693 research outputs found
Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections
This work focuses on the iterative solution of sequences of KKT linear
systems arising in interior point methods applied to large convex quadratic
programming problems. This task is the computational core of the interior point
procedure and an efficient preconditioning strategy is crucial for the
efficiency of the overall method. Constraint preconditioners are very effective
in this context; nevertheless, their computation may be very expensive for
large-scale problems, and resorting to approximations of them may be
convenient. Here we propose a procedure for building inexact constraint
preconditioners by updating a "seed" constraint preconditioner computed for a
KKT matrix at a previous interior point iteration. These updates are obtained
through low-rank corrections of the Schur complement of the (1,1) block of the
seed preconditioner. The updated preconditioners are analyzed both
theoretically and computationally. The results obtained show that our updating
procedure, coupled with an adaptive strategy for determining whether to
reinitialize or update the preconditioner, can enhance the performance of
interior point methods on large problems.Comment: 22 page
Development of low-scaling electronic structure methods using rank factorizations and an attenuated Coulomb metric
Novel low-scaling techniques for molecular electronic structure and property calculations are introduced. Through the use of rank-revealing matrix factorizations, overheads compared to canonical molecular orbital-based formulations are virtually eliminated. Asymptotic computational complexity is linear or sub-linear (depending on the property) through the use of sparsity-preserving transformations throughout. For electron correlation energy calculations within the random phase approximation, these techniques are combined with an attenuated Coulomb metric in the resolution-of-the-identity to improve the accuracy over existing low-scaling methods and to reduce the scaling compared to existing canonical methods. For the resolution-of-the-identity itself, a novel method for the compression of auxiliary bases is introduced, powered by removal of the particle-hole-interaction nullspace through projection. Furthermore, efficient schemes for the calculation of molecular response properties at the Hartree–Fock and density functional theory levels are introduced: For the linear scaling calculation of vibrational frequencies, the exact cancellation of different long-range operator derivatives is employed in combination with Laplace-transformed and Cholesky-decomposed coupled-perturbed self-consistent field theories. Using related techniques, calculations of indirect nuclear spin-spin coupling constants with asymptotically constant time complexity are demonstrated and used to explore the dependence of spin-spin couplings in a peptide on the size of a surrounding solvent environment
Unconditionally stable integration of Maxwell's equations
Numerical integration of Maxwell’s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction
implicit – finite difference time domain scheme. In this paper, we discuss unconditionally stable integration for a general semidiscrete Maxwell system allowing non-Cartesian space grids as encountered in finite-element discretizations. Such grids exclude the alternating direction implicit approach. Particular attention
is given to the second-order trapezoidal rule implemented with preconditioned conjugate gradient iteration and to second-order exponential integration using Krylov subspace iteration for evaluating the arising ϕ-functions. A three-space dimensional test problem is used for numerical assessment and comparison with an economical second-order implicit–explicit integrator
Nonparametric Estimation of Multi-View Latent Variable Models
Spectral methods have greatly advanced the estimation of latent variable
models, generating a sequence of novel and efficient algorithms with strong
theoretical guarantees. However, current spectral algorithms are largely
restricted to mixtures of discrete or Gaussian distributions. In this paper, we
propose a kernel method for learning multi-view latent variable models,
allowing each mixture component to be nonparametric. The key idea of the method
is to embed the joint distribution of a multi-view latent variable into a
reproducing kernel Hilbert space, and then the latent parameters are recovered
using a robust tensor power method. We establish that the sample complexity for
the proposed method is quadratic in the number of latent components and is a
low order polynomial in the other relevant parameters. Thus, our non-parametric
tensor approach to learning latent variable models enjoys good sample and
computational efficiencies. Moreover, the non-parametric tensor power method
compares favorably to EM algorithm and other existing spectral algorithms in
our experiments
Solving mixed sparse-dense linear least-squares problems by preconditioned iterative methods
The efficient solution of large linear least-squares problems in which the system matrix A contains rows with very different densities is challenging. Previous work has focused on direct methods for problems in which A has a few relatively dense rows. These rows are initially ignored, a factorization of the sparse part is computed using a sparse direct solver, and then the solution is updated to take account of the omitted dense rows. In some practical applications the number of dense rows can be significant and for very large problems, using a direct solver may not be feasible. We propose processing rows that are identified as dense separately within a conjugate gradient method using an incomplete factorization preconditioner combined with the factorization of a dense matrix of size equal to the number
of dense rows. Numerical experiments on large-scale problems from real applications are used to illustrate the effectiveness of our approach. The results demonstrate that we can efficiently solve problems that could not be solved by a preconditioned conjugate gradient method without exploiting the dense rows
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