1,055 research outputs found
Transverse electric scattering on inhomogeneous objects: spectrum of integral operator and preconditioning
The domain integral equation method with its FFT-based matrix-vector products
is a viable alternative to local methods in free-space scattering problems.
However, it often suffers from the extremely slow convergence of iterative
methods, especially in the transverse electric (TE) case with large or negative
permittivity. We identify the nontrivial essential spectrum of the pertaining
integral operator as partly responsible for this behavior, and the main reason
why a normally efficient deflating preconditioner does not work. We solve this
problem by applying an explicit multiplicative regularizing operator, which
transforms the system to the form `identity plus compact', yet allows the
resulting matrix-vector products to be carried out at the FFT speed. Such a
regularized system is then further preconditioned by deflating an apparently
stable set of eigenvalues with largest magnitudes, which results in a robust
acceleration of the restarted GMRES under constraint memory conditions.Comment: 20 pages, 8 figure
A parallel nearly implicit time-stepping scheme
Across-the-space parallelism still remains the most mature, convenient and natural way to parallelize large scale problems. One of the major problems here is that implicit time stepping is often difficult to parallelize due to the structure of the system. Approximate implicit schemes have been suggested to circumvent the problem. These schemes have attractive stability properties and they are also very well parallelizable.\ud
The purpose of this article is to give an overall assessment of the parallelism of the method
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
Spatiotemporal dynamics in 2D Kolmogorov flow over large domains
Kolmogorov flow in two dimensions - the two-dimensional Navier-Stokes
equations with a sinusoidal body force - is considered over extended periodic
domains to reveal localised spatiotemporal complexity. The flow response
mimicks the forcing at small forcing amplitudes but beyond a critical value
develops a long wavelength instability. The ensuing state is described by a
Cahn-Hilliard-type equation and as a result coarsening dynamics are observed
for random initial data. After further bifurcations, this regime gives way to
multiple attractors, some of which possess spatially-localised time dependence.
Co-existence of such attractors in a large domain gives rise to interesting
collisional dynamics which is captured by a system of 5 (1-space and 1-time)
PDEs based on a long wavelength limit. The coarsening regime reinstates itself
at yet higher forcing amplitudes in the sense that only longest-wavelength
solutions remain attractors. Eventually, there is one global longest-wavelength
attractor which possesses two localised chaotic regions - a kink and antikink -
which connect two steady one-dimensional flow regions of essentially half the
domain width each. The wealth of spatiotemporal complexity uncovered presents a
bountiful arena in which to study the existence of simple invariant localised
solutions which presumably underpin all of the observed behaviour
Preconditioned fully implicit PDE solvers for monument conservation
Mathematical models for the description, in a quantitative way, of the
damages induced on the monuments by the action of specific pollutants are often
systems of nonlinear, possibly degenerate, parabolic equations. Although some
the asymptotic properties of the solutions are known, for a short window of
time, one needs a numerical approximation scheme in order to have a
quantitative forecast at any time of interest. In this paper a fully implicit
numerical method is proposed, analyzed and numerically tested for parabolic
equations of porous media type and on a systems of two PDEs that models the
sulfation of marble in monuments. Due to the nonlinear nature of the underlying
mathematical model, the use of a fixed point scheme is required and every step
implies the solution of large, locally structured, linear systems. A special
effort is devoted to the spectral analysis of the relevant matrices and to the
design of appropriate iterative or multi-iterative solvers, with special
attention to preconditioned Krylov methods and to multigrid procedures.
Numerical experiments for the validation of the analysis complement this
contribution.Comment: 26 pages, 13 figure
Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code
In this paper we present a rigorous derivation of the reduced MHD models with
and without parallel velocity that are implemented in the non-linear MHD code
JOREK. The model we obtain contains some terms that have been neglected in the
implementation but might be relevant in the non-linear phase. These are
necessary to guarantee exact conservation with respect to the full MHD energy.
For the second part of this work, we have replaced the linearized time stepping
of JOREK by a non-linear solver based on the Inexact Newton method including
adaptive time stepping. We demonstrate that this approach is more robust
especially with respect to numerical errors in the saturation phase of an
instability and allows to use larger time steps in the non-linear phase
- âŠ