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  • 1.
    Rubensson, Emanuel
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi (stängd 20110512).
    Sparse Matrices in Self-Consistent Field Methods2006Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis is part of an effort to enable large-scale Hartree-Fock/Kohn-Sham (HF/KS) calculations. The objective is to model molecules and materials containing thousands of atoms at the quantum mechanical level. HF/KS calculations are usually performed with the Self-Consistent Field (SCF) method. This method involves two computationally intensive steps. These steps are the construction of the Fock/Kohn-Sham potential matrix from a given electron density and the subsequent update of the electron density usually represented by the so-called density matrix. In this thesis the focus lies on the representation of potentials and electron density and on the density matrix construction step in the SCF method. Traditionally a diagonalization has been used for the construction of the density matrix. This diagonalization method is, however, not appropriate for large systems since the time complexity for this operation is σ(n3). Three types of alternative methods are described in this thesis; energy minimization, Chebyshev expansion, and density matrix purification. The efficiency of these methods relies on fast matrix-matrix multiplication. Since the occurring matrices become sparse when the separation between atoms exceeds some value, the matrix-matrix multiplication can be performed with complexity σ(n).

    A hierarchic sparse matrix data structure is proposed for the storage and manipulation of matrices. This data structure allows for easy development and implementation of algebraic matrix operations, particularly needed for the density matrix construction, but also for other parts of the SCF calculation. The thesis addresses also truncation of small elements to enforce sparsity, permutation and blocking of matrices, and furthermore calculation of the HOMO-LUMO gap and a few surrounding eigenpairs when density matrix purification is used instead of the traditional diagonalization method.

    Ladda ner fulltext (pdf)
    FULLTEXT01
  • 2.
    Rubensson, Emanuel H.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Matrix Algebra for Quantum Chemistry2008Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis concerns methods of reduced complexity for electronic structure calculations.  When quantum chemistry methods are applied to large systems, it is important to optimally use computer resources and only store data and perform operations that contribute to the overall accuracy. At the same time, precarious approximations could jeopardize the reliability of the whole calculation.  In this thesis, the self-consistent field method is seen as a sequence of rotations of the occupied subspace. Errors coming from computational approximations are characterized as erroneous rotations of this subspace. This viewpoint is optimal in the sense that the occupied subspace uniquely defines the electron density. Errors should be measured by their impact on the overall accuracy instead of by their constituent parts. With this point of view, a mathematical framework for control of errors in Hartree-Fock/Kohn-Sham calculations is proposed.  A unifying framework is of particular importance when computational approximations are introduced to efficiently handle large systems.

    An important operation in Hartree-Fock/Kohn-Sham calculations is the calculation of the density matrix for a given Fock/Kohn-Sham matrix. In this thesis, density matrix purification is used to compute the density matrix with time and memory usage increasing only linearly with system size. The forward error of purification is analyzed and schemes to control the forward error are proposed. The presented purification methods are coupled with effective methods to compute interior eigenvalues of the Fock/Kohn-Sham matrix also proposed in this thesis.New methods for inverse factorizations of Hermitian positive definite matrices that can be used for congruence transformations of the Fock/Kohn-Sham and density matrices are suggested as well.

    Most of the methods above have been implemented in the Ergo quantum chemistry program. This program uses a hierarchic sparse matrix library, also presented in this thesis, which is parallelized for shared memory computer architectures. It is demonstrated that the Ergo program is able to perform linear scaling Hartree-Fock calculations.

    Ladda ner fulltext (pdf)
    FULLTEXT01
  • 3.
    Rubensson, Emanuel H.
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Bock, Nicolas
    Theoretical Division, Los Alamos National Laboratory.
    Holmström, Erik
    Instituto de Física, Universidad Austral de Chile.
    Niklasson, Anders M. N.
    KTH, Skolan för industriell teknik och management (ITM), Materialvetenskap, Tillämpad materialfysik.
    Recursive inverse factorization2008Ingår i: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, nr 10, s. 104105-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A recursive algorithm for the inverse factorization S−1=ZZ* of Hermitian positive definite matrices S is proposed. The inverse factorization is based on iterative refinement [A.M.N. Niklasson, Phys. Rev. B 70, 193102 (2004)] combined with a recursive decomposition of S. As the computational kernel is matrix-matrix multiplication, the algorithm can be parallelized and the computational effort increases linearly with system size for systems with sufficiently sparse matrices. Recent advances in network theory are used to find appropriate recursive decompositions. We show that optimization of the so-called network modularity results in an improved partitioning compared to other approaches. In particular, when the recursive inverse factorization is applied to overlap matrices of irregularly structured three-dimensional molecules.

