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46
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 90 (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.
Local and parallel finite element algorithms based on twogrid discretizations
 Math. Comput
"... Abstract. A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components ..."
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Cited by 38 (13 self)
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Abstract. A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for finite element solutions on general shaperegular grids. Some numerical experiments are also presented to support the theory. 1.
FeaturePreserving Adaptive Mesh Generation for Molecular Shape Modeling and Simulation
, 2007
"... We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a n ..."
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Cited by 18 (7 self)
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We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a number of molecules are demonstrated, showing that our methods possess several desirable properties: featurepreservation, local adaptivity, high quality, and smoothness (for surface meshes). We also demonstrate an example of molecular simulation using the finite element method and the meshes generated by our method. The approaches presented and their implementations are also applicable to other types of inputs such as 3D scalar volumes and triangular surface meshes with low quality, and hence can be used for generation/improvment of meshes in a broad range of applications.
Parallel geometric multigrid
 Numerical Solution of Partial Differential Equations on Parallel Computers, volume 51 of LNCSE, chapter 5
, 2005
"... Summary. Multigrid methods are among the fastest numerical algorithms for the solution of large sparse systems of linear equations. While these algorithms exhibit asymptotically optimal computational complexity, their efficient parallelisation is hampered by the poor computationtocommunication rat ..."
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Cited by 15 (6 self)
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Summary. Multigrid methods are among the fastest numerical algorithms for the solution of large sparse systems of linear equations. While these algorithms exhibit asymptotically optimal computational complexity, their efficient parallelisation is hampered by the poor computationtocommunication ratio on the coarse grids. Our contribution discusses parallelisation techniques for geometric multigrid methods. It covers both theoretical approaches as well as practical implementation issues that may guide code development. 1
A New Parallel Domain Decomposition Method for the Adaptive Finite Element Solution of Elliptic Partial Differential Equations
, 1999
"... We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solv ..."
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Cited by 13 (10 self)
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We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain  although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in [5], it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus...
Partitioning and Dynamic Load Balancing for the Numerical Solution of Partial Differential Equations
 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS ON PARALLEL COMPUTERS
, 2005
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S.: A new minimization protocol for solving nonlinear PoissonBoltzmann mortar finite element equation
 BIT
, 2007
"... The nonlinear Poisson–Boltzmann equation (PBE) is a widelyused implicit solvent model in biomolecular simulations. This paper formulates a new PBE nonlinear algebraic system from a mortar finite element approximation, and proposes a new minimization protocol to solve it efficiently. In particular ..."
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Cited by 10 (1 self)
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The nonlinear Poisson–Boltzmann equation (PBE) is a widelyused implicit solvent model in biomolecular simulations. This paper formulates a new PBE nonlinear algebraic system from a mortar finite element approximation, and proposes a new minimization protocol to solve it efficiently. In particular, the PBE mortar nonlinear algebraic system is proved to have a unique solution, and is equivalent to a unconstrained minimization problem. It is then solved as the unconstrained minimization problem by the subspace trust region Newton method. Numerical results show that the new minimization protocol is more efficient than the traditional merit least squares approach in solving the nonlinear system. At least 80 percent of the total CPU time was saved for a PBE model problem. AMS subject classification (2000): 65N30, 65H10, 65K10, 9208. Key words: Poisson–Boltzmann equation, mortar finite element, nonlinear system,
HighFidelity Geometric Modeling for Biomedical Applications
"... We describe a combination of algorithms for high fidelity geometric modeling and mesh generation. Although our methods and implementations are applicationneutral, our primary target application is multiscale biomedical models that range in scales across the molecular, cellular, and organ levels. Ou ..."
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Cited by 8 (5 self)
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We describe a combination of algorithms for high fidelity geometric modeling and mesh generation. Although our methods and implementations are applicationneutral, our primary target application is multiscale biomedical models that range in scales across the molecular, cellular, and organ levels. Our software toolchain implementing these algorithms is general in the sense that it can take as input a molecule in PDB/PQR forms, a 3D scalar volume, or a userdefined triangular surface mesh that may have very low quality. The main goal of our work presented is to generate high quality and smooth surface triangulations from the aforementioned inputs, and to reduce the mesh sizes by mesh coarsening. Tetrahedral meshes are also generated for finite element analysis in biomedical applications. Experiments on a number of biostructures are demonstrated, showing that our approach possesses several desirable properties: featurepreservation, local adaptivity, high quality, and smoothness (for surface meshes). The availability of this software toolchain will give researchers in computational biomedicine and other modeling areas access to higherfidelity geometric models.
A domain decomposition solver for a parallel adaptive meshing paradigm
 SIAM J. SCI. COMPUT
"... We describe a domain decomposition algorithm for use in the parallel adaptive meshing paradigm of Bank and Holst. Our algorithm has low communication, makes extensive use of existing sequential solvers, and exploits in several important ways data generated as part of the adaptive meshing paradigm. ..."
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Cited by 6 (1 self)
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We describe a domain decomposition algorithm for use in the parallel adaptive meshing paradigm of Bank and Holst. Our algorithm has low communication, makes extensive use of existing sequential solvers, and exploits in several important ways data generated as part of the adaptive meshing paradigm. Numerical examples illustrate the effectiveness of the procedure.
A parallel algorithm for adaptive local refinement of tetrahedral meshes using bisection
 Numerical Mathematics: Theory, Methods and Applications
"... Local mesh refinement is one of the key steps in implementations of adaptive finite element methods. This paper presents a parallel algorithm for distributed memory parallel computers for adaptive local refinement of tetrahedral meshes using bisection. The algorithm is part of PHG, Parallel Hierarch ..."
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Cited by 5 (2 self)
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Local mesh refinement is one of the key steps in implementations of adaptive finite element methods. This paper presents a parallel algorithm for distributed memory parallel computers for adaptive local refinement of tetrahedral meshes using bisection. The algorithm is part of PHG, Parallel Hierarchical Grid, a toolbox under development for parallel adaptive multigrid solution of PDEs. The algorithm proposed is characterized by allowing simultaneous refinement of submeshes to arbitrary levels before synchronization between submeshes and without the need of a central coordinator process for managing new vertices. Some general properties on local refinement of conforming tetrahedral meshes using bisection are also discussed which are useful in analysing and validating the parallel refinement algorithm as well as in simplifying the implementation.