Orbitally discrete coarse spaces

<p>Given a coarse space (X, E), we endow X with the discrete topology and denote X ♯ = {p ∈ βG : each member P ∈ p is unbounded }. For p, q ∈ X ♯ , p||q means that there exists an entourage E ∈ E such that E[P] ∈ q for each P ∈ p. We say that (X, E) is orbitally discrete if, for every p ∈ X ♯ , the orbit p = {q ∈ X ♯ : p||q} is discrete in βG. We prove that every orbitally discrete space is almost finitary and scattered.</p>

Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients

In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend on the coefficient of the PDE. This poses a significant challenge for stochastic coefficients as it is computationally expensive to solve the local spectral problems for every realization of the coefficient. To tackle this computational burden, we propose a machine learning approach. Our method is based on the use of a deep neural network (DNN) to approximate the relation between the stochastic coefficients and the coarse spaces. For the input of the DNN, we apply the Karhunen–Loève expansion and use the first few dominant terms in the expansion. The output of the DNN is the resulting coarse space, which is then applied with the standard adaptive BDDC algorithm. We will present some numerical results with oscillatory and high contrast coefficients to show the efficiency and robustness of the proposed scheme.

Selectors and orderings of coarse spaces

Given a coarse space $(X, \mathcal{E})$, we consider linear orders on $X$ compatible with the coarse structure $\mathcal E$ and explore interplays between these orders and macro-uniform selectors of $(X, \mathcal{E})$.

Coarse structures on groups defined by conjugations

For a group G, we denote by G↔ the coarse space on G endowed with the coarse structure with the base {{(x,y)∈G×G:y∈xF}:F∈[G]<ω}, xF={z−1xz:z∈F}. Our goal is to explore interplays between algebraic properties of G and asymptotic properties of G↔. In particular, we show that asdim G↔=0 if and only if G/ZG is locally finite, ZG is the center of G. For an infinite group G, the coarse space of subgroups of G is discrete if and only if G is a Dedekind group.

Coarse cohomology with twisted coefficients

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

An adaptively enriched coarse space for Schwarz preconditioners for P1 discontinuous Galerkin multiscale finite element problems

In this paper, we propose a two-level additive Schwarz domain decomposition preconditioner for the symmetric interior penalty Galerkin method for a second-order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.

Closeness and linkness in balleans

A set $X$ endowed with a coarse structure is called ballean or coarse space. For a ballean $(X, \mathcal{E})$, we say that two subsets $A$, $B$ of $X$ are close (linked) if there exists an entourage $E\in \mathcal{E}$ such that $A\subseteq E [B]$, $B\subseteq E[A]$ (either $A, B$ are bounded or contain unbounded close subsets). We explore the following general question: which information about a ballean is contained and can be extracted from the relations of closeness and linkness.