83,212 research outputs found
Gradient based iterative solutions for general linear matrix equations
AbstractIn this paper, we present a gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle. Convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices. Two numerical examples are provided to show that the proposed algorithm is effective
Quantum annealing for systems of polynomial equations
Numerous scientific and engineering applications require numerically solving
systems of equations. Classically solving a general set of polynomial equations
requires iterative solvers, while linear equations may be solved either by
direct matrix inversion or iteratively with judicious preconditioning. However,
the convergence of iterative algorithms is highly variable and depends, in
part, on the condition number. We present a direct method for solving general
systems of polynomial equations based on quantum annealing, and we validate
this method using a system of second-order polynomial equations solved on a
commercially available quantum annealer. We then demonstrate applications for
linear regression, and discuss in more detail the scaling behavior for general
systems of linear equations with respect to problem size, condition number, and
search precision. Finally, we define an iterative annealing process and
demonstrate its efficacy in solving a linear system to a tolerance of
.Comment: 11 pages, 4 figures. Added example for a system of quadratic
equations. Supporting code is available at
https://github.com/cchang5/quantum_poly_solver . This is a post-peer-review,
pre-copyedit version of an article published in Scientific Reports. The final
authenticated version is available online at:
https://www.nature.com/articles/s41598-019-46729-
A conjugate gradient algorithm for the astrometric core solution of Gaia
The ESA space astrometry mission Gaia, planned to be launched in 2013, has
been designed to make angular measurements on a global scale with
micro-arcsecond accuracy. A key component of the data processing for Gaia is
the astrometric core solution, which must implement an efficient and accurate
numerical algorithm to solve the resulting, extremely large least-squares
problem. The Astrometric Global Iterative Solution (AGIS) is a framework that
allows to implement a range of different iterative solution schemes suitable
for a scanning astrometric satellite. In order to find a computationally
efficient and numerically accurate iteration scheme for the astrometric
solution, compatible with the AGIS framework, we study an adaptation of the
classical conjugate gradient (CG) algorithm, and compare it to the so-called
simple iteration (SI) scheme that was previously known to converge for this
problem, although very slowly. The different schemes are implemented within a
software test bed for AGIS known as AGISLab, which allows to define, simulate
and study scaled astrometric core solutions. After successful testing in
AGISLab, the CG scheme has been implemented also in AGIS. The two algorithms CG
and SI eventually converge to identical solutions, to within the numerical
noise (of the order of 0.00001 micro-arcsec). These solutions are independent
of the starting values (initial star catalogue), and we conclude that they are
equivalent to a rigorous least-squares estimation of the astrometric
parameters. The CG scheme converges up to a factor four faster than SI in the
tested cases, and in particular spatially correlated truncation errors are much
more efficiently damped out with the CG scheme.Comment: 24 pages, 16 figures. Accepted for publication in Astronomy &
Astrophysic
Greedy low-rank algorithm for spatial connectome regression
Recovering brain connectivity from tract tracing data is an important
computational problem in the neurosciences. Mesoscopic connectome
reconstruction was previously formulated as a structured matrix regression
problem (Harris et al., 2016), but existing techniques do not scale to the
whole-brain setting. The corresponding matrix equation is challenging to solve
due to large scale, ill-conditioning, and a general form that lacks a
convergent splitting. We propose a greedy low-rank algorithm for connectome
reconstruction problem in very high dimensions. The algorithm approximates the
solution by a sequence of rank-one updates which exploit the sparse and
positive definite problem structure. This algorithm was described previously
(Kressner and Sirkovi\'c, 2015) but never implemented for this connectome
problem, leading to a number of challenges. We have had to design judicious
stopping criteria and employ efficient solvers for the three main sub-problems
of the algorithm, including an efficient GPU implementation that alleviates the
main bottleneck for large datasets. The performance of the method is evaluated
on three examples: an artificial "toy" dataset and two whole-cortex instances
using data from the Allen Mouse Brain Connectivity Atlas. We find that the
method is significantly faster than previous methods and that moderate ranks
offer good approximation. This speedup allows for the estimation of
increasingly large-scale connectomes across taxa as these data become available
from tracing experiments. The data and code are available online
On Algorithms Based on Joint Estimation of Currents and Contrast in Microwave Tomography
This paper deals with improvements to the contrast source inversion method
which is widely used in microwave tomography. First, the method is reviewed and
weaknesses of both the criterion form and the optimization strategy are
underlined. Then, two new algorithms are proposed. Both of them are based on
the same criterion, similar but more robust than the one used in contrast
source inversion. The first technique keeps the main characteristics of the
contrast source inversion optimization scheme but is based on a better
exploitation of the conjugate gradient algorithm. The second technique is based
on a preconditioned conjugate gradient algorithm and performs simultaneous
updates of sets of unknowns that are normally processed sequentially. Both
techniques are shown to be more efficient than original contrast source
inversion.Comment: 12 pages, 12 figures, 5 table
Fast iterative solvers for convection-diffusion control problems
In this manuscript, we describe effective solvers for the optimal control of stabilized convection-diffusion problems. We employ the local projection stabilization, which we show to give the same matrix system whether the discretize-then-optimize or optimize-then-discretize approach for this problem is used. We then derive two effective preconditioners for this problem, the �first to be used with MINRES and the second to be used with the Bramble-Pasciak Conjugate Gradient method. The key components of both preconditioners are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to enact this latter approximation. We present numerical results to demonstrate that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the mesh size h, and the regularization parameter β, for a range of problems
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