  • 4.
    Rubensson, Emanuel H.
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi (stängd 20110512).
    Rudberg, Elias
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi (stängd 20110512).
    Salek, Pawel
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi (stängd 20110512).
    A hierarchic sparse matrix data structure for large-scale Hartree-Fock/Kohn-Sham calculations2007Ingår i: Journal of Computational Chemistry, ISSN 0192-8651, E-ISSN 1096-987X, Vol. 28, nr 16, s. 2531-2537Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A hierarchic sparse matrix data structure for Hartree-Fock/Kohn-Sham calculations is presented. The data structure makes the implementation of matrix manipulations needed for large systems faster, easier, and more maintainable without loss of performance. Algorithms for symmetric matrix square and inverse Cholesky decomposition within the hierarchic framework are also described. The presented data structure is general; in addition to its use in HartreeFock/Kohn-Sham calculations, it may also be used in other research areas where matrices with similar properties are encountered. The applicability of the data structure to ab initio calculations is shown with help of benchmarks on water droplets and graphene nanoribbons.

  • 5.
    Rubensson, Emanuel H.
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Rudberg, Elias
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Salek, Pawel
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Density matrix purification with rigorous error control2008Ingår i: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, nr 7Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Density matrix purification, although being a powerful tool for linear scaling construction of the density matrix in electronic structure calculations, has been limited by uncontrolled error accumulation. In this article, a strategy for the removal of small matrix elements in density matrix purification is proposed with which the forward error can be rigorously controlled. The total forward error is separated into two parts, the error in eigenvalues and the error in the occupied invariant subspace. We use the concept of canonical angles to measure and control differences between exact and approximate occupied subspaces. We also analyze the conditioning of the density matrix construction problem and propose a method for calculation of interior eigenvalues to be used together with density matrix purification. (C) 2008 American Institute of Physics.

  • 6.
    Rubensson, Emanuel H.
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Rudberg, Elias
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Salek, Pawel
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Rotations of occupied invariant subspaces in self-consistent field calculations2008Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 49, nr 3, s. 032103-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this article, the self-consistent field (SCF) procedure as used in Hartree-Fock and Kohn-Sham calculations is viewed as a sequence of rotations of the so-called occupied invariant subspace of the potential and density matrices. Computational approximations are characterized as erroneous rotations of this subspace. Differences between subspaces are measured and controlled by the canonical angles between them. With this approach, a first step is taken toward a method where errors from computational approximations are rigorously controlled and threshold values are directly related to the accuracy of the current trial density, thus eliminating the use of ad hoc threshold values. Then, the use of computational resources can be kept down as much as possible without impairment of the SCF convergence. (C) 2008 American Institute of Physics.

  • 7.
    Rubensson, Emanuel H.
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Rudberg, Elias
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Salek, Pawel
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Truncation of Small Matrix Elements Based on the Euclidean Norm for Blocked Data Structures2009Ingår i: Journal of Computational Chemistry, ISSN 0192-8651, E-ISSN 1096-987X, Vol. 30, nr 6, s. 974-977Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Methods for the removal of small symmetric matrix elements based on the Euclidean norm of the error matrix are presented in this article. In large scale Hartree-Fock and Kohn-Sham calculations it is important to be able to enforce matrix sparsity while keeping errors under control. Truncation based on some unitary-invariant norm allows for control of errors in the occupied subspace as described in (Rubensson et al. J Math Phys 49, 032103). The Euclidean norm is unitary-invariant and does not grow intrinsically with system size and is thus suitable for error control in large scale calculations. The presented truncation schemes repetitively use the Lanczos method to compute the Euclidean norms of the error matrix candidates. Ritz value convergence patterns are utilized to reduce the total number of Lanczos iterations.

  • 8.
    Rubensson, Emanuel H.
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Zahedi, Sara
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Computation of interior eigenvalues in electronic structure calculations facilitated by density matrix purification2008Ingår i: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, nr 17, s. 176101-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Density matrix purification, is in this work, used to facilitate the computation of eigenpairs around the highest occupied and the lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) in electronic structure calculations. The ability of purification to give large separation between eigenvalues close to the HOMO-LUMO gap is used to accelerate convergence of the Lanczos method. Illustrations indicate that a new eigenpair is found more often than every second Lanczos iteration when the proposed methods are used.

  • 9.
    Rubensson, Emanuel
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Jensen, Hans-jørgen
    Determination of the chemical potential and HOMO/LUMO orbitals in density purification methods2006Ingår i: Chemical Physics Letters, ISSN 0009-2614, E-ISSN 1873-4448, Vol. 432, nr 4-6, s. 591-594Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Several density purification methods have been proposed to achieve linear scaling in Hartree-Fock and Kohn-Sham calculations. However, only the density is found, while in the traditional diagonalization method the orbitals are also obtained. This could be seen as a drawback as in many cases one would like at least the HOMO and LUMO orbitals and their orbital energies. In this letter, we show how a value for the chemical potential can be obtained as a by-product of density purification methods at negligible cost. Once the chemical potential is known, MO's around the HOMO-LUMO gap can be calculated with the Spectral Transformation Lanczos method.

