[Show abstract][Hide abstract] ABSTRACT: In this paper we motivate, discuss the implementation and present the
resulting numerics for a new definition of strength of connection which is
based on the notion of algebraic distance. This algebraic distance measure,
combined with compatible relaxation, is used to choose suitable coarse grids
and accurate interpolation operators for algebraic multigrid algorithms. The
main tool of the proposed measure is the least squares functional defined using
a set of relaxed test vectors. The motivating application is the anisotropic
diffusion problem, in particular problems with non-grid aligned anisotropy. We
demonstrate numerically that the measure yields a robust technique for
determining strength of connectivity among variables, for both two-grid and
multigrid solvers. %We illustrate the use of the measure to construct, in
addition, an adaptive aggregation form of interpolation for the targeted
anisotropic problems. %Our approach is not a two-level approach -- we provide
preliminary results that show its extendability to multigrid. The proposed
algebraic distance measure can also be used in any other coarsening procedure,
assuming a rich enough set of the near-kernel components of the matrix for the
targeted system is known or computed.
[Show abstract][Hide abstract] ABSTRACT: This paper provides an overview of the main ideas driving the bootstrap
algebraic multigrid methodology, including compatible relaxation and algebraic
distances for defining effective coarsening strategies, the least squares
method for computing accurate prolongation operators and the bootstrap cycles
for computing the test vectors that are used in the least squares process. We
review some recent research in the development, analysis and application of
bootstrap algebraic multigrid and point to open problems in these areas.
Results from our previous research as well as some new results for some model
diffusion problems with highly oscillatory diffusion coefficient are presented
to illustrate the basic components of the BAMG algorithm.
Numerical Mathematics Theory Methods and Applications 06/2014; 8(01). DOI:10.4208/nmtma.2015.w06si · 0.71 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a fully numerical multigrid approach for solving the all-electron Kohn–Sham equation in molecules. The equation is represented on a hierarchy of Cartesian grids, from coarse ones that span the entire molecule to very fine ones that describe only a small volume around each atom. This approach is adaptable to any type of geometry. We demonstrate it for a variety of small molecules and obtain high accuracy agreement with results obtained previously for diatomic molecules using a prolate-spheroidal grid. We provide a detailed presentation of the numerical methodology and discuss possible extensions of this approach.
Journal of Chemical Theory and Computation 10/2013; 9(11):4744–4760. DOI:10.1021/ct400479u · 5.50 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Laplacian matrices of graphs arise in large-scale computational applications
such as semi-supervised machine learning; spectral clustering of images,
genetic data and web pages; transportation network flows; electrical resistor
circuits; and elliptic partial differential equations discretized on
unstructured grids with finite elements. A Lean Algebraic Multigrid (LAMG)
solver of the symmetric linear system Ax=b is presented, where A is a graph
Laplacian. LAMG's run time and storage are empirically demonstrated to scale
linearly with the number of edges.
LAMG consists of a setup phase during which a sequence of
increasingly-coarser Laplacian systems is constructed, and an iterative solve
phase using multigrid cycles. General graphs pose algorithmic challenges not
encountered in traditional multigrid applications. LAMG combines a lean
piecewise-constant interpolation, judicious node aggregation based on a new
node proximity measure (the affinity), and an energy correction of coarse-level
systems. This results in fast convergence and substantial setup and memory
savings. A serial LAMG implementation scaled linearly for a diverse set of 3774
real-world graphs with up to 47 million edges, with no parameter tuning. LAMG
was more robust than the UMFPACK direct solver and Combinatorial Multigrid
(CMG), although CMG was faster than LAMG on average. Our methodology is
extensible to eigenproblems and other graph computations.
