15,185 research outputs found
Interior-point solver for convex separable block-angular problems
Constraints matrices with block-angular structures are pervasive in Optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solved the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based on a power series expansion which approximates the inverse of the matrix of the linking constraints system. In this work we present an efficient solver based on this algorithm. Some of its features are: it solves linearly constrained convex separable problems (linear, quadratic or nonlinear); both Newton and second-order predictor-corrector directions can be used, either with the Cholesky+PCG scheme or with a Cholesky factorization of normal equations; the preconditioner
may include any number of terms of the power series; for any number of these terms, it estimates the spectral radius of the matrix in the power series (which is instrumental for the quality of the precondi-
tioner). The solver has been hooked to SML, a structure-conveying modelling language based on the popular AMPL modeling language. Computational results are reported for some large and/or difficult instances in the literature: (1) multicommodity flow problems; (2) minimum congestion problems; (3) statistical data protection problems using l1 and l2 distances (which are linear and quadratic problems, respectively), and the pseudo-Huber function, a nonlinear approximation to l1 which improves the preconditioner. In the largest instances, of up to 25 millions of variables and 300000 constraints, this approach is from two to three orders of magnitude faster than state-of-the-art linear and quadratic optimization solvers.Preprin
An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations
An a posteriori verification method is proposed for the generalized
real-symmetric eigenvalue problem and is applied to densely clustered
eigenvalue problems in large-scale electronic state calculations. The proposed
method is realized by a two-stage process in which the approximate solution is
computed by existing numerical libraries and is then verified in a moderate
computational time. The procedure returns intervals containing one exact
eigenvalue in each interval. Test calculations were carried out for organic
device materials, and the verification method confirms that all exact
eigenvalues are well separated in the obtained intervals. This verification
method will be integrated into EigenKernel (https://github.com/eigenkernel/),
which is middleware for various parallel solvers for the generalized eigenvalue
problem. Such an a posteriori verification method will be important in future
computational science.Comment: 15 pages, 7 figure
Cortical spatio-temporal dimensionality reduction for visual grouping
The visual systems of many mammals, including humans, is able to integrate
the geometric information of visual stimuli and to perform cognitive tasks
already at the first stages of the cortical processing. This is thought to be
the result of a combination of mechanisms, which include feature extraction at
single cell level and geometric processing by means of cells connectivity. We
present a geometric model of such connectivities in the space of detected
features associated to spatio-temporal visual stimuli, and show how they can be
used to obtain low-level object segmentation. The main idea is that of defining
a spectral clustering procedure with anisotropic affinities over datasets
consisting of embeddings of the visual stimuli into higher dimensional spaces.
Neural plausibility of the proposed arguments will be discussed
GHOST: Building blocks for high performance sparse linear algebra on heterogeneous systems
While many of the architectural details of future exascale-class high
performance computer systems are still a matter of intense research, there
appears to be a general consensus that they will be strongly heterogeneous,
featuring "standard" as well as "accelerated" resources. Today, such resources
are available as multicore processors, graphics processing units (GPUs), and
other accelerators such as the Intel Xeon Phi. Any software infrastructure that
claims usefulness for such environments must be able to meet their inherent
challenges: massive multi-level parallelism, topology, asynchronicity, and
abstraction. The "General, Hybrid, and Optimized Sparse Toolkit" (GHOST) is a
collection of building blocks that targets algorithms dealing with sparse
matrix representations on current and future large-scale systems. It implements
the "MPI+X" paradigm, has a pure C interface, and provides hybrid-parallel
numerical kernels, intelligent resource management, and truly heterogeneous
parallelism for multicore CPUs, Nvidia GPUs, and the Intel Xeon Phi. We
describe the details of its design with respect to the challenges posed by
modern heterogeneous supercomputers and recent algorithmic developments.
Implementation details which are indispensable for achieving high efficiency
are pointed out and their necessity is justified by performance measurements or
predictions based on performance models. The library code and several
applications are available as open source. We also provide instructions on how
to make use of GHOST in existing software packages, together with a case study
which demonstrates the applicability and performance of GHOST as a component
within a larger software stack.Comment: 32 pages, 11 figure
Practical Inexact Proximal Quasi-Newton Method with Global Complexity Analysis
Recently several methods were proposed for sparse optimization which make
careful use of second-order information [10, 28, 16, 3] to improve local
convergence rates. These methods construct a composite quadratic approximation
using Hessian information, optimize this approximation using a first-order
method, such as coordinate descent and employ a line search to ensure
sufficient descent. Here we propose a general framework, which includes
slightly modified versions of existing algorithms and also a new algorithm,
which uses limited memory BFGS Hessian approximations, and provide a novel
global convergence rate analysis, which covers methods that solve subproblems
via coordinate descent
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