KSSOLV—a MATLAB toolbox for solving the Kohn-Sham equations

Yang, Chao; Meza, Juan C.; Lee, Byounghak; Wang, Lin‐Wang

doi:10.1145/1499096.1499099

Cited by 105 publications

(100 citation statements)

References 50 publications

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“…The term E rep measures the degree of repulsiveness of the local pseudo-potential with a term that corresponds to the non-singular part of ion-ion potential energy and the detail of E Ewald can be found in [59]. Hence, the discretized minimization problem is (5.7) min E total (X) s.t.…”

Section: Low-rank Nearest Correlationmentioning

confidence: 99%

A feasible method for optimization with orthogonality constraints

2012

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Abstract. Minimization with orthogonality constraints (e.g., X X = I) and/or spherical constraints (e.g., x 2 = 1) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we propose to use a Crank-Nicolson-like update scheme to preserve the constraints and based on it, develop curvilinear search algorithms with lower per-iteration cost compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842% to the best known solution on the largest problem "tai256c" in QAPLIB can be reached in 5 minutes on a typical laptop.

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Section: Low-rank Nearest Correlationmentioning

confidence: 99%

A feasible method for optimization with orthogonality constraints

2012

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“…which proves that the left-hand side of (20) is no greater than the right hand side of (20). Furthermore, the lower and upper bounds in (24) are attained at the n × k rank-one matrices S = [0 · · · 0 q k+1 ] and S = [q n 0 · · · 0], respectively.…”

Section: The Hessian At Solutionmentioning

confidence: 70%

“…We now demonstrate that the performance of EigPen can be improved by preconditioning using two matrices "Benzene" and "SiH4" generated from KSSOLV [20] -a MATLAB toolbox for solving the KohnSham equations in electronic structure calculation. The Kohn-Sham problem is discretized by a planewave expansion of the eigenfunctions.…”

Section: Preconditioning For Eigpenmentioning

confidence: 97%

Trace-Penalty Minimization for Large-scale Eigenspace Computation

Wen¹,

Yang²,

Liu³

et al. 2013

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The Rayleigh-Ritz (RR) procedure, including orthogonalization, constitutes a major bottleneck in computing relatively high-dimensional eigenspaces of large sparse matrices. Although operations involved in RR steps can be parallelized to a certain level, their parallel scalability, which is limited by some inherent sequential steps, is lower than dense matrix-matrix multiplications. The primary motivation of this paper is to develop a methodology that reduces the use of the RR procedure in exchange for matrix-matrix multiplications. We propose an unconstrained trace-penalty minimization model and establish its equivalence to the eigenvalue problem. With a suitably chosen penalty parameter, this model possesses far fewer undesirable full-rank stationary points than the classic trace minimization model. More importantly, it enables us to deploy algorithms that makes heavy use of dense matrixmatrix multiplications. Although the proposed algorithm does not necessarily reduce the total number of arithmetic operations, it leverages highly optimized operations on modern high performance computers to achieve parallel scalability. Numerical results based on a preliminary implementation, parallelized using OpenMP, show that our approach is promising.

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“…For possibly modifying m k , our implementation follows a strategy used by Yang et al [45] that is intended to maintain acceptable conditioning of the least-squares problem. In this, the condition number of the least-squares coefficient matrix (which is just the condition number of R in the QR decomposition) is monitored, and left-most columns of the matrix are dropped (and the QR decomposition updated) as necessary to keep the condition number below a prescribed threshold.…”

Section: Practical Considerationsmentioning

confidence: 99%

“…This category includes structurally similar methods developed for electronic structure computations, notably those by Pulay [35], [36] (known as Pulay mixing within the materials community and direct inversion on the iterative subspace, or DIIS, among computational chemists), and a number of other "mixing" methods; see Kresse and Furthmüller [28], [29], Le Bris [5], and Yang et al [45] for overviews. (In these applications, "mixing" derives from "charge mixing," and Anderson acceleration is known as "Anderson mixing.")…”

mentioning

confidence: 99%

Anderson Acceleration for Fixed-Point Iterations

Walker¹,

Ni²

2011

SIAM J. Numer. Anal.

422

233

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Abstract. This paper concerns an acceleration method for fixed-point iterations that originated in work of D. G. Anderson [J. Assoc. Comput. Mach., 12 (1965), pp. 547-560], which we accordingly call Anderson acceleration here. This method has enjoyed considerable success and wide usage in electronic structure computations, where it is known as Anderson mixing; however, it seems to have been untried or underexploited in many other important applications. Moreover, while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities, this method has received relatively little attention from these communities over the years. A recent paper by H. Fang and Y. Saad [Numer. Linear Algebra Appl., 16 (2009), pp. 197-221] has clarified a remarkable relationship of Anderson acceleration to quasi-Newton (secant updating) methods and extended it to define a broader Anderson family of acceleration methods. In this paper, our goals are to shed additional light on Anderson acceleration and to draw further attention to its usefulness as a general tool. We first show that, on linear problems, Anderson acceleration without truncation is "essentially equivalent" in a certain sense to the generalized minimal residual (GMRES) method. We also show that the Type 1 variant in the Fang-Saad Anderson family is similarly essentially equivalent to the Arnoldi (full orthogonalization) method. We then discuss practical considerations for implementing Anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications.

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KSSOLV—a MATLAB toolbox for solving the Kohn-Sham equations

Cited by 105 publications

References 50 publications

A feasible method for optimization with orthogonality constraints

A feasible method for optimization with orthogonality constraints

Trace-Penalty Minimization for Large-scale Eigenspace Computation

Anderson Acceleration for Fixed-Point Iterations

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