[Show abstract][Hide abstract] ABSTRACT: Algebraic multigrid is an iterative method that is often optimal for solving the matrix equations that arise in a wide variety of applications, including discretized partial differential equations. It automatically constructs a sequence of increasingly smaller matrix problems that enable efficient resolution of all scales present in the solution. One of the main components of the method is an adequate choice of coarse grids. The current coarsening methodology is based on measuring how a so-called algebraically smooth error value at one point depends on the error values at its neighbors. Such a concept of strength of connection is well understood for operators whose principal part is an M-matrix; however, the strength concept for more general matrices is not yet clearly understood, and this lack of knowledge limits the scope of AMG applicability. The purpose of this paper is to motivate a general definition of strength of connection, based on the notion of algebraic distances, discuss its implementation, and present the results of initial numerical experiments. The algebraic distance measure, we propose, uses as its main tool a least squares functional, which is also applied to define interpolation.
[Show abstract][Hide abstract] ABSTRACT: We present a bottom-up aggregation approach to image segmentation. Beginning with an image, we execute a sequence of steps in which pixels are gradually merged to produce larger and larger regions. In each step, we consider pairs of adjacent regions and provide a probability measure to assess whether or not they should be included in the same segment. Our probabilistic formulation takes into account intensity and texture distributions in a local area around each region. It further incorporates priors based on the geometry of the regions. Finally, posteriors based on intensity and texture cues are combined using “a mixture of experts” formulation. This probabilistic approach is integrated into a graph coarsening scheme, providing a complete hierarchical segmentation of the image. The algorithm complexity is linear in the number of the image pixels and it requires almost no user-tuned parameters. In addition, we provide a novel evaluation scheme for image segmentation algorithms, attempting to avoid human semantic considerations that are out of scope for segmentation algorithms. Using this novel evaluation scheme, we test our method and provide a comparison to several existing segmentation algorithms.
[Show abstract][Hide abstract] ABSTRACT: This work concerns the development of an algebraic multilevel method for computing state vectors of Markov chains. We present an efficient bootstrap algebraic multigrid (AMG) method for this task. In our proposed approach, we employ a multilevel eigensolver, with interpolation built using ideas based on compatible relaxation, algebraic distances, and least squares fitting of test vectors. Our adaptive variational strategy for computation of the state vector of a given Markov chain is then a combination of this multilevel eigensolver and an associated additive multilevel preconditioned correction process. We show that the bootstrap AMG eigensolver by itself can efficiently compute accurate approximations to the steady state vector. An additional benefit of the bootstrap approach is that it yields an accurate interpolation operator for many other eigenmodes. This in turn allows for the use of the resulting multigrid hierarchy as a preconditioner to accelerate the generalized minimal residual (GMRES) iteration for computing an additive correction equation for the approximation to the steady state vector. Unlike other existing multilevel methods for Markov chains, our method does not employ any special processing of the coarse-level systems to ensure that stochastic properties of the fine-level system are maintained there. The proposed approach is applied to a range of test problems involving nonsymmetric M-matrices arising from stochastic matrices and showing promising results.
[Show abstract][Hide abstract] ABSTRACT: We develop an algebraic multigrid (AMG) setup scheme based on the bootstrap framework for multiscale scientific computation. Our approach uses a weighted least squares definition of interpolation, based on a set of test vectors that are computed by a bootstrap setup cycle and then improved by a multigrid eigensolver and a local residual-based adaptive relaxation process. To emphasize the robustness, efficiency, and flexibility of the individual components of the proposed approach, we include extensive numerical results of the method applied to scalar elliptic partial differential equations discretized on structured meshes. As a first test problem, we consider the Laplace equation discretized on a uniform quadrilateral mesh, a problem for which multigrid is well understood. Then, we consider various more challenging variable coefficient systems coming from covariant finite-difference approximations of the two-dimensional gauge Laplacian system, a commonly used model problem in AMG algorithm development for linear systems arising in lattice field theory computations.
