Three-level BDDC in three dimensions

This is the published version, also available here: http://dx.doi.org/10.1137/050629902.Balancing domain decomposition by constraints (BDDC) methods are nonoverlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from the discretization of elliptic boundary value problems. Their coarse problems are given in terms of a small number of continuity constraints for each subdomain, which are enforced across the interface. The coarse problem matrix is generated and factored by a direct solver at the beginning of the computation and it can ultimately become a bottleneck if the number of subdomains is very large. In this paper, two three-level BDDC methods are introduced for solving the coarse problem approximately for problems in three dimensions. This is an extension of previous work for the two-dimensional case. Edge constraints are considered in this work since vertex constraints alone, which work well in two dimensions, result in a noncompetitive algorithm in three dimensions. Some new technical tools are then needed in the analysis and this makes the three-dimensional case more complicated. Estimates of the condition numbers are provided for two three-level BDDC methods, and numerical experiments are also discussed

A Nonoverlapping Domain Decomposition Method for Incompressible Stokes Equations with Continuous Pressures

This is the publisher's version, also available electronically from http://epubs.siam.org/doi/abs/10.1137/120861503A nonoverlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semidefinite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments demonstrate the convergence rate of the proposed algorithm

Parameter estimation by implicit sampling

Implicit sampling is a weighted sampling method that is used in data assimilation, where one sequentially updates estimates of the state of a stochastic model based on a stream of noisy or incomplete data. Here we describe how to use implicit sampling in parameter estimation problems, where the goal is to find parameters of a numerical model, e.g.~a partial differential equation (PDE), such that the output of the numerical model is compatible with (noisy) data. We use the Bayesian approach to parameter estimation, in which a posterior probability density describes the probability of the parameter conditioned on data and compute an empirical estimate of this posterior with implicit sampling. Our approach generates independent samples, so that some of the practical difficulties one encounters with Markov Chain Monte Carlo methods, e.g.~burn-in time or correlations among dependent samples, are avoided. We describe a new implementation of implicit sampling for parameter estimation problems that makes use of multiple grids (coarse to fine) and BFGS optimization coupled to adjoint equations for the required gradient calculations. The implementation is "dimension independent", in the sense that a well-defined finite dimensional subspace is sampled as the mesh used for discretization of the PDE is refined. We illustrate the algorithm with an example where we estimate a diffusion coefficient in an elliptic equation from sparse and noisy pressure measurements. In the example, dimension\slash mesh-independence is achieved via Karhunen-Lo\`{e}ve expansions

A FETI-DP TYPE DOMAIN DECOMPOSITION ALGORITHM FOR THREE-DIMENSIONAL INCOMPRESSIBLE STOKES EQUATIONS

The FETI-DP (dual-primal finite element tearing and interconnecting) algorithms, proposed by the authors in [SIAM J. Numer. Anal., 51 (2013), pp. 1235–1253] and [Internat. J. Numer. Methods Engrg., 94 (2013), pp. 128–149] for solving incompressible Stokes equations, are extended to three-dimensional problems. A new analysis of the condition number bound for using the Dirichlet preconditioner is given. The algorithm and analysis are valid for mixed finite elements with both continuous and discontinuous pressures. An advantage of this new analysis is that the numerous coarse level velocity components, required in the previous analysis to enforce the divergence-free subdomain boundary velocity conditions, are no longer needed. This greatly reduces the size of the coarse level problem in the algorithm, especially for three-dimensional problems. The coarse level velocity space can be chosen as simple as those coarse spaces for solving scalar elliptic problems corresponding to each velocity component. Both the Dirichlet and lumped preconditioners are analyzed using the same framework in this new analysis. Their condition number bounds are proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments in both two and three dimensions, using mixed finite elements with both continuous and discontinuous pressures, demonstrate the convergence rate of the algorithms