ..."
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Cited by 26 (13 self)
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Abstract. A suite of scalable atomistic simulation programs has been developed for materials research based on spacetime multiresolution algorithms. Design and analysis of parallel algorithms are presented for molecular dynamics (MD) simulations and quantummechanical (QM) calculations based on the density functional theory. Performance tests have been carried out on 1,088processor Cray T3E and 1,280processor IBM SP3 computers. The linearscaling algorithms have enabled 6.44billionatom MD and 111,000atom QM calculations on 1,024 SP3 processors with parallel efficiency well over 90%. The productionquality programs also feature waveletbased computationalspace decomposition for adaptive load balancing, spacefillingcurvebased adaptive data compression with userdefined error bound for scalable I/O, and octreebased fast visibility culling for immersive and interactive visualization of massive simulation data.
Numerical methods for electronic structure calculations of materials
, 2006
"... The goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scienti£c computing audience. For this reason, we assume the r ..."
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Cited by 25 (1 self)
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The goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scienti£c computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and on the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful, but approximate, versions of this equation, which allow one to study nontrivial systems, took about £ve or six decades to develop. In particular, the last two decades saw a ¤urry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as Density Functional Theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an ef£cient way the ground state con£guration for many materials. This article will emphasize pseudopotentialdensity
Parallel selfconsistentfield calculations using chebyshevfiltered subspace acceleration
 Physical Review E
, 2006
"... Solving the KohnSham eigenvalue problem constitutes the most computationally expensive part in selfconsistent density functional theory (DFT) calculations. A nonlinear Chebyshevfiltered subspace iteration is developed which avoids computing explicit eigenvectors, except at the first SCF iteration ..."
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Cited by 21 (5 self)
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Solving the KohnSham eigenvalue problem constitutes the most computationally expensive part in selfconsistent density functional theory (DFT) calculations. A nonlinear Chebyshevfiltered subspace iteration is developed which avoids computing explicit eigenvectors, except at the first SCF iteration. The method may be viewed as an approach to solve the original nonlinear KohnSham equation by a nonlinear subspace iteration technique, without emphasizing the intermediate linearized KohnSham eigenvalue problems. The method reaches selfconsistency within a similar number of SCF iterations as eigensolverbased approaches. However, replacing the standard diagonalization at each SCF iteration by a Chebyshev subspace filtering step results in a significant speedup over methods based on standard dagonalization. Algorithmic details of a parallel implementation of this method are discussed. Numerical results are presented to show that the method enables to perform a class of highly challenging DFT calculations that were not feasible before.
Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems
 Commun. Math. Sci
"... Abstract. We propose an algorithm for extracting the diagonal of the inverse matrices arising from electronic structure calculation. The proposed algorithm uses a hierarchical decomposition of the computational domain. It first constructs hierarchical Schur complements of the interior points for the ..."
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Cited by 21 (8 self)
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Abstract. We propose an algorithm for extracting the diagonal of the inverse matrices arising from electronic structure calculation. The proposed algorithm uses a hierarchical decomposition of the computational domain. It first constructs hierarchical Schur complements of the interior points for the blocks of the domain in a bottomup pass and then extracts the diagonal entries efficiently in a topdown pass by exploiting the hierarchical local dependence of the inverse matrices. The overall cost of our algorithm is O(N3/2) for a two dimensional problem with N degrees of freedom. Numerical results in electronic structure calculation illustrate the efficiency and accuracy of the proposed algorithm. Key words. Diagonal extraction, hierarchical Schur complement, electronic structure calculation. AMS subject classifications. 65F30, 65Z05. 1.
DECAY BOUNDS AND O(n) ALGORITHMS FOR APPROXIMATING FUNCTIONS OF SPARSE MATRICES
, 2007
"... We establish decay bounds for the entries of f(A), where A is a sparse (in particular, banded) n × n diagonalizable matrix and f is smooth on a subset of the complex plane containing the spectrum of A. Combined with techniques from approximation theory, the bounds are used to compute sparse (or band ..."
