8,842 research outputs found
Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients
We present a robust and scalable preconditioner for the solution of
large-scale linear systems that arise from the discretization of elliptic PDEs
amenable to rank compression. The preconditioner is based on hierarchical
low-rank approximations and the cyclic reduction method. The setup and
application phases of the preconditioner achieve log-linear complexity in
memory footprint and number of operations, and numerical experiments exhibit
good weak and strong scalability at large processor counts in a distributed
memory environment. Numerical experiments with linear systems that feature
symmetry and nonsymmetry, definiteness and indefiniteness, constant and
variable coefficients demonstrate the preconditioner applicability and
robustness. Furthermore, it is possible to control the number of iterations via
the accuracy threshold of the hierarchical matrix approximations and their
arithmetic operations, and the tuning of the admissibility condition parameter.
Together, these parameters allow for optimization of the memory requirements
and performance of the preconditioner.Comment: 24 pages, Elsevier Journal of Computational and Applied Mathematics,
Dec 201
Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods
Many problems in computational science and engineering are simultaneously
characterized by the following challenging issues: uncertainty, nonlinearity,
nonstationarity and high dimensionality. Existing numerical techniques for such
models would typically require considerable computational and storage
resources. This is the case, for instance, for an optimization problem governed
by time-dependent Navier-Stokes equations with uncertain inputs. In particular,
the stochastic Galerkin finite element method often leads to a prohibitively
high dimensional saddle-point system with tensor product structure. In this
paper, we approximate the solution by the low-rank Tensor Train decomposition,
and present a numerically efficient algorithm to solve the optimality equations
directly in the low-rank representation. We show that the solution of the
vorticity minimization problem with a distributed control admits a
representation with ranks that depend modestly on model and discretization
parameters even for high Reynolds numbers. For lower Reynolds numbers this is
also the case for a boundary control. This opens the way for a reduced-order
modeling of the stochastic optimal flow control with a moderate cost at all
stages.Comment: 29 page
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
This work derives a residual-based a posteriori error estimator for reduced
models learned with non-intrusive model reduction from data of high-dimensional
systems governed by linear parabolic partial differential equations with
control inputs. It is shown that quantities that are necessary for the error
estimator can be either obtained exactly as the solutions of least-squares
problems in a non-intrusive way from data such as initial conditions, control
inputs, and high-dimensional solution trajectories or bounded in a
probabilistic sense. The computational procedure follows an offline/online
decomposition. In the offline (training) phase, the high-dimensional system is
judiciously solved in a black-box fashion to generate data and to set up the
error estimator. In the online phase, the estimator is used to bound the error
of the reduced-model predictions for new initial conditions and new control
inputs without recourse to the high-dimensional system. Numerical results
demonstrate the workflow of the proposed approach from data to reduced models
to certified predictions
Space-time domain decomposition for advection-diffusion problems in mixed formulations
This paper is concerned with the numerical solution of porous-media flow and
transport problems , i. e. heterogeneous, advection-diffusion problems. Its aim
is to investigate numerical schemes for these problems in which different time
steps can be used in different parts of the domain. Global-in-time,
non-overlapping domain-decomposition methods are coupled with operator
splitting making possible the different treatment of the advection and
diffusion terms. Two domain-decomposition methods are considered: one uses the
time-dependent Steklov--Poincar{\'e} operator and the other uses optimized
Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For
each method, a mixed formulation of an interface problem on the space-time
interface is derived, and different time grids are employed to adapt to
different time scales in the subdomains. A generalized Neumann-Neumann
preconditioner is proposed for the first method. To illustrate the two methods
numerical results for two-dimensional problems with strong heterogeneities are
presented. These include both academic problems and more realistic prototypes
for simulations for the underground storage of nuclear waste
Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh--B\'enard convection
By utilizing diffusion maps embedding and transition matrix analysis we
investigate sparse temperature measurement time-series data from
Rayleigh--B\'enard convection experiments in a cylindrical container of aspect
ratio between its diameter () and height (). We consider
the two cases of a cylinder at rest and rotating around its cylinder axis. We
find that the relative amplitude of the large-scale circulation (LSC) and its
orientation inside the container at different points in time are associated to
prominent geometric features in the embedding space spanned by the two dominant
diffusion-maps eigenvectors. From this two-dimensional embedding we can measure
azimuthal drift and diffusion rates, as well as coherence times of the LSC. In
addition, we can distinguish from the data clearly the single roll state (SRS),
when a single roll extends through the whole cell, from the double roll state
(DRS), when two counter-rotating rolls are on top of each other. Based on this
embedding we also build a transition matrix (a discrete transfer operator),
whose eigenvectors and eigenvalues reveal typical time scales for the stability
of the SRS and DRS as well as for the azimuthal drift velocity of the flow
structures inside the cylinder. Thus, the combination of nonlinear dimension
reduction and dynamical systems tools enables to gain insight into turbulent
flows without relying on model assumptions
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