We study the problem of decentralized optimization over time-varying networks with strongly convex smooth cost functions. In our approach, nodes run a multi-step gossip procedure after making each gradient update, thus ensuring approximate consensus at each iteration, while the outer loop is based on accelerated Nesterov scheme. The algorithm achieves precision ε > 0 in O( √ κ g χ log 2 (1/ε))communication steps and O( √ κ g log(1/ε)) gradient computations at each node, where κ g is the global function number and χ characterizes connectivity of the communication network. In the case of a static network, χ = 1/γ where γ denotes the normalized spectral gap of communication matrix W. The complexity bound includes κ g , which can be significantly better than the worst-case condition number among the nodes.
The paper presents a workflow for fast pore-scale simulation of singlephase flow in tight reservoirs typically characterized by low, multiscale porosity. Multiscale porosity implies that the computational domain contains porous voxels (unresolved porosity) in addition to pure fluid voxels. In this case, the Stokes-Brinkman equations govern the flow, with the Darcy term needed to account for the flow in the porous voxels. As the central part of our workflow, robust and efficient solvers for Stokes and Stokes-Brinkman equations are presented. The solvers are customized for low-porosity binary and multiclass images, respectively. Another essential component of the workflow is a preprocessing module for classifying images with respect to the connectivity of the multiscale pore space. Particularly, an approximation of the Stokes-Brinkman problem, namely, the Darcy problem, is investigated for the images that do not have pure fluid percolation paths. Thorough computational experiments demonstrate efficiency and robustness of the workflow for simulations on images from tight reservoirs. Raw files describing the used CT images are provided as supplementary materials to enable other researchers to use them.
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