arXiv subjects

Andrew Knyazev

Publications and source records attributed to Andrew Knyazev.

24 records · Page 2Linked to original sources

arXiv2015

Conjugate Gradient Acceleration of Non-Linear Smoothing Filters

The most efficient signal edge-preserving smoothing filters, e.g., for denoising, are non-linear. Thus, their acceleration is challenging and is often performed in practice by tuning filter parameters, such as by increasing the width of the local smoothing neighborhood, resulting in more aggressive smoothing of a single sweep at the cost of increased edge blurring. We propose an alternative technology, accelerating the original filters without tuning, by running them through a special conjugate gradient method, not affecting their quality. The filter non-linearity is dealt with by careful freezing and restarting. Our initial numerical experiments on toy one-dimensional signals demonstrate 20x acceleration of the classical bilateral filter and 3-5x acceleration of the recently developed guided filter.

cs.CV↗

arXiv2015

Chebyshev and Conjugate Gradient Filters for Graph Image Denoising

In 3D image/video acquisition, different views are often captured with varying noise levels across the views. In this paper, we propose a graph-based image enhancement technique that uses a higher quality view to enhance a degraded view. A depth map is utilized as auxiliary information to match the perspectives of the two views. Our method performs graph-based filtering of the noisy image by directly computing a projection of the image to be filtered onto a lower dimensional Krylov subspace of the graph Laplacian. We discuss two graph spectral denoising methods: first using Chebyshev polynomials, and second using iterations of the conjugate gradient algorithm. Our framework generalizes previously known polynomial graph filters, and we demonstrate through numerical simulations that our proposed technique produces subjectively cleaner images with about 1-3 dB improvement in PSNR over existing polynomial graph filters.

cs.CV↗

arXiv2015

Preconditioned Continuation Model Predictive Control

Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. A Continuation/GMRES Method for NMPC, suggested by T. Ohtsuka in 2004, uses the GMRES iterative algorithm to solve a forward difference approximation $Ax=b$ of the Continuation NMPC (CNMPC) equations on every time step. The coefficient matrix $A$ of the linear system is often ill-conditioned, resulting in poor GMRES convergence, slowing down the on-line computation of the control by CNMPC, and reducing control quality. We adopt CNMPC for challenging minimum-time problems, and improve performance by introducing efficient preconditioning, utilizing parallel computing, and substituting MINRES for GMRES.

math.OC↗

arXiv2014

Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices

We propose a Preconditioned Locally Harmonic Residual (PLHR) method for computing several interior eigenpairs of a generalized Hermitian eigenvalue problem, without traditional spectral transformations, matrix factorizations, or inversions. PLHR is based on a short-term recurrence, easily extended to a block form, computing eigenpairs simultaneously. PLHR can take advantage of Hermitian positive definite preconditioning, e.g., based on an approximate inverse of an absolute value of a shifted matrix, introduced in [SISC, 35 (2013), pp. A696-A718]. Our numerical experiments demonstrate that PLHR is efficient and robust for certain classes of large-scale interior eigenvalue problems, involving Laplacian and Hamiltonian operators, especially if memory requirements are tight.

math.NA↗

arXiv2010

Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods

We define angles from-to and between infinite dimensional subspaces of a Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general canonical correlations of stochastic processes. The spectral theory of selfadjoint operators is used to investigate the properties of the angles, e.g., to establish connections between the angles corresponding to orthogonal complements. The classical gaps and angles of Dixmier and Friedrichs are characterized in terms of the angles. We introduce principal invariant subspaces and prove that they are connected by an isometry that appears in the polar decomposition of the product of corresponding orthogonal projectors. Point angles are defined by analogy with the point operator spectrum. We bound the Hausdorff distance between the sets of the squared cosines of the angles corresponding to the original subspaces and their perturbations. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh-Ritz method, where the former subspace serves as a trial subspace and the orthogonal projector of the latter subspace serves as an operator in the Rayleigh-Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the trial subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. Finally, we consider the classical alternating projectors method and propose its ultimate acceleration, using the conjugate gradient approach. The corresponding convergence rate estimate is obtained in terms of the angles. We illustrate a possible acceleration for the domain decomposition method with a small overlap for the 1D diffusion equation.

math.NA↗

arXiv2007

Large scale ab initio calculations based on three levels of parallelization

We suggest and implement a parallelization scheme based on an efficient multiband eigenvalue solver, called the locally optimal block preconditioned conjugate gradient LOBPCG method, and using an optimized three-dimensional (3D) fast Fourier transform (FFT) in the ab initio}plane-wave code ABINIT. In addition to the standard data partitioning over processors corresponding to different k-points, we introduce data partitioning with respect to blocks of bands as well as spatial partitioning in the Fourier space of coefficients over the plane waves basis set used in ABINIT. This k-points-multiband-FFT parallelization avoids any collective communications on the whole set of processors relying instead on one-dimensional communications only. For a single k-point, super-linear scaling is achieved for up to 100 processors due to an extensive use of hardware optimized BLAS, LAPACK, and SCALAPACK routines, mainly in the LOBPCG routine. We observe good performance up to 200 processors. With 10 k-points our three-way data partitioning results in linear scaling up to 1000 processors for a practical system used for testing.

cond-mat.mtrl-sci↗