19,491 research outputs found
Adaptive BDDC in Three Dimensions
The adaptive BDDC method is extended to the selection of face constraints in
three dimensions. A new implementation of the BDDC method is presented based on
a global formulation without an explicit coarse problem, with massive
parallelism provided by a multifrontal solver. Constraints are implemented by a
projection and sparsity of the projected operator is preserved by a generalized
change of variables. The effectiveness of the method is illustrated on several
engineering problems.Comment: 28 pages, 9 figures, 9 table
Computing the eigenvalues of symmetric H2-matrices by slicing the spectrum
The computation of eigenvalues of large-scale matrices arising from finite
element discretizations has gained significant interest in the last decade.
Here we present a new algorithm based on slicing the spectrum that takes
advantage of the rank structure of resolvent matrices in order to compute m
eigenvalues of the generalized symmetric eigenvalue problem in operations, where is a small constant
A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs
A self-learning algebraic multigrid method for dominant and minimal singular
triplets and eigenpairs is described. The method consists of two multilevel
phases. In the first, multiplicative phase (setup phase), tentative singular
triplets are calculated along with a multigrid hierarchy of interpolation
operators that approximately fit the tentative singular vectors in a collective
and self-learning manner, using multiplicative update formulas. In the second,
additive phase (solve phase), the tentative singular triplets are improved up
to the desired accuracy by using an additive correction scheme with fixed
interpolation operators, combined with a Ritz update. A suitable generalization
of the singular value decomposition is formulated that applies to the coarse
levels of the multilevel cycles. The proposed algorithm combines and extends
two existing multigrid approaches for symmetric positive definite eigenvalue
problems to the case of dominant and minimal singular triplets. Numerical tests
on model problems from different areas show that the algorithm converges to
high accuracy in a modest number of iterations, and is flexible enough to deal
with a variety of problems due to its self-learning properties.Comment: 29 page
A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem
In this paper, we consider the tensor eigenvalue complementarity problem
which is closely related to the optimality conditions for polynomial
optimization, as well as a class of differential inclusions with nonconvex
processes. By introducing an NCP-function, we reformulate the tensor eigenvalue
complementarity problem as a system of nonlinear equations. We show that this
function is strongly semismooth but not differentiable, in which case the
classical smoothing methods cannot apply. Furthermore, we propose a damped
semismooth Newton method for tensor eigenvalue complementarity problem. A new
procedure to evaluate an element of the generalized Jocobian is given, which
turns out to be an element of the B-subdifferential under mild assumptions. As
a result, the convergence of the damped semismooth Newton method is guaranteed
by existing results. The numerical experiments also show that our method is
efficient and promising
Rational minimax approximation via adaptive barycentric representations
Computing rational minimax approximations can be very challenging when there
are singularities on or near the interval of approximation - precisely the case
where rational functions outperform polynomials by a landslide. We show that
far more robust algorithms than previously available can be developed by making
use of rational barycentric representations whose support points are chosen in
an adaptive fashion as the approximant is computed. Three variants of this
barycentric strategy are all shown to be powerful: (1) a classical Remez
algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares,
and (3) a differential correction algorithm. Our preferred combination,
implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and
then switch to (1) for generically quadratic convergence. By such methods we
can calculate approximations up to type (80, 80) of on in
standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan,
and Carpenter required 200-digit extended precision.Comment: 29 pages, 11 figure
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