The reduced reliability of next generation exascale systems means that the resiliency properties of a numerical algorithm will become an important factor in both the choice of algorithm, and in its analysis. The multi...
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ISBN:
(纸本)9783319399294;9783319399270
The reduced reliability of next generation exascale systems means that the resiliency properties of a numerical algorithm will become an important factor in both the choice of algorithm, and in its analysis. The multigrid algorithm is the workhorse for the distributed solution of linear systems but little is known about its resiliency properties and convergence behavior in a fault- prone environment. In the current work, we propose a probabilistic model for the effect of faults involving random diagonal matrices. We summarize results of the theoretical analysis of the model for the rate of convergence of fault-prone multigrid methods which show that the standard multigrid method will not be resilient. Finally, we present a modification of the standard multigrid algorithm that will be resilient.
In this article, we propose the sparse grid combination technique for the second moment analysis of elliptic partial differential equations on random domains. By employing shape sensitivity analysis, we linearize the ...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
In this article, we propose the sparse grid combination technique for the second moment analysis of elliptic partial differential equations on random domains. By employing shape sensitivity analysis, we linearize the influence of the random domain perturbation on the solution. We derive deterministic partial differential equations to approximate the random solution's mean and its covariance with leading order in the amplitude of the random domain perturbation. The partial differential equation for the covariance is a tensor product Dirichlet problem which can efficiently be determined by Galerkin's method in the sparse tensor product space. We show that this Galerkin approximation coincides with the solution derived from the combination technique if the detail spaces in the related multiscale hierarchy are constructed with respect to Galerkin projections. This means that the combination technique does not impose an additional error in our construction. Numerical experiments quantify and qualify the proposed method.
A finite element (FE) model, that explicitly discretizes a single 3D spherulite is proposed. A spherulite is a two-phase microstructure consisting of amorphous and crystalline regions. Crystalline regions, that grow f...
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ISBN:
(纸本)9783319399294;9783319399270
A finite element (FE) model, that explicitly discretizes a single 3D spherulite is proposed. A spherulite is a two-phase microstructure consisting of amorphous and crystalline regions. Crystalline regions, that grow from a central nucleus in the form of lamellae, have particular lattice orientations. In the FE analyses, 8-chain and crystal viscoplasticity constitutive models are employed. Stress-strain distributions and slip system activities in the spherulite microstructure are studied and found to be in good agreement with the literature. Influences of the crystallinity ratio on the yield stress and the initial Young's modulus are also investigated.
In this paper, a Riemannian BFGS method is defined for minimizing a smooth function on a Riemannian manifold endowed with a retraction and a vector transport. The method is based on a Riemannian generalization of a ca...
ISBN:
(纸本)9783319399294;9783319399270
In this paper, a Riemannian BFGS method is defined for minimizing a smooth function on a Riemannian manifold endowed with a retraction and a vector transport. The method is based on a Riemannian generalization of a cautious update and a weak line search condition. It is shown that, the Riemannian BFGS method converges (i) globally to a stationary point without assuming that the objective function is convex and (ii) superlinearly to a nondegenerate minimizer. The weak line search condition removes completely the need to consider the differentiated retraction. The joint diagonalization problem is used to demonstrate the performance of the algorithm with various parameters, line search conditions, and pairs of retraction and vector transport.
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