487 research outputs found
The INTERNODES method for applications in contact mechanics and dedicated preconditioning techniques
The mortar finite element method is a well-established method for the numerical solution of partial differential equations on domains displaying non-conforming interfaces. The method is known for its application in computational contact mechanics. However, its implementation remains challenging as it relies on geometrical projections and unconventional quadrature rules. The INTERNODES (INTERpolation for NOn-conforming DEcompositionS) method, instead, could overcome the implementation difficulties thanks to flexible interpolation techniques. Moreover, it was shown to be at least as accurate as the mortar method making it a very promising alternative for solving problems in contact mechanics. Unfortunately, in such situations the method requires solving a sequence of ill-conditioned linear systems. In this paper, preconditioning techniques are designed and implemented for the efficient solution of those linear systems
Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems
We propose a verified computation method for partial eigenvalues of a
Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a
contour integral-type eigensolver, can reduce a given eigenproblem into a
generalized eigenproblem of block Hankel matrices whose entries consist of
complex moments. In this study, we evaluate all errors in computing the complex
moments. We derive a truncation error bound of the quadrature. Then, we take
numerical errors of the quadrature into account and rigorously enclose the
entries of the block Hankel matrices. Each quadrature point gives rise to a
linear system, and its structure enables us to develop an efficient technique
to verify the approximate solution. Numerical experiments show that the
proposed method outperforms a standard method and infer that the proposed
method is potentially efficient in parallel.Comment: 15 pages, 4 figures, 1 tabl
Efficient ab initio auxiliary-field quantum Monte Carlo calculations in Gaussian bases via low-rank tensor decomposition
We describe an algorithm to reduce the cost of auxiliary-field quantum Monte
Carlo (AFQMC) calculations for the electronic structure problem. The technique
uses a nested low-rank factorization of the electron repulsion integral (ERI).
While the cost of conventional AFQMC calculations in Gaussian bases scales as
where is the size of the basis, we show that
ground-state energies can be computed through tensor decomposition with reduced
memory requirements and sub-quartic scaling. The algorithm is applied to
hydrogen chains and square grids, water clusters, and hexagonal BN. In all
cases we observe significant memory savings and, for larger systems, reduced,
sub-quartic simulation time.Comment: 14 pages, 13 figures, expanded dataset and tex
GMRES-Accelerated ADMM for Quadratic Objectives
We consider the sequence acceleration problem for the alternating direction
method-of-multipliers (ADMM) applied to a class of equality-constrained
problems with strongly convex quadratic objectives, which frequently arise as
the Newton subproblem of interior-point methods. Within this context, the ADMM
update equations are linear, the iterates are confined within a Krylov
subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its
ability to accelerate convergence. The basic ADMM method solves a
-conditioned problem in iterations. We give
theoretical justification and numerical evidence that the GMRES-accelerated
variant consistently solves the same problem in iterations
for an order-of-magnitude reduction in iterations, despite a worst-case bound
of iterations. The method is shown to be competitive against
standard preconditioned Krylov subspace methods for saddle-point problems. The
method is embedded within SeDuMi, a popular open-source solver for conic
optimization written in MATLAB, and used to solve many large-scale semidefinite
programs with error that decreases like , instead of ,
where is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on
Optimization (SIOPT
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