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Cited by 21 (2 self)
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We establish decay bounds for the entries of f(A), where A is a sparse (in particular, banded) n × n diagonalizable matrix and f is smooth on a subset of the complex plane containing the spectrum of A. Combined with techniques from approximation theory, the bounds are used to compute sparse (or banded) approximations to f(A), resulting in algorithms that under appropriate conditions have linear complexity in the matrix dimension. Applications to various types of problems are discussed and illustrated by numerical examples.
Selfconsistentfield calculations using Chebyshevfiltered subspace iteration
 Journal of Computational Physics
"... Abstract The power of density functional theory is often limited by the high computational demand in solving an eigenvalue problem at each selfconsistentfield (SCF) iteration. The method presented in this paper replaces the explicit eigenvalue calculations by an approximation of the wanted invari ..."
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Cited by 19 (3 self)
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Abstract The power of density functional theory is often limited by the high computational demand in solving an eigenvalue problem at each selfconsistentfield (SCF) iteration. The method presented in this paper replaces the explicit eigenvalue calculations by an approximation of the wanted invariant subspace, obtained with the help of wellselected Chebyshev polynomial filters. In this approach, only the initial SCF iteration requires solving an eigenvalue problem, in order to provide a good initial subspace. In the remaining SCF iterations, no iterative eigensolvers are involved. Instead, Chebyshev polynomials are used to refine the subspace. The subspace iteration at each step is easily five to ten times faster than solving a corresponding eigenproblem by the most efficient eigenalgorithms. Moreover, the subspace iteration reaches selfconsistency within roughly the same number of steps as an eigensolverbased approach. This results in a significantly faster SCF iteration.
THE EXPONENTIALLY CONVERGENT TRAPEZOIDAL RULE
"... Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods ..."
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Cited by 17 (3 self)
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Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
DECAY PROPERTIES OF SPECTRAL PROJECTORS WITH APPLICATIONS TO ELECTRONIC STRUCTURE
, 2010
"... Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the ..."
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Cited by 16 (2 self)
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Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential offdiagonal decay (‘nearsightedness’) for the density matrix of gapped systems at zero electronic temperature in both orthogonal and nonorthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for nonmetallic systems. Our theory also allows us to treat the case of density matrices for arbitrary systems at finite electronic temperature, including metals. Other possible applications are also discussed.
Multiresolution quantum chemistry in multiwavelet bases: timedependent density functional theory with asymptotically corrected potentials in local density and generalized gradient approximations
 Department of Applied Mathematics, University of Colorado
"... Abstract. Multiresolution analysis in multiwavelet bases is being investigated as an alternative computational framework for molecular electronic structure calculations. The features that make it attractive include an orthonormal basis, fast algorithms with guaranteed precision and sparse represe ..."
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Cited by 14 (6 self)
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Abstract. Multiresolution analysis in multiwavelet bases is being investigated as an alternative computational framework for molecular electronic structure calculations. The features that make it attractive include an orthonormal basis, fast algorithms with guaranteed precision and sparse representations of many operators (e.g., Green functions). In this paper, we discuss the multiresolution formulation of quantum chemistry including application to density functional theory and developments that make practical computation in three and higher dimensions. 1
Nonperiodic finiteelement formulation of kohnsham density functional theory
 Journal of the Mechanics and Physics of Solids 58
, 2010
"... We present a realspace, nonperiodic, finiteelement formulation for KohnSham Density Functional Theory (KSDFT). We transform the original variational problem into a local saddlepoint problem, and show its wellposedness by proving the existence of minimizers. Further, we prove the convergence o ..."
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Cited by 14 (2 self)
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We present a realspace, nonperiodic, finiteelement formulation for KohnSham Density Functional Theory (KSDFT). We transform the original variational problem into a local saddlepoint problem, and show its wellposedness by proving the existence of minimizers. Further, we prove the convergence of finiteelement approximations including numerical quadratures. Based on domain decomposition, we develop a parallel finiteelement implementation of this formulation capable of performing both allelectron and pseudopotential calculations. We assess the accuracy of the formulation through selected test cases and demonstrate good agreement with the literature. We also evaluate the numerical performance of the implementation with regard to its scalability and convergence rates. We view this work as a step towards developing a method that can accurately study defects like vacancies, dislocations and crack tips using Density Functional Theory (DFT) at reasonable computational cost by retaining electronic resolution where it is necessary and seamlessly coarsegraining far away.