[Show abstract][Hide abstract] ABSTRACT: A fast multilevel algorithm (MuST) for evaluating an $n$-sample sinc interpolant at $mn$ points is presented. For uniform grids, its complexity is $25mn\log(1/\delta)$ flops for the sinc kernel and $75mn\log(1/\delta)$ for the sincd kernel, where $\delta$ is the target evaluation accuracy. MuST is faster than FFT- and FMM-based evaluations for large $n$ and/or for large $\delta$. It is also applicable to nonuniform grids and to other kernels. Numerical experiments demonstrating the algorithm's practicality are presented.
[Show abstract][Hide abstract] ABSTRACT: Multigrid solvers proved very efficient for solving massive systems of equations in various fields. These solvers are based on iterative relaxation schemes together with the approximation of the "smooth" error function on a coarser level (grid). We present two efficient multilevel eigensolvers for solving massive eigenvalue problems that emerge in data analysis tasks. The first solver, a version of classical algebraic multigrid (AMG), is applied to eigenproblems arising in clustering, image segmentation, and dimensionality reduction, demonstrating an order of magnitude speedup compared to the popular Lanczos algorithm. The second solver is based on a new, much more accurate interpolation scheme. It enables calculating a large number of eigenvectors very inexpensively.
[Show abstract][Hide abstract] ABSTRACT: This work concerns the development of an Algebraic Multilevel method for computing stationary vectors of Markov chains. We present an efficient Bootstrap Algebraic Multilevel method for this task. In our proposed approach, we employ a multilevel eigensolver, with interpolation built using ideas based on compatible relaxation, algebraic distances, and least squares fitting of test vectors. Our adaptive variational strategy for computation of the state vector of a given Markov chain is then a combination of this multilevel eigensolver and associated multilevel preconditioned GMRES iterations. We show that the Bootstrap AMG eigensolver by itself can efficiently compute accurate approximations to the state vector. An additional benefit of the Bootstrap approach is that it yields an accurate interpolation operator for many other eigenmodes. This in turn allows for the use of the resulting AMG hierarchy to accelerate the MLE steps using standard multigrid correction steps. The proposed approach is applied to a range of test problems, involving non-symmetric stochastic M-matrices, showing promising results for all problems considered. Comment: 16 pages, 7 figures, submitted to SISC April 2010.
[Show abstract][Hide abstract] ABSTRACT: In this paper we generalize and improve the multiscale organization of graphs by introducing a new measure that quantifies the "closeness" between two nodes. The calculation of the measure is linear in the number of edges in the graph and involves just a small number of relaxation sweeps. A similar notion of distance is then calculated and used at each coarser level. We demonstrate the use of this measure in multiscale methods for several important combinatorial optimization problems and discuss the multiscale graph organization.
SIAM Journal on Multiscale Modeling and Simulation 04/2010; DOI:10.1137/100791142 · 1.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We introduce a multiscale approach that combines segmentation with classification to detect abnormal brain structures in medical imagery, and demonstrate its utility in automatically detecting multiple sclerosis (MS) lesions in 3-D multichannel magnetic resonance (MR) images. Our method uses segmentation to obtain a hierarchical decomposition of a multichannel, anisotropic MR scans. It then produces a rich set of features describing the segments in terms of intensity, shape, location, neighborhood relations, and anatomical context. These features are then fed into a decision forest classifier, trained with data labeled by experts, enabling the detection of lesions at all scales. Unlike common approaches that use voxel-by-voxel analysis, our system can utilize regional properties that are often important for characterizing abnormal brain structures. We provide experiments on two types of real MR images: a multichannel proton-density-, T2-, and T1-weighted dataset of 25 MS patients and a single-channel fluid attenuated inversion recovery (FLAIR) dataset of 16 MS patients. Comparing our results with lesion delineation by a human expert and with previously extensively validated results shows the promise of the approach.
