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55
The Adaptive Multilevel Finite Element Solution of the PoissonBoltzmann Equation on Massively Parallel Computers
 J. COMPUT. CHEM
, 2000
"... Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element soluti ..."
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Cited by 89 (17 self)
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Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element solution of the PoissonBoltzmann equation for a microtubule on the NPACI IBM Blue Horizon supercomputer. The microtubule system is 40 nm in length and 24 nm in diameter, consists of roughly 600,000 atoms, and has a net charge of1800 e. PoissonBoltzmann calculations are performed for several processor configurations and the algorithm shows excellent parallel scaling.
Adaptive numerical treatment of elliptic systems on manifolds
 Advances in Computational Mathematics, 15(1):139
, 2001
"... ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element ..."
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Cited by 57 (26 self)
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ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2 and 3manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2 and 3manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unitybased method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4component covariant elliptic system on a Riemannian 3manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasioptimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.
The Auxiliary Space Method And Optimal Multigrid Preconditioning Techniques For Unstructured Grids
 Computing
, 1996
"... . An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxi ..."
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Cited by 53 (5 self)
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. An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique make it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris ...
Domain decomposition for multiscale PDEs
 Numer. Math
"... We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises of ..."
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Cited by 50 (19 self)
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We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in heterogeneous porous media, in both the deterministic and (MonteCarlo simulated) stochastic cases. We consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid. We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains,
Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes, Electron
 MR 95i:65173 Zbl 0852.65108
, 1994
"... Abstract. Multigrid and domain decomposition methods have proven to be versatile methods for the iterative solution of linear and nonlinear systems of equations arising from the discretization of partial differential equations. The efficiency of these methods derives from the use of a grid hierarchy ..."
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Cited by 41 (0 self)
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Abstract. Multigrid and domain decomposition methods have proven to be versatile methods for the iterative solution of linear and nonlinear systems of equations arising from the discretization of partial differential equations. The efficiency of these methods derives from the use of a grid hierarchy. In some applications to problems on unstructured grids, however, no natural multilevel structure of the grid is available and thus must be generated as part of the solution procedure. In this paper, we consider the problem of generating a multilevel grid hierarchy when only a fine, unstructured grid is given. We restrict attention to problems in two dimensions. Our techniques generate a sequence of coarser grids by first forming a maximal independent set of the graph of the grid or its dual and then applying a Cavendish type algorithm to form the coarser triangulation. Iterates on the different levels are combined using standard interpolation and restriction operators. Numerical tests indicate that convergence using this approach can be as fast as standard multigrid and domain decomposition methods on a structured mesh.
An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advectiondiffusion problems
 Math. Comp
"... Abstract. We consider a scalar advection{diusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous nite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, nonsymmetric linear s ..."
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Cited by 30 (0 self)
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Abstract. We consider a scalar advection{diusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous nite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, nonsymmetric linear system, we propose and study an additive, two{level overlapping Schwarz preconditioner, consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems dened on a family of subdomains. This is a generalization of the corresponding overlapping method for approximations on continuous nite element spaces. Related to the lack of continuity of our approximation spaces, some interesting new features arise in our generalization, which have no analog in the conforming case. We prove an upper bound for the number of iterations obtained by using this preconditioner with GMRES, which is independent of the number of degrees of freedom of the original problem and the number of subdomains. The performance of the method is illustrated by several numerical experiments for dierent test problems, using linear nite elements in two dimensions.
THE USE OF POINTWISE INTERPOLATION IN DOMAIN DECOMPOSITION METHODS WITH NONNESTED MESHES
, 1995
"... In this paper, we develop a new technique, and a corresponding theory, for Schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from nite element discretization of elliptic partial di erential equations. The theory provides an optimal convergence ..."
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Cited by 29 (5 self)
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In this paper, we develop a new technique, and a corresponding theory, for Schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from nite element discretization of elliptic partial di erential equations. The theory provides an optimal convergence of an additive Schwarz algorithm that is constructed with a nonnested coarse space, and a not necessarily shape regular subdomain partitioning. The theory is also applicable to the graph partitioning algorithms recently developed, [5, 15], for problems de ned on unstructured meshes.
Additive Schwarz Domain Decomposition Methods For Elliptic Problems On Unstructured Meshes
 Numerical Algorithms
, 1994
"... . We give several additive Schwarz domain decomposition methods for solving finite element problems which arise from the discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Our theory requires no assumption (for the main results) on the substructures whic ..."
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Cited by 28 (13 self)
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. We give several additive Schwarz domain decomposition methods for solving finite element problems which arise from the discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Our theory requires no assumption (for the main results) on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be nonnested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasiuniform. In this general setting, our algorithms have the same optimal convergence rate of the usual domain decomposition methods on structured meshes. The condition numbers of the preconditoned systems depend only on the (possibly small) overlap of the substructures and the size of the coarse grid, but is independent of the sizes of the subdomains. Key Words. Unstructured meshes, nonnested coarse meshes, additive ...
Parallel Computation of Flow in Heterogeneous Media Modelled by Mixed Finite Elements
"... In this paper we describe a fast parallel method for highly illconditioned saddlepoint systems arising from mixed finite element simulations of stochastic partial differential equations (PDEs) modelling flow in heterogeneous media. Each realisation of these stochastic PDEs requires the solution of ..."
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Cited by 25 (20 self)
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In this paper we describe a fast parallel method for highly illconditioned saddlepoint systems arising from mixed finite element simulations of stochastic partial differential equations (PDEs) modelling flow in heterogeneous media. Each realisation of these stochastic PDEs requires the solution of the linear firstorder velocitypressure system comprising Darcy's law coupled with an incompressibility constraint. The chief difficulty is that the permeability may be highly variable, especially when the statistical model has a large variance and a small correlation length. For reasonable accuracy, the discretisation has to be extremely fine. We solve these problems by first reducing the saddlepoint formulation to a symmetric positive definite (SPD) problem using a suitable basis for the space of divergencefree velocities. The reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive Schwarz domain decomposition preconditioner. The resu...
Iterative Choices of Regularization Parameters in Linear Inverse Problems
 Inverse Problem
, 1998
"... We investigate possibilities of choosing reasonable regularization parameters in an efficient manner and of estimating the observation errors in linear inverse problems. Numerical experiments are presented to illustrate the efficiency of the proposed algorithms. Mathematics Subject Classification ( ..."
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Cited by 23 (4 self)
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We investigate possibilities of choosing reasonable regularization parameters in an efficient manner and of estimating the observation errors in linear inverse problems. Numerical experiments are presented to illustrate the efficiency of the proposed algorithms. Mathematics Subject Classification (1991): 35R30, 49K 1 Introduction In this paper, we consider inverse problems of the form Tf = z; (1.1) where T is a bounded operator mapping the parameter space X into the observation space Y . Here z 2 Y are the observation data which may be corrupted by error. The noisy data with noise level ffi are denoted by z ffi . The above problem is often illposed due to lack of a continuous inverse of T so that small perturbations in the data can have large effects on the solution f of (1.1). To make a numerical resolution feasible some type of regularization has to be introduced. Here we focus on one of the most frequently used methods given by expressing the inverse problem as the penalized ...