We consider the problem of computing PageRank. The matrix involved is large and cannot be factored, and hence techniques based on matrix-vector products must be applied. A variant of the restarted refined Arnoldi method is proposed, which does not involve Ritz value computations. Numerical examples illustrate the performance and convergence behavior of the algorithm. (2000): 65F15, 65C40. AMS subject classification

We consider 2 × 2 block indefinite linear systems whose (2, 2) block is zero. Such systems arise in many applications. We discuss two techniques that are based on modifying the (1, 1) block in a way that makes the system easier to solve. The main part of the paper focuses on an augmented Lagrangian approach: a technique that modifies the (1,1) block without changing the system size. The choice of the parameter involved, the spectrum of the linear system, and its condition number are discussed, and some analytical observations are provided. A technique of deflating the (1,1) block is then introduced. Finally, numerical experiments that validate the analysis are presented.

We introduce an 1 -sparse method for the reconstruction of a piecewise smooth point set surface. The technique is motivated by recent advancements in sparse signal reconstruction. The assumption underlying our work is that common objects, even geometrically complex ones, can typically be characterized by a rather small number of features. This, in turn, naturally lends itself to incorporating the powerful notion of sparsity into the model. The sparse reconstruction principle gives rise to a reconstructed point set surface that consists mainly of smooth modes, with the residual of the objective function strongly concentrated near sharp features. Our technique is capable of recovering orientation and positions of highly noisy point sets. The global nature of the optimization yields a sparse solution and avoids local minima. Using an interior-point log-barrier solver with a customized preconditioning scheme, the solver for the corresponding convex optimization problem is competitive and the results are of high quality.

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SUMMARYWe introduce a new preconditioning technique for iteratively solving linear systems arising from finite element discretization of the mixed formulation of the time-harmonic Maxwell equations. The preconditioners are motivated by spectral equivalence properties of the discrete operators, but are augmentation free and Schur complement free. We provide a complete spectral analysis, and show that the eigenvalues of the preconditioned saddle point matrix are strongly clustered. The analytical observations are accompanied by numerical results that demonstrate the scalability of the proposed approach.

Most visual effects fluid solvers use a time-splitting approach where velocity is first advected in the flow, then projected to be incompressible with pressure. Even if a highly accurate advection scheme is used, the self-advection step typically transfers some kinetic energy from divergence-free modes into divergent modes, which are then projected out by pressure, losing energy noticeably for large time steps. Instead of taking smaller time steps or using significantly more complex time integration, we propose a new scheme called IVOCK (Integrated Vorticity of Convective Kinematics) which cheaply captures much of what is lost in self-advection by identifying it as a violation of the vorticity equation. We measure vorticity on the grid before and after advection, taking into account vortex stretching, and use a cheap multigrid V-cycle approximation to a vector potential whose curl will correct the vorticity error. IVOCK works independently of the advection scheme (we present examples with various semi-Lagrangian methods and FLIP), works independently of how boundary conditions are applied (it just corrects error in advection, leaving pressure etc. to take care of boundaries and other forces), and other solver parameters (we provide smoke, fire, and water examples). For 10 ∼ 25% extra computation time per step much larger steps can be used, while producing detailed vorticial structures and convincing turbulence that are lost without correction. Table 1: Algorithm abbreviations used through out this paper. IVOCK The computational routine (Alg.1) correcting vorticity for advection SF Classic Stable Fluids advection [Stam 1999] SF-IVOCK IVOCK with SF advection SL3 Semi-Lagrangian with RK3 path tracing and clamped cubic interpolation BFECC Kim et al.'s scheme [2005], with extrema clamping([Selle et al. 2008]) BFECC-IVOCK IVOCK with BFECC advection MC Selle et al.'s MacCormack method [2008] MC-IVOCK IVOCK with MacCormack FLIP Zhu and Bridson's incompressible variant of FLIP [2005] FLIP-IVOCK FLIP advection of velocity and density, SL3 for vorticity in IVOCK.

We present a new iterative scheme for PageRank computation. The algorithm is applied to the linear system formulation of the problem, using inner-outer stationary iterations. It is simple, can be easily implemented and parallelized, and requires minimal storage overhead. Our convergence analysis shows that the algorithm is effective for a crude inner tolerance and is not sensitive to the choice of the parameters involved. The same idea can be used as a preconditioning technique for nonstationary schemes. Numerical examples featuring matrices of dimensions exceeding 100,000,000 in sequential and parallel environments demonstrate the merits of our technique. Our code is available online for viewing and testing, along with several large scale examples.

We first explore methods for approximating the commute time and Katz score between a pair of nodes. These methods are based on the approach of matrices, moments, and quadrature developed in the numerical linear algebra community. They rely on the Lanczos process and provide upper and lower bounds on an estimate of the pair-wise scores. We also explore methods to approximate the commute times and Katz scores from a node to all other nodes in the graph. Here, our approach for the commute times is based on a variation of the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adopt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pair-wise commute time method and column-wise Katz algorithm both have attractive theoretical properties and empirical performance.