3,461 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
Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media
This paper proposes to address the issue of complexity reduction for the
numerical simulation of multiscale media in a quasi-periodic setting. We
consider a stationary elliptic diffusion equation defined on a domain such
that is the union of cells and we
introduce a two-scale representation by identifying any function defined
on with a bi-variate function , where relates to the
index of the cell containing the point and relates to a local
coordinate in a reference cell . We introduce a weak formulation of the
problem in a broken Sobolev space using a discontinuous Galerkin
framework. The problem is then interpreted as a tensor-structured equation by
identifying with a tensor product space of
functions defined over the product set . Tensor numerical methods
are then used in order to exploit approximability properties of quasi-periodic
solutions by low-rank tensors.Comment: Changed the choice of test spaces V(D) and X (with regard to
regularity) and the argumentation thereof. Corrected proof of proposition 3.
Corrected wrong multiplicative factor in proposition 4 and its proof (was 2
instead of 1). Added remark 6 at the end of section 2. Extended remark 7.
Added references. Some minor improvements (typos, typesetting
Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient
This paper addresses a multi-scale finite element method for second order
linear elliptic equations with arbitrarily rough coefficient. We propose a
local oversampling method to construct basis functions that have optimal local
approximation property. Our methodology is based on the compactness of the
solution operator restricted on local regions of the spatial domain, and does
not depend on any scale-separation or periodicity assumption of the
coefficient. We focus on a special type of basis functions that are harmonic on
each element and have optimal approximation property. We first reduce our
problem to approximating the trace of the solution space on each edge of the
underlying mesh, and then achieve this goal through the singular value
decomposition of an oversampling operator. Rigorous error estimates can be
obtained through thresholding in constructing the basis functions. Numerical
results for several problems with multiple spatial scales and high contrast
inclusions are presented to demonstrate the compactness of the local solution
space and the capacity of our method in identifying and exploiting this compact
structure to achieve computational savings
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