On the initial estimate of interface forces in FETI methods

The Balanced Domain Decomposition (BDD) method and the Finite Element Tearing and Interconnecting (FETI) method are two commonly used non-overlapping domain decomposition methods. Due to strong theoretical and numerical similarities, these two methods are generally considered as being equivalently efficient. However, for some particular cases, such as for structures with strong heterogeneities, FETI requires a large number of iterations to compute the solution compared to BDD. In this paper, the origin of the bad efficiency of FETI in these particular cases is traced back to poor initial estimates of the interface stresses. To improve the estimation of interface forces a novel strategy for splitting interface forces between neighboring substructures is proposed. The additional computational cost incurred is not significant. This yields a new initialization for the FETI method and restores numerical efficiency which makes FETI comparable to BDD even for problems where FETI was performing poorly. Various simple test problems are presented to discuss the efficiency of the proposed strategy and to illustrate the so-obtained numerical equivalence between the BDD and FETI solvers

Edge-promoting reconstruction of absorption and diffusivity in optical tomography

In optical tomography a physical body is illuminated with near-infrared light and the resulting outward photon flux is measured at the object boundary. The goal is to reconstruct internal optical properties of the body, such as absorption and diffusivity. In this work, it is assumed that the imaged object is composed of an approximately homogeneous background with clearly distinguishable embedded inhomogeneities. An algorithm for finding the maximum a posteriori estimate for the absorption and diffusion coefficients is introduced assuming an edge-preferring prior and an additive Gaussian measurement noise model. The method is based on iteratively combining a lagged diffusivity step and a linearization of the measurement model of diffuse optical tomography with priorconditioned LSQR. The performance of the reconstruction technique is tested via three-dimensional numerical experiments with simulated measurement data.Comment: 18 pages, 6 figure

A parallel Newton-Krylov-FETI-DP Solver based on FEAP: Large-scale applications and scalability for problems in the mechanics of soft biological tissues in arterial wall structures

An MPI-parallel Newton-Krylov-FETI-DP solver based on FEAP is presented together with applications to nonlinear problems in the quasi-static biomechanics of soft biological tissues. The formulation is based on highly nonlinear hyperelastic anisotropic and poly-convex models. High-resolution computations of the wall stresses in patient-specific arterial wall structures subjected to an interior normal pressure in the physiological regime of the blood pressure (up to 500 [mmHg]) are reported together with results on strong scalability. The weak scalability of Newton-Krylov-FETI-DP is investigated for up to 140 million degrees of freedom using 4096 processor cores on a Cray XT6m supercomputer in a series of simple tension tests. An implementation of a new FEAP-interface called libfw is presented which allows for the flexible unified integration of FEAP into other software packages, e.g., into LifeV. The modifications done to FEAP are dissected and discussed in detail as a case study in order to illustrate possible approaches for the integration of different code components or applications in similar scenarios

Large-scale applications and scalability for problems in the mechanics of soft biological tissues in arterial wall structures

Der Zugriff auf den Volltext ist gesperrt, neue Version unter DuEPublico-ID 37999 An MPI-parallel Newton-Krylov-FETI-DP solver based on FEAP is presented together with applications to nonlinear problems in the quasi-static biomechanics of soft biological tissues. The formulation is based on highly nonlinear hyperelastic anisotropic and poly-convex models. High-resolution computations of the wall stresses in patient-specific arterial wall structures subjected to an interior normal pressure in the physiological regime of the blood pressure (up to 500 [mmHg]) are reported together with results on strong scalability. The weak scalability of Newton-Krylov-FETI-DP is investigated for up to 140 million degrees of freedom using 4096 processor cores on a Cray XT6m supercomputer in a series of simple tension tests. An implementation of a new FEAP-interface called libfw is presented which allows for the flexible unified integration of FEAP into other software packages, e.g., into LifeV. The modifications done to FEAP are dissected and discussed in detail as a case study in order to illustrate possible approaches for the integration of different code components or applications in similar scenarios

A Multigrid Method for the Efficient Numerical Solution of Optimization Problems Constrained by Partial Differential Equations

We study the minimization of a quadratic functional subject to constraints given by a linear or semilinear elliptic partial differential equation with distributed control. Further, pointwise inequality constraints on the control are accounted for. In the linear-quadratic case, the discretized optimality conditions yield a large, sparse, and indefinite system with saddle point structure. One main contribution of this thesis consists in devising a coupled multigrid solver which avoids full constraint elimination. To this end, we define a smoothing iteration incorporating elements from constraint preconditioning. A local mode analysis shows that for discrete optimality systems, we can expect smoothing rates close to those obtained with respect to the underlying constraint PDE. Our numerical experiments include problems with constraints where standard pointwise smoothing is known to fail for the underlying PDE. In particular, we consider anisotropic diffusion and convection-diffusion problems. The framework of our method allows to include line smoothers or ILU-factorizations, which are suitable for such problems. In all cases, numerical experiments show that convergence rates do not depend on the mesh size of the finest level and discrete optimality systems can be solved with a small multiple of the computational cost which is required to solve the underlying constraint PDE. Employing the full multigrid approach, the computational cost is proportional to the number of unknowns on the finest grid level. We discuss the role of the regularization parameter in the cost functional and show that the convergence rates are robust with respect to both the fine grid mesh size and the regularization parameter under a mild restriction on the next to coarsest mesh size. Incorporating spectral filtering for the reduced Hessian in the control smoothing step allows us to weaken the mesh size restriction. As a result, problems with near-vanishing regularization parameter can be treated efficiently with a negligible amount of additional computational work. For fine discretizations, robust convergence is obtained with rates which are independent of the regularization parameter, the coarsest mesh size, and the number of levels. In order to treat linear-quadratic problems with pointwise inequality constraints on the control, the multigrid approach is modified to solve subproblems generated by a primal-dual active set strategy (PDAS). Numerical experiments demonstrate the high efficiency of this approach due to mesh-independent convergence of both the outer PDAS method and the inner multigrid solver. The PDAS-multigrid method is incorporated in the sequential quadratic programming (SQP) framework. Inexact Newton techniques further enhance the computational efficiency. Globalization is implemented with a line search based on the augmented Lagrangian merit function. Numerical experiments highlight the efficiency of the resulting SQP-multigrid approach. In all cases, locally superlinear convergence of the SQP method is observed. In combination with the mesh-independent convergence rate of the inner solver, a solution method with optimal efficiency is obtained