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Register a new userRandomized methods to characterize large-scale vortical flow network
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Added by
Zhe Bai
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We demonstrate the effective use of randomized methods for linear algebra to perform network-based analysis of complex vortical flows. Network theoretic approaches can reveal the connectivity structures among a set of vortical elements and analyze their collective dynamics. These approaches have recently been generalized to analyze high-dimensional turbulent flows, for which network computations can become prohibitively expensive. In this work, we propose efficient methods to approximate network quantities, such as the leading eigendecomposition of the adjacency matrix, using randomized methods. Specifically, we use the Nystrom method to approximate the leading eigenvalues and eigenvectors, achieving significant computational savings and reduced memory requirements. The effectiveness of the proposed technique is demonstrated on two high-dimensional flow fields: two-dimensional flow past an airfoil and two-dimensional turbulence. We find that quasi-uniform column sampling outperforms uniform column sampling, while both feature the same computational complexity.
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The material point method (MPM) has been increasingly used for the simulation of large deformation processes in fluid-infiltrated porous materials. For undrained poromechanical problems, however, standard MPMs are numerically unstable because they use low-order interpolation functions that violate the inf-sup stability condition. In this work, we develop stabilized MPM formulations for dynamic and quasi-static poromechanics that permit the use of standard low-order interpolation functions notwithstanding the drainage condition. For the stabilization of both dynamic and quasi-static formulations, we utilize the polynomial pressure projection method whereby a stabilization term is augmented to the balance of mass. The stabilization term can be implemented with both the original and generalized interpolation material point (GIMP) methods, and it is compatible with existing time-integration methods. Here we use fully-implicit methods for both dynamic and quasi-static poromechanical problems, aided by a block-preconditioned Newton-Krylov solver. The stabilized MPMs are verified and investigated through several numerical examples under dynamic and quasi-static conditions. Results show that the proposed MPM formulations allow standard low-order interpolation functions to be used for both the solid displacement and pore pressure fields of poromechanical formulations, from undrained to drained conditions, and from dynamic to quasi-static conditions.
In this paper, we study a multi-scale deep neural network (MscaleDNN) as a meshless numerical method for computing oscillatory Stokes flows in complex domains. The MscaleDNN employs a multi-scale structure in the design of its DNN using radial scalings to convert the approximation of high frequency components of the highly oscillatory Stokes solution to one of lower frequencies. The MscaleDNN solution to the Stokes problem is obtained by minimizing a loss function in terms of L2 normof the residual of the Stokes equation. Three forms of loss functions are investigated based on vorticity-velocity-pressure, velocity-stress-pressure, and velocity-gradient of velocity-pressure formulations of the Stokes equation. We first conduct a systematic study of the MscaleDNN methods with various loss functions on the Kovasznay flow in comparison with normal fully connected DNNs. Then, Stokes flows with highly oscillatory solutions in a 2-D domain with six randomly placed holes are simulated by the MscaleDNN. The results show that MscaleDNN has faster convergence and consistent error decays in the simulation of Kovasznay flow for all four tested loss functions. More importantly, the MscaleDNN is capable of learning highly oscillatory solutions when the normal DNNs fail to converge.
This work considers the problem of computing the CANDECOMP/PARAFAC (CP) decomposition of large tensors. One popular way is to translate the problem into a sequence of overdetermined least squares subproblems with Khatri-Rao product (KRP) structure. In this work, for tensor with different levels of importance in each fiber, combining stochastic optimization with randomized sampling, we present a mini-batch stochastic gradient descent algorithm with importance sampling for those special least squares subproblems. Four different sampling strategies are provided. They can avoid forming the full KRP or corresponding probabilities and sample the desired fibers from the original tensor directly. Moreover, a more practical algorithm with adaptive step size is also given. For the proposed algorithms, we present their convergence properties and numerical performance. The results on synthetic data show that our algorithms outperform the existing algorithms in terms of accuracy or the number of iterations.
Production in an economy is a set of firms activities as suppliers and customers; a firm buys goods from other firms, puts value added and sells products to others in a giant network of production. Empirical study is lacking despite the fact that the structure of the production network is important to understand and make models for many aspects of dynamics in economy. We study a nation-wide production network comprising a million firms and millions of supplier-customer links by using recent statistical methods developed in physics. We show in the empirical analysis scale-free degree distribution, disassortativity, correlation of degree to firm-size, and community structure having sectoral and regional modules. Since suppliers usually provide credit to their customers, who supply it to theirs in turn, each link is actually a creditor-debtor relationship. We also study chains of failures or bankruptcies that take place along those links in the network, and corresponding avalanche-size distribution.
A weakly admissible mesh (WAM) on a continuum real-valued domain is a sequence of discrete grids such that the discrete maximum norm of polynomials on the grid is comparable to the supremum norm of polynomials on the domain. The asymptotic rate of growth of the grid sizes and of the comparability constant must grow in a controlled manner. In this paper we generalize the notion of a WAM to a hierarchical subspaces of not necessarily polynomial functions, and we analyze particular strategies for random sampling as a technique for generating WAMs. Our main results show that WAMs and their stronger variant, admissible meshes, can be generated by random sampling, and our analysis provides concrete estimates for growth of both the meshes and the discrete-continuum comparability constants.
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