3,117 research outputs found
Hierarchical interpolative factorization for elliptic operators: differential equations
This paper introduces the hierarchical interpolative factorization for
elliptic partial differential equations (HIF-DE) in two (2D) and three
dimensions (3D). This factorization takes the form of an approximate
generalized LU/LDL decomposition that facilitates the efficient inversion of
the discretized operator. HIF-DE is based on the multifrontal method but uses
skeletonization on the separator fronts to sparsify the dense frontal matrices
and thus reduce the cost. We conjecture that this strategy yields linear
complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity
in 3D can be achieved by skeletonizing the compressed fronts themselves, which
amounts geometrically to a recursive dimensional reduction scheme. Numerical
experiments support our claims and further demonstrate the performance of our
algorithm as a fast direct solver and preconditioner. MATLAB codes are freely
available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math.
arXiv admin note: substantial text overlap with arXiv:1307.266
Hierarchical interpolative factorization for elliptic operators: integral equations
This paper introduces the hierarchical interpolative factorization for
integral equations (HIF-IE) associated with elliptic problems in two and three
dimensions. This factorization takes the form of an approximate generalized LU
decomposition that permits the efficient application of the discretized
operator and its inverse. HIF-IE is based on the recursive skeletonization
algorithm but incorporates a novel combination of two key features: (1) a
matrix factorization framework for sparsifying structured dense matrices and
(2) a recursive dimensional reduction strategy to decrease the cost. Thus,
higher-dimensional problems are effectively mapped to one dimension, and we
conjecture that constructing, applying, and inverting the factorization all
have linear or quasilinear complexity. Numerical experiments support this claim
and further demonstrate the performance of our algorithm as a generalized fast
multipole method, direct solver, and preconditioner. HIF-IE is compatible with
geometric adaptivity and can handle both boundary and volume problems. MATLAB
codes are freely available.Comment: 39 pages, 14 figures, 13 tables; to appear, Comm. Pure Appl. Mat
Butterfly Factorization
The paper introduces the butterfly factorization as a data-sparse
approximation for the matrices that satisfy a complementary low-rank property.
The factorization can be constructed efficiently if either fast algorithms for
applying the matrix and its adjoint are available or the entries of the matrix
can be sampled individually. For an matrix, the resulting
factorization is a product of sparse matrices, each with
non-zero entries. Hence, it can be applied rapidly in operations.
Numerical results are provided to demonstrate the effectiveness of the
butterfly factorization and its construction algorithms
Testing the consequences of specifications in modal µ
Journal ArticleIn a companion paper in these proceedings [6], we introduced the CCS notation and explained how to write specifications succinctly in CCS using the composition operator. In this paper we explain how one may associate a process logic with CCS and use it to resolve deadlock, safety, liveness, and fairness properties of specifications by static testing
Slip history of the 2003 San Simeon earthquake constrained by combining 1-Hz GPS, strong motion, and teleseismic data
The slip history of the 2003 San Simeon earthquake is constrained by combining strong motion and teleseismic data, along with GPS static offsets and 1-Hz GPS observations. Comparisons of a 1-Hz GPS time series and a co-located strong motion data are in very good agreement, demonstrating a new application of GPS. The inversion results for this event indicate that the rupture initiated at a depth of 8.5 km and propagated southeastwards with a speed ~3.0 km/sec, with rake vectors forming a fan structure around the hypocenter. We obtained a peak slip of 2.8 m and total seismic moment of 6.2 × 10^(18) Nm. We interpret the slip distribution as indicating that the hanging wall rotates relative to the footwall around the hypocenter, in a sense that appears consistent with the shape of the mapped fault trace
When chiral photons meet chiral fermions - Photoinduced anomalous Hall effects in Weyl semimetals
The Weyl semimetal is characterized by three-dimensional linear band touching
points called Weyl nodes. These nodes come in pairs with opposite chiralities.
We show that the coupling of circularly polarized photons with these chiral
electrons generates a Hall conductivity without any applied magnetic field in
the plane orthogonal to the light propagation. This phenomenon comes about
because with all three Pauli matrices exhausted to form the three-dimensional
linear dispersion, the Weyl nodes cannot be gapped. Rather, the net influence
of chiral photons is to shift the positions of the Weyl nodes. Interestingly,
the momentum shift is tightly correlated with the chirality of the node to
produce a net anomalous Hall signal. Application of our proposal to the
recently discovered TaAs family of Weyl semimetals leads to an
order-of-magnitude estimate of the photoinduced Hall conductivity which is
within the experimentally accessible range.Comment: 9 pages, 4 figure
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