  • 10.
    Rubensson, Emanuel
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Rudberg, Elias
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Near-Idempotent MatricesManuskript (Övrigt vetenskapligt)
  • 11.
    Rubensson, Emanuel
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Rudberg, Elias
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Salek, Pawel
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Sparse matrix algebra for quantum modeling of large systems2007Ingår i: Applied Parallel Computing - STATE OF THE ART IN SCIENTIFIC COMPUTING     / [ed] Kagstrom B, Elmroth E, Dongarra J, Wasniewski J, Berlin, Germany: Springer-Verlag , 2007, s. 90-99Konferensbidrag (Refereegranskat)
    Abstract [en]

    Matrices appearing in Hartree-Fock or density functional theory coming from discretization with help of atom-centered local basis sets become sparse when the separation between atoms exceeds some system-dependent threshold value. Efficient implementation of sparse matrix algebra is therefore essential in large-scale quantum calculations. We describe a unique combination of algorithms and data representation that provides high performance and strict error control in blocked sparse matrix algebra. This has applications to matrix-rnatrix multiplication, the Trace-Correcting Purification algorithm and the entire self-consistent field calculation.

  • 12.
    Rubensson, Emanuel
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Salek, Pawel
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Systematic sparse matrix error control for linear scaling electronic structure calculations2005Ingår i: Journal of Computational Chemistry, ISSN 0192-8651, E-ISSN 1096-987X, Vol. 26, nr 15, s. 1628-1637Artikel i tidskrift (Refereegranskat)
    Abstract [en]

     Efficient truncation criteria used in multiatom blocked sparse matrix operations for ab initio calculations are proposed. As system size increases, so does the need to stay on top of errors and still achieve high performance. A variant of a blocked sparse matrix algebra to achieve strict error control with good performance is proposed. The presented idea is that the condition to drop a certain submatrix should depend not only on the magnitude of that particular submatrix, but also on which other submatrices that are dropped. The decision to remove a certain submatrix is based on the contribution the removal would cause to the error in the chosen norm. We study the effect of an accumulated truncation error in iterative algorithms like trace correcting density matrix purification. One way to reduce the initial exponential growth of this error is presented. The presented error control for a sparse blocked matrix toolbox allows for achieving optimal performance by performing only necessary operations needed to maintain the requested level of accuracy.

  • 13.
    Rudberg, Elias
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Rubensson, Emanuel H.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Salek, Pawel
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Automatic Selection of Integral Thresholds by Extrapolation in Coulomb and Exchange Matrix Constructions2009Ingår i: Journal of Chemical Theory and Computation, ISSN 1549-9618, E-ISSN 1549-9626, Vol. 5, nr 1, s. 80-85Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a method to compute Coulomb and exchange matrices with predetermined accuracy as measured by a matrix norm. The computation of these matrices is fundamental in Hartree-Fock and Kohn-Sham electronic structure calculations. We show numerically that, when modern algorithms for Coulomb and exchange matrix evaluation are applied, the Euclidean norm of the error matrix e is related to the threshold value tau as epsilon C tau(alpha). The presented extrapolation method automatically selects the integral thresholds so that the Euclidean norm of the error matrix is at the requested accuracy. This approach is demonstrated for a variety of systems, including protein-like systems, water clusters, and graphene sheets. The proposed method represents an important step toward complete error control throughout the self-consistent field calculation as described in [J. Math. Phys. 2008, 49, 032103].

  • 14.
    Rudberg, Elias
    et al.
    KTH, Skolan för bioteknologi (BIO).
    Rubensson, Emanuel H.
    KTH, Skolan för bioteknologi (BIO).
    Salek, Pawel
    KTH, Skolan för bioteknologi (BIO).
    Estimation of errors in Coulomb and exchange matrix constructionManuskript (Övrigt vetenskapligt)
  • 15.
    Rudberg, Elias
    et al.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Rubensson, Emanuel H.
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Salek, Pawel
    KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
    Hartree-Fock calculations with linearly scaling memory usage2008Ingår i: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, nr 18Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present an implementation of a set of algorithms for performing Hartree-Fock calculations with resource requirements in terms of both time and memory directly proportional to the system size. In particular, a way of directly computing the Hartree-Fock exchange matrix in sparse form is described which gives only small addressing overhead. Linear scaling in both time and memory is demonstrated in benchmark calculations for system sizes up to 11 650 atoms and 67 204 Gaussian basis functions on a single computer with 32 Gbytes of memory. The sparsity of overlap, Fock, and density matrices as well as band gaps are also shown for a wide range of system sizes, for both linear and three-dimensional systems. (C) 2008 American Institute of Physics.

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