Recursively accelerated multilevel aggregation for markov chains

Abstract. A recursive acceleration method is proposed for multiplicative multilevel aggregation algorithms that calculate the stationary probability vector of large, sparse, and irreducible Markov chains. Pairs of consecutive iterates at all branches and levels of a multigrid W cycle with simple, nonoverlapping aggregation are recombined to produce improved iterates at those levels. This is achieved by solving quadratic programming problems with inequality constraints: the linear combination of the two iterates is sought that has a minimal two-norm residual, under the constraint that all vector components are nonnegative. It is shown how the two-dimensional quadratic programming problems can be solved explicitly in an efficient way. The method is further enhanced by windowed top-level acceleration of the W cycles using the same constrained quadratic programming approach. Recursive acceleration is an attractive alternative to smoothing the restriction and interpolation operators, since the operator complexity is better controlled and the probabilistic interpretation of coarse-level operators is maintained on all levels. Numerical results are presented showing that the resulting recursively accelerated multilevel aggregation cycles for Markov chains, combined with top-level acceleration, converge significantly faster than W cycles and lead to close-to-linear computational complexity for challenging test problems

Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective

A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach

Патогенетическое значение эндотелиальной дисфункции при первичной глаукоме

В эксперименте на модели глаукомы установлено, что дизрегуляция эндотелиальных факторов сосудистого тонуса (ЭТ-1, NO) вызывала формирование дисфункции эндотелия, которая усиливалась гиперхолестеринемией и дислипопротеидемией, а также патологическим уровнем перекисного окисления липидов. Эндотелиальная дисфункция способствовала дизрегуляции гемо- и гидродинамики глаз и развитию ишемично-реперфузионного повреждения нервных тканей глаза, что позволяет считать этот фактор патогенетическим в развитии глаукомы.The experiment with glaucoma model showed that dysregulation of endothelial factors of vascular tone (ET-1 and NO) determined the development of endothelial dysfunction. This process was intensified by hypercholesteremia, dyslipoproteidemia and high level of lipid peroxidation. Endothelial dysfunction induced changes in the eye hemo- and hydrodynamics as a development of ischemia-reperfusion injury of the eye nerve tissues. It let us consider this process to be pathogenetic one in glaucoma pathology

Better Safe than Sorry - Socio-Spatial Group Structure Emerges from Individual Variation in Fleeing, Avoidance or Velocity in an Agent-Based Model

In group-living animals, such as primates, the average spatial group structure often reflects the dominance hierarchy, with central dominants and peripheral subordinates. This central-peripheral group structure can arise by self-organization as a result of subordinates fleeing from dominants after losing a fight. However, in real primates, subordinates often avoid interactions with potentially aggressive group members, thereby preventing aggression and subsequent fleeing. Using agent-based modeling, we investigated which spatial and encounter structures emerge when subordinates also avoid known potential aggressors at a distance as compared with the model which only included fleeing after losing a fight (fleeing model). A central-peripheral group structure emerged in most conditions. When avoidance was employed at small or intermediate distances, centrality of dominants emerged similar to the fleeing model, but in a more pronounced way. This result was also found when fleeing after a fight was made independent of dominance rank, i.e. occurred randomly. Employing avoidance at larger distances yielded more spread out groups. This provides a possible explanation of larger group spread in more aggressive species. With avoidance at very large distances, spatially and socially distinct subgroups emerged. We also investigated how encounters were distributed amongst group members. In the fleeing model all individuals encountered all group members equally often, whereas in the avoidance model encounters occurred mostly among similar-ranking individuals. Finally, we also identified a very general and simple mechanism causing a central-peripheral group structure: when individuals merely differed in velocity, faster individuals automatically ended up at the periphery. In summary, a central-peripheral group pattern can easily emerge from individual variation in different movement properties in general, such as fleeing, avoidance or velocity. Moreover, avoidance behavior also affects the encounter structure and can lead to subgroup formation

Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs

We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algorithm in forming aggregates and the resulting coarse level hierarchy is then used in a K-cycle iteration solve phase with a $\ell^1$-Jacobi smoother. Numerical tests of a parallel implementation of the method for graphics processors are presented to demonstrate its effectiveness.Comment: 18 pages, 3 figure

Pressure balance at the magnetopause: Experimental studies

The pressure balance at the magnetopause is formed by magnetic field and plasma in the magnetosheath, on one side, and inside the magnetosphere, on the other side. In the approach of dipole earth's magnetic field configuration and gas-dynamics solar wind flowing around the magnetosphere, the pressure balance predicts that the magnetopause distance R depends on solar wind dynamic pressure Pd as a power low R ~ Pd^alpha, where the exponent alpha=-1/6. In the real magnetosphere the magnetic filed is contributed by additional sources: Chapman-Ferraro current system, field-aligned currents, tail current, and storm-time ring current. Net contribution of those sources depends on particular magnetospheric region and varies with solar wind conditions and geomagnetic activity. As a result, the parameters of pressure balance, including power index alpha, depend on both the local position at the magnetopause and geomagnetic activity. In addition, the pressure balance can be affected by a non-linear transfer of the solar wind energy to the magnetosheath, especially for quasi-radial regime of the subsolar bow shock formation proper for the interplanetary magnetic field vector aligned with the solar wind plasma flow.Comment: 8 pages, 2 figure

Biomass partitioning and root morphology of savanna trees across a water gradient.

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