[Show abstract][Hide abstract] ABSTRACT: The two-dimensional layout optimization problem reinforced by the efficient space utilization demand has a wide spectrum of practical applications. Formulating the problem as a nonlinear minimization problem under planar equality and/or inequality density constraints, we present a linear time multigrid algorithm for solving correction to this problem. The method is demonstrated on various graph drawing (visualization) instances.
SIAM Journal on Multiscale Modeling and Simulation 03/2009; 8(5). DOI:10.1137/090771995 · 1.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Linear ordering problems are combinatorial optimization problems which deal with the minimization of dierent functionals in which the graph vertices are mapped onto (1,2,...,n). These problems are widely used and studied in many practical and theoretical applications. In this paper we present a variety of linear-time algorithms for these problems inspired by the Algebraic Multi- grid approach which is based on weighted edge contraction. The experimental result for four such problems turned out to be better than every known result in almost all cases, while the short running time of the algorithms enables testing very large graphs.
Journal of Experimental Algorithmics 09/2008; 13. DOI:10.1145/1412228.1412232
[Show abstract][Hide abstract] ABSTRACT: We present a novel automatic multiscale algorithm applied to segmentation of anatomical structures in brain MRI. The algorithm which is derived from algebraic multigrid, uses a graph representation of the image and performs a coarsening process that produces a full hierarchy of segments. Our main contribution is the incorporation of prior knowledge information into the multiscale framework through a Bayesian formulation. The probabilistic information is based on an atlas prior and on a likelihood function estimated from a manually labeled training set. The significance of our new approach is that the constructed pyramid, reflects the prior knowledge formulated. This leads to an accurate and efficient methodology for detection of various anatomical structures simultaneously. Quantitative validation results on gold standard MRI show the benefit of our approach.
[Show abstract][Hide abstract] ABSTRACT: Linear ordering problems are combinatorial optimization problems which deal with the minimization of different functionals in which the graph vertices are mapped onto (1, 2, ..., n). These problems are widely used and studied in many practical and theoret-ical applications. In this review we summarize a variety of linear-time algorithms for these problems inspired by the Algebraic Multigrid approach which is based on weighted edge contraction. The experimental results for four such problems turned out to be better than ev-ery known results in almost all cases, while the short running time of the algorithms enables applications on very large graphs.
[Show abstract][Hide abstract] ABSTRACT: We present a novel approach that allows us to reliably compute many useful properties of a silhouette. Our approach assigns, for every internal point of the silhouette, a value reflecting the mean time required for a random walk beginning at the point to hit the boundaries. This function can be computed by solving Poisson's equation, with the silhouette contours providing boundary conditions. We show how this function can be used to reliably extract various shape properties including part structure and rough skeleton, local orientation and aspect ratio of different parts, and convex and concave sections of the boundaries. In addition to this, we discuss properties of the solution and show how to efficiently compute this solution using multigrid algorithms. We demonstrate the utility of the extracted properties by using them for shape classification and retrieval.
IEEE Transactions on Pattern Analysis and Machine Intelligence 01/2007; 28(12):1991-2005. DOI:10.1109/TPAMI.2006.253 · 5.78 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present an algorithm for edge detection suitable for both natural as well as noisy images. Our method is based on efficient multiscale utilization of elongated filters measuring the difference of oriented means of various lengths and orientations, along with a theoretical estimation of the effect of noise on the response of such filters. We use a scale adaptive threshold along with a recursive decision process to reveal the significant edges of all lengths and orientations and to localize them accurately even in low-contrast and very noisy images. We further use this algorithm for fiber detection and enhancement by utilizing stochastic completion-like process from both sides of a fiber. Our algorithm relies on an efficient multiscale algorithm for computing all "significantly different" oriented means in an image in O(N log rho), where N is the number of pixels, and p is the length of the longest structure of interest. Experimental results on both natural and noisy images are presented.
IEEE 11th International Conference on Computer Vision, ICCV 2007, Rio de Janeiro, Brazil, October 14-20, 2007; 01/2007