3,021 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
Fast and robust curve skeletonization for real-world elongated objects
We consider the problem of extracting curve skeletons of three-dimensional,
elongated objects given a noisy surface, which has applications in agricultural
contexts such as extracting the branching structure of plants. We describe an
efficient and robust method based on breadth-first search that can determine
curve skeletons in these contexts. Our approach is capable of automatically
detecting junction points as well as spurious segments and loops. All of that
is accomplished with only one user-adjustable parameter. The run time of our
method ranges from hundreds of milliseconds to less than four seconds on large,
challenging datasets, which makes it appropriate for situations where real-time
decision making is needed. Experiments on synthetic models as well as on data
from real world objects, some of which were collected in challenging field
conditions, show that our approach compares favorably to classical thinning
algorithms as well as to recent contributions to the field.Comment: 47 pages; IEEE WACV 2018, main paper and supplementary materia
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
A fast semi-direct least squares algorithm for hierarchically block separable matrices
We present a fast algorithm for linear least squares problems governed by
hierarchically block separable (HBS) matrices. Such matrices are generally
dense but data-sparse and can describe many important operators including those
derived from asymptotically smooth radial kernels that are not too oscillatory.
The algorithm is based on a recursive skeletonization procedure that exposes
this sparsity and solves the dense least squares problem as a larger,
equality-constrained, sparse one. It relies on a sparse QR factorization
coupled with iterative weighted least squares methods. In essence, our scheme
consists of a direct component, comprised of matrix compression and
factorization, followed by an iterative component to enforce certain equality
constraints. At most two iterations are typically required for problems that
are not too ill-conditioned. For an HBS matrix with
having bounded off-diagonal block rank, the algorithm has optimal complexity. If the rank increases with the spatial dimension as is
common for operators that are singular at the origin, then this becomes
in 1D, in 2D, and
in 3D. We illustrate the performance of the method on
both over- and underdetermined systems in a variety of settings, with an
emphasis on radial basis function approximation and efficient updating and
downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App
SkelFMM: A Simplified Fast Multipole Method Based on Recursive Skeletonization
This work introduces the kernel-independent multi-level algorithm "skelFMM"
for evaluating all pairwise interactions between points connected through a
kernel such as the fundamental solution of the Laplace or the Helmholtz
equations. The method is based on linear algebraic tools such as randomized low
rank approximation and "skeleton representations" of far-field interactions.
The work is related to previously proposed linear algebraic reformulations of
the fast multipole method (FMM), but is distinguished by relying on simpler
data structures. In particular, skelFMM does not require an "interaction list",
as it relies instead on algebraically-modified kernel interactions between
near-neighbors at every level. Like other kernel independent algorithms, it
only requires evaluation of the kernel function, allowing the methodology to
easily be extended to a range of different kernels in 2D and 3D. The simplicity
of the algorithm makes it particularly amenable to parallel implementation on
heterogeneous hardware architectures.
The performance of the algorithm is demonstrated through numerical
experiments conducted on uniform and non-uniform point distributions in 2D and
3D, involving Laplace and (low frequency) Helmholtz kernels. The algorithm
relies on a precomputation stage that constructs a tailored representation for
a given geometry of points. Once the precomputation has completed, the
matrix-vector multiplication attains high speed through GPU acceleration that
leverages batched linear algebra
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