4,736 research outputs found

    A GPU-accelerated Direct-sum Boundary Integral Poisson-Boltzmann Solver

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    In this paper, we present a GPU-accelerated direct-sum boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov subspace based linear algebraic solver such as the GMRES. The molecular surfaces are discretized with flat triangles and centroid collocation. To speed up our method, we take advantage of the parallel nature of the boundary integral formulation and parallelize the schemes within CUDA shared memory architecture on GPU. The schemes use only 11N+6Nc11N+6N_c size-of-double device memory for a biomolecule with NN triangular surface elements and NcN_c partial charges. Numerical tests of these schemes show well-maintained accuracy and fast convergence. The GPU implementation using one GPU card (Nvidia Tesla M2070) achieves 120-150X speed-up to the implementation using one CPU (Intel L5640 2.27GHz). With our approach, solving PB equations on well-discretized molecular surfaces with up to 300,000 boundary elements will take less than about 10 minutes, hence our approach is particularly suitable for fast electrostatics computations on small to medium biomolecules

    Triangulation of uniform particle systems: its application to the implicit surface texturing

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    Particle systems, as originally presented by Witkin and Heckbert [32], offer an elegant solution to sample implicit surfaces of arbitrary genus, while providing an extremely regular distribution of samples over the surface. In this paper, we present an ef cient technique that uses particle systems to rapidly generate a triangular mesh over an implicit surface, where each triangle is almost equilateral. The major advantage of such a triangulation is that it minimizes the deformations between the mesh and the underlying implicit surface. We exploit this property by using few triangular texture samples mapped in a non-periodic fashion as presented by Neyret and Cani [16]. The result is a pattern-based texturing method that maps homogeneous non-periodic textures to arbitrary implicit surfaces, with almost no deformation

    Finite element methods for surface PDEs

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    In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples

    The century of the incomplete revolution: searching for general relativistic quantum field theory

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    In fundamental physics, this has been the century of quantum mechanics and general relativity. It has also been the century of the long search for a conceptual framework capable of embracing the astonishing features of the world that have been revealed by these two ``first pieces of a conceptual revolution''. I discuss the general requirements on the mathematics and some specific developments towards the construction of such a framework. Examples of covariant constructions of (simple) generally relativistic quantum field theories have been obtained as topological quantum field theories, in nonperturbative zero-dimensional string theory and its higher dimensional generalizations, and as spin foam models. A canonical construction of a general relativistic quantum field theory is provided by loop quantum gravity. Remarkably, all these diverse approaches have turn out to be related, suggesting an intriguing general picture of general relativistic quantum physics.Comment: To appear in the Journal of Mathematical Physics 2000 Special Issu

    Improvements to the APBS biomolecular solvation software suite

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    The Adaptive Poisson-Boltzmann Solver (APBS) software was developed to solve the equations of continuum electrostatics for large biomolecular assemblages that has provided impact in the study of a broad range of chemical, biological, and biomedical applications. APBS addresses three key technology challenges for understanding solvation and electrostatics in biomedical applications: accurate and efficient models for biomolecular solvation and electrostatics, robust and scalable software for applying those theories to biomolecular systems, and mechanisms for sharing and analyzing biomolecular electrostatics data in the scientific community. To address new research applications and advancing computational capabilities, we have continually updated APBS and its suite of accompanying software since its release in 2001. In this manuscript, we discuss the models and capabilities that have recently been implemented within the APBS software package including: a Poisson-Boltzmann analytical and a semi-analytical solver, an optimized boundary element solver, a geometry-based geometric flow solvation model, a graph theory based algorithm for determining pKaK_a values, and an improved web-based visualization tool for viewing electrostatics

    Dynamical Optimal Transport on Discrete Surfaces

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    We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows
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