### Local approximation estimators for algebraic multigrid.

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We describe a parallel implementation of the Balancing Domain Decomposition by Constraints (BDDC) method enhanced by an adaptive construction of coarse problem. The method is designed for numerically difficult problems, where standard choice of continuity of arithmetic averages across faces and edges of subdomains fails to maintain the low condition number of the preconditioned system. Problems of elasticity analysis of bodies consisting of different materials with rapidly changing stiffness may...

We study the convergence of the iterations in a Hilbert space $V,{x}_{k+1}=W\left(P\right){x}_{k},W\left(P\right)z=w=T(Pw+(I-P)z)$, where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$, if the operator $W\left(P\right)$ is continuous and the Lipschitz constant $\u2225(I-P)W\left(P\right)\u2225<1$. If an operator $W\left({P}_{1}\right)$ satisfies these assumptions and ${P}_{2}$ is an orthogonal projection such that ${P}_{1}{P}_{2}={P}_{2}{P}_{1}={P}_{1}$, then the operator $W\left({P}_{2}\right)$ is defined and continuous in $V$ and satisfies $\u2225(I-{P}_{2})W\left({P}_{2}\right)\u2225\le \u2225(I-{P}_{1})W\left({P}_{1}\right)\u2225$.

We prove nearly uniform convergence bounds for the BPX preconditioner based on smoothed aggregation under the assumption that the mesh is regular. The analysis is based on the fact that under the assumption of regular geometry, the coarse-space basis functions form a system of macroelements. This property tends to be satisfied by the smoothed aggregation bases formed for unstructured meshes.

Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and ${L}^{p}$ bounds on the ensemble then give ${L}^{p}$ convergence.

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