122,963 research outputs found
A distributed-memory parallel algorithm for discretized integral equations using Julia
Boundary value problems involving elliptic PDEs such as the Laplace and the
Helmholtz equations are ubiquitous in physics and engineering. Many such
problems have alternative formulations as integral equations that are
mathematically more tractable than their PDE counterparts. However, the
integral equation formulation poses a challenge in solving the dense linear
systems that arise upon discretization. In cases where iterative methods
converge rapidly, existing methods that draw on fast summation schemes such as
the Fast Multipole Method are highly efficient and well established. More
recently, linear complexity direct solvers that sidestep convergence issues by
directly computing an invertible factorization have been developed. However,
storage and compute costs are high, which limits their ability to solve
large-scale problems in practice. In this work, we introduce a
distributed-memory parallel algorithm based on an existing direct solver named
``strong recursive skeletonization factorization.'' The analysis of its
parallel scalability applies generally to a class of existing methods that
exploit the so-called strong admissibility. Specifically, we apply low-rank
compression to certain off-diagonal matrix blocks in a way that minimizes data
movement. Given a compression tolerance, our method constructs an approximate
factorization of a discretized integral operator (dense matrix), which can be
used to solve linear systems efficiently in parallel. Compared to iterative
algorithms, our method is particularly suitable for problems involving
ill-conditioned matrices or multiple right-hand sides. Large-scale numerical
experiments are presented to demonstrate the performance of our implementation
using the Julia language
Multilevel quasiseparable matrices in PDE-constrained optimization
Optimization problems with constraints in the form of a partial differential
equation arise frequently in the process of engineering design. The
discretization of PDE-constrained optimization problems results in large-scale
linear systems of saddle-point type. In this paper we propose and develop a
novel approach to solving such systems by exploiting so-called quasiseparable
matrices. One may think of a usual quasiseparable matrix as of a discrete
analog of the Green's function of a one-dimensional differential operator. Nice
feature of such matrices is that almost every algorithm which employs them has
linear complexity. We extend the application of quasiseparable matrices to
problems in higher dimensions. Namely, we construct a class of preconditioners
which can be computed and applied at a linear computational cost. Their use
with appropriate Krylov methods leads to algorithms of nearly linear
complexity
NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions
This article describes the implementation in the software package NumGfun of
classical algorithms that operate on solutions of linear differential equations
or recurrence relations with polynomial coefficients, including what seems to
be the first general implementation of the fast high-precision numerical
evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our
descriptions contain improvements over existing algorithms. We also provide
references to relevant ideas not currently used in NumGfun
Hardness Results for Structured Linear Systems
We show that if the nearly-linear time solvers for Laplacian matrices and
their generalizations can be extended to solve just slightly larger families of
linear systems, then they can be used to quickly solve all systems of linear
equations over the reals. This result can be viewed either positively or
negatively: either we will develop nearly-linear time algorithms for solving
all systems of linear equations over the reals, or progress on the families we
can solve in nearly-linear time will soon halt
Sparse Solution of Underdetermined Linear Equations via Adaptively Iterative Thresholding
Finding the sparset solution of an underdetermined system of linear equations
has attracted considerable attention in recent years. Among a large
number of algorithms, iterative thresholding algorithms are recognized as one
of the most efficient and important classes of algorithms. This is mainly due
to their low computational complexities, especially for large scale
applications. The aim of this paper is to provide guarantees on the global
convergence of a wide class of iterative thresholding algorithms. Since the
thresholds of the considered algorithms are set adaptively at each iteration,
we call them adaptively iterative thresholding (AIT) algorithms. As the main
result, we show that as long as satisfies a certain coherence property, AIT
algorithms can find the correct support set within finite iterations, and then
converge to the original sparse solution exponentially fast once the correct
support set has been identified. Meanwhile, we also demonstrate that AIT
algorithms are robust to the algorithmic parameters. In addition, it should be
pointed out that most of the existing iterative thresholding algorithms such as
hard, soft, half and smoothly clipped absolute deviation (SCAD) algorithms are
included in the class of AIT algorithms studied in this paper.Comment: 33 pages, 1 figur
- …