1,397 research outputs found
Low-rank updates and a divide-and-conquer method for linear matrix equations
Linear matrix equations, such as the Sylvester and Lyapunov equations, play
an important role in various applications, including the stability analysis and
dimensionality reduction of linear dynamical control systems and the solution
of partial differential equations. In this work, we present and analyze a new
algorithm, based on tensorized Krylov subspaces, for quickly updating the
solution of such a matrix equation when its coefficients undergo low-rank
changes. We demonstrate how our algorithm can be utilized to accelerate the
Newton method for solving continuous-time algebraic Riccati equations. Our
algorithm also forms the basis of a new divide-and-conquer approach for linear
matrix equations with coefficients that feature hierarchical low-rank
structure, such as HODLR, HSS, and banded matrices. Numerical experiments
demonstrate the advantages of divide-and-conquer over existing approaches, in
terms of computational time and memory consumption
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
Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio
Many problems in fluid modelling require the efficient solution of highly
anisotropic elliptic partial differential equations (PDEs) in "flat" domains.
For example, in numerical weather- and climate-prediction an elliptic PDE for
the pressure correction has to be solved at every time step in a thin spherical
shell representing the global atmosphere. This elliptic solve can be one of the
computationally most demanding components in semi-implicit semi-Lagrangian time
stepping methods which are very popular as they allow for larger model time
steps and better overall performance. With increasing model resolution,
algorithmically efficient and scalable algorithms are essential to run the code
under tight operational time constraints. We discuss the theory and practical
application of bespoke geometric multigrid preconditioners for equations of
this type. The algorithms deal with the strong anisotropy in the vertical
direction by using the tensor-product approach originally analysed by B\"{o}rm
and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the
analysis to three dimensions under slightly weakened assumptions, and
numerically demonstrate its efficiency for the solution of the elliptic PDE for
the global pressure correction in atmospheric forecast models. For this we
compare the performance of different multigrid preconditioners on a
tensor-product grid with a semi-structured and quasi-uniform horizontal mesh
and a one dimensional vertical grid. The code is implemented in the Distributed
and Unified Numerics Environment (DUNE), which provides an easy-to-use and
scalable environment for algorithms operating on tensor-product grids. Parallel
scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR
supercomputer.Comment: 22 pages, 6 Figures, 2 Table
Row Compression and Nested Product Decomposition of a Hierarchical Representation of a Quasiseparable Matrix
This research introduces a row compression and nested product decomposition of an nxn hierarchical representation of a rank structured matrix A, which extends the compression and nested product decomposition of a quasiseparable matrix. The hierarchical parameter extraction algorithm of a quasiseparable matrix is efficient, requiring only O(nlog(n))operations, and is proven backward stable. The row compression is comprised of a sequence of small Householder transformations that are formed from the low-rank, lower triangular, off-diagonal blocks of the hierarchical representation. The row compression forms a factorization of matrix A, where A = QC, Q is the product of the Householder transformations, and C preserves the low-rank structure in both the lower and upper triangular parts of matrix A. The nested product decomposition is accomplished by applying a sequence of orthogonal transformations to the low-rank, upper triangular, off-diagonal blocks of the compressed matrix C. Both the compression and decomposition algorithms are stable, and require O(nlog(n)) operations. At this point, the matrix-vector product and solver algorithms are the only ones fully proven to be backward stable for quasiseparable matrices. By combining the fast matrix-vector product and system solver, linear systems involving the hierarchical representation to nested product decomposition are directly solved with linear complexity and unconditional stability. Applications in image deblurring and compression, that capitalize on the concepts from the row compression and nested product decomposition algorithms, will be shown
An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling
We present a sparse linear system solver that is based on a multifrontal
variant of Gaussian elimination, and exploits low-rank approximation of the
resulting dense frontal matrices. We use hierarchically semiseparable (HSS)
matrices, which have low-rank off-diagonal blocks, to approximate the frontal
matrices. For HSS matrix construction, a randomized sampling algorithm is used
together with interpolative decompositions. The combination of the randomized
compression with a fast ULV HSS factorization leads to a solver with lower
computational complexity than the standard multifrontal method for many
applications, resulting in speedups up to 7 fold for problems in our test
suite. The implementation targets many-core systems by using task parallelism
with dynamic runtime scheduling. Numerical experiments show performance
improvements over state-of-the-art sparse direct solvers. The implementation
achieves high performance and good scalability on a range of modern shared
memory parallel systems, including the Intel Xeon Phi (MIC). The code is part
of a software package called STRUMPACK -- STRUctured Matrices PACKage, which
also has a distributed memory component for dense rank-structured matrices
Convolutional Dictionary Learning: Acceleration and Convergence
Convolutional dictionary learning (CDL or sparsifying CDL) has many
applications in image processing and computer vision. There has been growing
interest in developing efficient algorithms for CDL, mostly relying on the
augmented Lagrangian (AL) method or the variant alternating direction method of
multipliers (ADMM). When their parameters are properly tuned, AL methods have
shown fast convergence in CDL. However, the parameter tuning process is not
trivial due to its data dependence and, in practice, the convergence of AL
methods depends on the AL parameters for nonconvex CDL problems. To moderate
these problems, this paper proposes a new practically feasible and convergent
Block Proximal Gradient method using a Majorizer (BPG-M) for CDL. The
BPG-M-based CDL is investigated with different block updating schemes and
majorization matrix designs, and further accelerated by incorporating some
momentum coefficient formulas and restarting techniques. All of the methods
investigated incorporate a boundary artifacts removal (or, more generally,
sampling) operator in the learning model. Numerical experiments show that,
without needing any parameter tuning process, the proposed BPG-M approach
converges more stably to desirable solutions of lower objective values than the
existing state-of-the-art ADMM algorithm and its memory-efficient variant do.
Compared to the ADMM approaches, the BPG-M method using a multi-block updating
scheme is particularly useful in single-threaded CDL algorithm handling large
datasets, due to its lower memory requirement and no polynomial computational
complexity. Image denoising experiments show that, for relatively strong
additive white Gaussian noise, the filters learned by BPG-M-based CDL
outperform those trained by the ADMM approach.Comment: 21 pages, 7 figures, submitted to IEEE Transactions on Image
Processin
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