896 research outputs found
Geometry-Oblivious FMM for Compressing Dense SPD Matrices
We present GOFMM (geometry-oblivious FMM), a novel method that creates a
hierarchical low-rank approximation, "compression," of an arbitrary dense
symmetric positive definite (SPD) matrix. For many applications, GOFMM enables
an approximate matrix-vector multiplication in or even time,
where is the matrix size. Compression requires storage and work.
In general, our scheme belongs to the family of hierarchical matrix
approximation methods. In particular, it generalizes the fast multipole method
(FMM) to a purely algebraic setting by only requiring the ability to sample
matrix entries. Neither geometric information (i.e., point coordinates) nor
knowledge of how the matrix entries have been generated is required, thus the
term "geometry-oblivious." Also, we introduce a shared-memory parallel scheme
for hierarchical matrix computations that reduces synchronization barriers. We
present results on the Intel Knights Landing and Haswell architectures, and on
the NVIDIA Pascal architecture for a variety of matrices.Comment: 13 pages, accepted by SC'1
A distributed-memory package for dense Hierarchically Semi-Separable matrix computations using randomization
We present a distributed-memory library for computations with dense
structured matrices. A matrix is considered structured if its off-diagonal
blocks can be approximated by a rank-deficient matrix with low numerical rank.
Here, we use Hierarchically Semi-Separable representations (HSS). Such matrices
appear in many applications, e.g., finite element methods, boundary element
methods, etc. Exploiting this structure allows for fast solution of linear
systems and/or fast computation of matrix-vector products, which are the two
main building blocks of matrix computations. The compression algorithm that we
use, that computes the HSS form of an input dense matrix, relies on randomized
sampling with a novel adaptive sampling mechanism. We discuss the
parallelization of this algorithm and also present the parallelization of
structured matrix-vector product, structured factorization and solution
routines. The efficiency of the approach is demonstrated on large problems from
different academic and industrial applications, on up to 8,000 cores.
This work is part of a more global effort, the STRUMPACK (STRUctured Matrices
PACKage) software package for computations with sparse and dense structured
matrices. Hence, although useful on their own right, the routines also
represent a step in the direction of a distributed-memory sparse solver
Randomized Strong Recursive Skeletonization: Simultaneous compression and factorization of -matrices in the Black-Box Setting
The hierarchical matrix (-matrix) formalism provides a way
to reinterpret the Fast Multipole Method and related fast summation schemes in
linear algebraic terms. The idea is to tessellate a matrix into blocks in such
as way that each block is either small or of numerically low rank; this enables
the storage of the matrix and the application of it to a vector in linear or
close to linear complexity. A key motivation for the reformulation is to extend
the range of dense matrices that can be represented. Additionally,
-matrices in principle also extend the range of operations
that can be executed to include matrix inversion and factorization. While such
algorithms can be highly efficient for certain specialized formats (such as
HBS/HSS matrices based on ``weak admissibility''), inversion algorithms for
general -matrices tend to be based on nested recursions and
recompressions, making them challenging to implement efficiently. An exception
is the \textit{strong recursive skeletonization (SRS)} algorithm by Minden, Ho,
Damle, and Ying, which involves a simpler algorithmic flow. However, SRS
greatly increases the number of blocks of the matrix that need to be stored
explicitly, leading to high memory requirements. This manuscript presents the
\textit{randomized strong recursive skeletonization (RSRS)} algorithm, which is
a reformulation of SRS that incorporates the randomized SVD (RSVD) to
simultaneously compress and factorize an -matrix. RSRS is a
``black box'' algorithm that interacts with the matrix to be compressed only
via its action on vectors; this extends the range of the SRS algorithm (which
relied on the ``proxy source'' compression technique) to include dense matrices
that arise in sparse direct solvers
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
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
Improved analysis of the subsampled randomized Hadamard transform
This paper presents an improved analysis of a structured dimension-reduction
map called the subsampled randomized Hadamard transform. This argument
demonstrates that the map preserves the Euclidean geometry of an entire
subspace of vectors. The new proof is much simpler than previous approaches,
and it offers---for the first time---optimal constants in the estimate on the
number of dimensions required for the embedding.Comment: 8 pages. To appear, Advances in Adaptive Data Analysis, special issue
"Sparse Representation of Data and Images." v2--v4 include minor correction
- …