We apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior Dirichlet boundary value problem for the two dimensional Poisson equation. Adopting biorthogonal wavelet matrix compression to the boundary terms with N degrees of freedom, we show that the resulting compression strategy fits the optimal convergence rate of the coupling Galerkin methods, while the number of nonzero entries in the corresponding stiffness matrices is considerably smaller than N 2 .
SUMMARYWe apply multiscale methods to the coupling of ÿnite and boundary element methods to solve an exterior two-dimensional Laplacian. The matrices belonging to the boundary terms of the coupled FEM-BEM system are compressed by using biorthogonal wavelet bases developed from A. Cohen, I. Daubechies and J.-C. Feauveau (Comm. Proc. Appl. Math. 1992; 45:485). The coupling yields a linear equation system which corresponds to a saddle point problem. As favourable solver, the Bramble-Pasciak-CG (Math. Comp. 1988; 50:1) is utilized. A suitable preconditioner is developed by combining the BPX (Math. Comp. 1990; 55:1) with the wavelet preconditioning (Numer. Math. 1992; 63:315). Through numerical experiments we provide results which corroborate the theory of the present paper.
Three dimensional (3D) geometrical models are not only a powerful tool for quantitatively characterizing complex tissues but also useful for probing structure-function relationships in a tissue. However, these models are generally incomplete due to experimental limitations in acquiring multiple (>4) fluorescent channels simultaneously. Indeed, predictive geometrical and functional models of the liver have been restricted to few tissue and cellular components, excluding important cellular populations such as hepatic stellate cells (HSCs) and Kupffer cells (KCs). Here, we performed deep-tissue immunostaining, multiphoton microscopy, deep- learning techniques, and 3D image processing to computationally expand the number of simultaneously reconstructed tissue structures. We then generated a spatio-temporal single- cell atlas of hepatic architecture (Hep3D), including all main tissue and cellular components at different stages of post-natal development in mice. We used Hep3D to quantitatively study 1) hepatic morphodynamics from early post-natal development to adulthood, and 2) the structural role of KCs in the murine liver homeostasis. In addition to a complete description of bile canaliculi and sinusoidal network remodeling, our analysis uncovered unexpected spatiotemporal patterns of non-parenchymal cells and hepatocytes differing in size, number of nuclei, and DNA content. Surprisingly, we found that the specific depletion of KCs alters the number and morphology of the HSCs. These findings reveal novel characteristics of liver heterogeneity and have important implications for both the structural organization of liver tissue and its function. Our next-gen 3D single-cell atlas is a powerful tool to understand liver tissue architecture, under both physiological and pathological conditions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.