Michael Griebel

Bio: Michael Griebel is an academic researcher from University of Bonn. The author has contributed to research in topics: Sparse grid & Discretization. The author has an hindex of 49, co-authored 214 publications receiving 9748 citations. Previous affiliations of Michael Griebel include Fraunhofer Society & Technische Universität München.

Numerical integration using sparse grids

Abstract: We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49] In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction

Molecular Simulation of the Influence of Chemical Cross-Links on the Shear Strength of Carbon Nanotube-Polymer Interfaces

Abstract: The influence of chemical cross-links between a single-walled fullerene nanotube and a polymer matrix on the matrix−nanotube shear strength has been studied using molecular dynamics simulations. A (10,10) nanotube embedded in either a crystalline or amorphous polyethylene matrix is used as a model for a nonbonded interface (in the absence of cross-links). The simulations predict that shear strengths and critical lengths required for load transfer can be enhanced and decreased, respectively, by over an order of magnitude with the formation of cross-links involving less than 1% of the nanotube carbon atoms. At this level of chemical functionalization, calculations also predict that there is a negligible change in tensile modulus for a (10,10) nanotube.

Dimension-adaptive tensor-product quadrature

Abstract: We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate-dimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.

Numerical Simulation in Fluid Dynamics: A Practical Introduction

Abstract: Notation 1. Numerical simulation - a key technology of the future 2. The mathematical description of flows 3. The numerical treatment of the Navier-Stokes equations 4. Visualization techniques 5. Example applications 6. Free boundary value problems 7. Example applications for free boundary value problems 8. Parallelization 9. Energy transport 10. Turbulence 11. Extension to Three dimensions 12. Concluding remarks Appendix A Appendix B Bibliography Index.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Fast parallel algorithms for short-range molecular dynamics

Abstract: Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

PACKMOL: a package for building initial configurations for molecular dynamics simulations.

Abstract: Adequate initial configurations for molecular dynamics simulations consist of arrangements of molecules distributed in space in such a way to approximately represent the system's overall structure. In order that the simulations are not disrupted by large van der Waals repulsive interactions, atoms from different molecules must keep safe pairwise distances. Obtaining such a molecular arrangement can be considered a packing problem: Each type molecule must satisfy spatial constraints related to the geometry of the system, and the distance between atoms of different molecules must be greater than some specified tolerance. We have developed a code able to pack millions of atoms, grouped in arbitrarily complex molecules, inside a variety of three-dimensional regions. The regions may be intersections of spheres, ellipses, cylinders, planes, or boxes. The user must provide only the structure of one molecule of each type and the geometrical constraints that each type of molecule must satisfy. Building complex mixtures, interfaces, solvating biomolecules in water, other solvents, or mixtures of solvents, is straightforward. In addition, different atoms belonging to the same molecule may also be restricted to different spatial regions, in such a way that more ordered molecular arrangements can be built, as micelles, lipid double-layers, etc. The packing time for state-of-the-art molecular dynamics systems varies from a few seconds to a few minutes in a personal computer. The input files are simple and currently compatible with PDB, Tinker, Molden, or Moldy coordinate files. The package is distributed as free software and can be downloaded from http://www.ime.unicamp.br/~martinez/packmol/.