6,203 research outputs found
Constraint interface preconditioning for topology optimization problems
The discretization of constrained nonlinear optimization problems arising in
the field of topology optimization yields algebraic systems which are
challenging to solve in practice, due to pathological ill-conditioning, strong
nonlinearity and size. In this work we propose a methodology which brings
together existing fast algorithms, namely, interior-point for the optimization
problem and a novel substructuring domain decomposition method for the ensuing
large-scale linear systems. The main contribution is the choice of interface
preconditioner which allows for the acceleration of the domain decomposition
method, leading to performance independent of problem size.Comment: To be published in SIAM J. Sci. Com
Composing Scalable Nonlinear Algebraic Solvers
Most efficient linear solvers use composable algorithmic components, with the
most common model being the combination of a Krylov accelerator and one or more
preconditioners. A similar set of concepts may be used for nonlinear algebraic
systems, where nonlinear composition of different nonlinear solvers may
significantly improve the time to solution. We describe the basic concepts of
nonlinear composition and preconditioning and present a number of solvers
applicable to nonlinear partial differential equations. We have developed a
software framework in order to easily explore the possible combinations of
solvers. We show that the performance gains from using composed solvers can be
substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table
A multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential
equations (PDEs). The solver is based on pseudo-spectral collocation with
domain decomposition and can handle one- to three-dimensional problems. It has
three distinct features. First, the combined problem of solving the PDE,
satisfying the boundary conditions, and matching between different subdomains
is cast into one set of equations readily accessible to standard linear and
nonlinear solvers. Second, touching as well as overlapping subdomains are
supported; both rectangular blocks with Chebyshev basis functions as well as
spherical shells with an expansion in spherical harmonics are implemented.
Third, the code is very flexible: The domain decomposition as well as the
distribution of collocation points in each domain can be chosen at run time,
and the solver is easily adaptable to new PDEs. The code has been used to solve
the equations of the initial value problem of general relativity and should be
useful in many other problems. We compare the new method to finite difference
codes and find it superior in both runtime and accuracy, at least for the
smooth problems considered here.Comment: 31 pages, 8 figure
The M\"obius Domain Wall Fermion Algorithm
We present a review of the properties of generalized domain wall Fermions,
based on a (real) M\"obius transformation on the Wilson overlap kernel,
discussing their algorithmic efficiency, the degree of explicit chiral
violations measured by the residual mass () and the Ward-Takahashi
identities. The M\"obius class interpolates between Shamir's domain wall
operator and Bori\c{c}i's domain wall implementation of Neuberger's overlap
operator without increasing the number of Dirac applications per conjugate
gradient iteration. A new scaling parameter () reduces chiral
violations at finite fifth dimension () but yields exactly the same
overlap action in the limit . Through the use of 4d
Red/Black preconditioning and optimal tuning for the scaling , we
show that chiral symmetry violations are typically reduced by an order of
magnitude at fixed . At large we argue that the observed scaling for
for Shamir is replaced by for the
properly tuned M\"obius algorithm with Comment: 59 pages, 11 figure
Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains
We describe a multidomain spectral-tau method for solving the
three-dimensional helically reduced wave equation on the type of two-center
domain that arises when modeling compact binary objects in astrophysical
applications. A global two-center domain may arise as the union of Cartesian
blocks, cylindrical shells, and inner and outer spherical shells. For each such
subdomain, our key objective is to realize certain (differential and
multiplication) physical-space operators as matrices acting on the
corresponding set of modal coefficients. We achieve sparse banded realizations
through the integration "preconditioning" of Coutsias, Hagstrom, Hesthaven, and
Torres. Since ours is the first three-dimensional multidomain implementation of
the technique, we focus on the issue of convergence for the global solver, here
the alternating Schwarz method accelerated by GMRES. Our methods may prove
relevant for numerical solution of other mixed-type or elliptic problems, and
in particular for the generation of initial data in general relativity.Comment: 37 pages, 3 figures, 12 table
Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption
In this paper we give new results on domain decomposition preconditioners for
GMRES when computing piecewise-linear finite-element approximations of the
Helmholtz equation , with
absorption parameter . Multigrid approximations of
this equation with are commonly used as preconditioners
for the pure Helmholtz case (). However a rigorous theory for
such (so-called "shifted Laplace") preconditioners, either for the pure
Helmholtz equation, or even the absorptive equation (), is
still missing. We present a new theory for the absorptive equation that
provides rates of convergence for (left- or right-) preconditioned GMRES, via
estimates of the norm and field of values of the preconditioned matrix. This
theory uses a - and -explicit coercivity result for the
underlying sesquilinear form and shows, for example, that if , then classical overlapping additive Schwarz will perform optimally for
the absorptive problem, provided the subdomain and coarse mesh diameters are
carefully chosen. Extensive numerical experiments are given that support the
theoretical results. The theory for the absorptive case gives insight into how
its domain decomposition approximations perform as preconditioners for the pure
Helmholtz case . At the end of the paper we propose a
(scalable) multilevel preconditioner for the pure Helmholtz problem that has an
empirical computation time complexity of about for
solving finite element systems of size , where we have
chosen the mesh diameter to avoid the pollution effect.
Experiments on problems with , i.e. a fixed number of grid points
per wavelength, are also given
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