Browsing by Subject "APPROXIMATIONS"

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  • Lehtola, Susi; Blockhuys, Frank; Van Alsenoy, Christian (2020)
    A uniform derivation of the self-consistent field equations in a finite basis set is presented. Both restricted and unrestricted Hartree-Fock (HF) theory as well as various density functional approximations are considered. The unitary invariance of the HF and density functional models is discussed, paving the way for the use of localized molecular orbitals. The self-consistent field equations are derived in a non-orthogonal basis set, and their solution is discussed also in the presence of linear dependencies in the basis. It is argued why iterative diagonalization of the Kohn-Sham-Fock matrix leads to the minimization of the total energy. Alternative methods for the solution of the self-consistent field equations via direct minimization as well as stability analysis are briefly discussed. Explicit expressions are given for the contributions to the Kohn-Sham-Fock matrix up to meta-GGA functionals. Range-separated hybrids and non-local correlation functionals are summarily reviewed.
  • Clason, Christian; Helin, Tapio; Kretschmann, Remo; Piiroinen, Petteri (2019)
    Uncertainty quantification requires efficient summarization of high- or even infinite-dimensional (i.e., nonparametric) distributions based on, e.g., suitable point estimates (modes) for posterior distributions arising from model-specific prior distributions. In this work, we consider nonparametric modes and maximum a posteriori (MAP) estimates for priors that do not admit continuous densities, for which previous approaches based on small ball probabilities fail. We propose a novel definition of generalized modes based on the concept of approximating sequences, which reduce to the classical mode in certain situations that include Gaussian priors but also exist for a more general class of priors. The latter includes the case of priors that impose strict bounds on the admissible parameters and in particular of uniform priors. For uniform priors defined by random series with uniformly distributed coefficients, we show that generalized MAP estimates but not classical MAP estimates can be characterized as minimizers of a suitable functional that plays the role of a generalized Onsager-Machlup functional. This is then used to show consistency of nonlinear Bayesian inverse problems with uniform priors and Gaussian noise.
  • Laitila, Jussi; Moilanen, Atte (2017)
    We present new tight performance guarantees for the greedy maximization of monotone submodular set functions. Our main result first provides a performance guarantee in terms of the overlap of the optimal and greedy solutions. As a consequence we improve performance guarantees of Nemhauser et al. (Math Program 14: 265-294, 1978) and Conforti and Cornuejols (Discr Appl Math 7: 251-274, 1984) for maximization over subsets, which are at least half the size of the problem domain. As a further application, we obtain a new tight approximation guarantee in terms of the cardinality of the problem domain.