221 research outputs found
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
Factor Fitting, Rank Allocation, and Partitioning in Multilevel Low Rank Matrices
We consider multilevel low rank (MLR) matrices, defined as a row and column
permutation of a sum of matrices, each one a block diagonal refinement of the
previous one, with all blocks low rank given in factored form. MLR matrices
extend low rank matrices but share many of their properties, such as the total
storage required and complexity of matrix-vector multiplication. We address
three problems that arise in fitting a given matrix by an MLR matrix in the
Frobenius norm. The first problem is factor fitting, where we adjust the
factors of the MLR matrix. The second is rank allocation, where we choose the
ranks of the blocks in each level, subject to the total rank having a given
value, which preserves the total storage needed for the MLR matrix. The final
problem is to choose the hierarchical partition of rows and columns, along with
the ranks and factors. This paper is accompanied by an open source package that
implements the proposed methods
A fast direct solver for nonlocal operators in wavelet coordinates
In this article, we consider fast direct solvers for nonlocal operators. The
pivotal idea is to combine a wavelet representation of the system matrix,
yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The
latter drastically reduces the fill-in during the factorization of the system
matrix by means of a Cholesky decomposition or an LU decomposition,
respectively. This way, we end up with the exact inverse of the compressed
system matrix with only a moderate increase of the number of nonzero entries in
the matrix.
To illustrate the efficacy of the approach, we conduct numerical experiments
for different highly relevant applications of nonlocal operators: We consider
(i) the direct solution of boundary integral equations in three spatial
dimensions, issuing from the polarizable continuum model, (ii) a parabolic
problem for the fractional Laplacian in integral form and (iii) the fast
simulation of Gaussian random fields
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
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