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The paper considers the time integration of frictionless dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Among the numerical integrators, interest focuses on the contact-stabilized Newmark method recently suggested by Deuflhard et al., which is compared to the classical Newmark method and an improved energy dissipative version due to Kane et al. In the absence of contact, any such variant is equivalent to the Störmer-Verlet scheme, which is well-known to have consistency order 2. In the presence of contact, however, the classical approach to discretization errors would not show consistency at all because of the discontinuity at the contact. Surprisingly, the question of consistency in the constrained situation has not been solved yet. The present paper fills this gap by means of a novel proof technique using specific norms based on earlier perturbation results due to the authors. The corresponding estimation of the local discretization error requires the bounded total variation of the solution. The results have consequences for the construction of an adaptive timestep control, which will be worked out subsequently in a forthcoming paper.
We present an extension module for the Dune system. This module, called dune-subgrid, allows to mark elements of another Dune hierarchical grid. The set of marked elements can then be accessed as a Dune grid in its own right. dune-subgrid is free software and is available for download. We describe the functionality and use of dune-subgrid, comment on its implementation, and give two example applications.
First, we show how dune-subgrid can be used for micro-FE simulations of trabecular bone. Then we present an algorithm that allows to use exact residuals for the adaptive solution of the spatial problems of time-discretized evolution equations.
We present a hierarchical a~posteriori error analysis for the
minimum value of the energy functional in symmetric obstacle
problems. The main result is that the energy of the exact solution
is, up to data oscillation, equivalent to an appropriate
hierarchical estimator. The proof of the main result does not
invoke any saturation assumption. Moreover, we prove an a
posteriori error estimate indicating that the estimator from
\cite{RHWHoppe_RKornhuber_1994a} is asymptotically reliable and we
give sufficient conditions for the validity of a saturation
assumption. Finally, we corroborate and complement our theoretical
results with numerical experiments.
We introduce geodesic finite elements as a new way to discretize
the nonlinear configuration space of a geometrically exact Cosserat rod.
These geodesic finite elements naturally generalize standard one-dimensional
finite elements to spaces of functions with values in a Riemannian manifold.
For the special orthogonal group, our approach reproduces the
interpolation formulas of [Crisfield/Jelenic:1999].
Geodesic finite elements are
conforming and lead to objective and path-independent problem formulations.
We introduce geodesic finite elements for general Riemannian manifolds,
discuss the relationship between geodesic finite elements and
coefficient vectors, and estimate the interpolation error.
Then we use them to find static equilibria of hyperelastic Cosserat rods.
Using the Riemannian trust-region algorithm of [Absil/Mahony/Sepulchre:2008]
we show numerically that the discretization error depends optimally on
the mesh size.
The identification of metastable conformations of molecules plays an
important role in computational drug design. One main difficulty is the
fact that the underlying dynamic processes take place in high dimensional
spaces. Although the restriction of degrees of freedom to a few dihedral
angles significantly reduces the complexity of the problem, the existing
algorithms are time-consuming. They are mainly based on the approximation
of a transfer operator by an extensive sampling of states according
to the Boltzmann distribution and short-time Hamiltonian dynamics simulations.
We present a method which can identify metastable conformations
without sampling the complete distribution. Our algorithm is based
on local transition rates and uses only pointwise information about the
potential energy surface. In order to apply the cluster algorithm PCCA+,
we compute a few eigenvectors of the rate matrix by the Jacobi-Davidson
method. Interpolation techniques are applied to approximate the thermodynamical
weights of the clusters. The concluding example illustrates
our approach for epigallocatechine, a molecule which can be described by
seven dihedral angles.
In order to compute the thermodynamic weights of the different metastable conformations
of a molecule, we want to approximate the molecule’s Boltzmann distribution in a reasonable
time. This is an essential issue in computational drug design. The energy landscape of active
biomolecules is generally very rough with a lot of high barriers and low regions. Many of the
algorithms that perform such samplings (e.g. the hybrid Monte Carlo method) have difficulties
with such landscapes. They are trapped in low-energy regions for a very long time and cannot
overcome high barriers. Moving from one low-energy region to another is a very rare event. For
these reasons, the distribution of the generated sampling points converges very slowly against
the thermodynamically correct distribution of the molecule.
The idea of ConfJump is to use a priori knowledge of the localization of low-energy regions
to enhance the sampling with artificial jumps between these low-energy regions. The artificial
jumps are combined with the hybrid Monte Carlo method. This allows the computation of
some dynamical properties of the molecule. In ConfJump, the detailed balance condition is
satisfied and the mathematically correct molecular distribution is sampled.
Biochemical interactions are determined by the 3D-structure of the involved components –
thus the identification of conformations is a key for many applications in rational drug design.
ConFlow is a new multilevel approach to conformational analysis with main focus on
completeness in investigation of conformational space.
In contrast to known conformational analysis, the starting point for design is a space-based
description of conformational areas. A tight integration of sampling and analysis leads to an
identification of conformational areas simultaneously during sampling. An incremental
decomposition of high-dimensional conformational space is used to guide the analysis. A new
concept for the description of conformations and their path connected components based on
convex hulls and Hypercubes is developed. The first results of the ConFlow application
constitute a ‘proof of concept’ and are further more highly encouraging. In comparison to
conventional industrial applications, ConFlow achieves higher accuracy and a specified
degree of completeness with comparable effort.
In this paper, we investigate the interconversion processes of the major flame retardant - 1,2,5,6,9,10-hexabromocyclododecane (HBCD) - by the means of statistical thermodynamics based on classical force-fields. Three ideas will be presented. First, the application of classical hybrid Monte-Carlo simulations for quantum mechanical processes will be justified. Second, the problem of insufficient convergence properties of hybrid Monte-Carlo methods for the generation of low temperature canonical ensembles will be solved by an interpolation approach. Furthermore, it will be shown how free energy differences can be used for a rate matrix computation. The results of our numerical simulations will be compared to experimental results.
Keywords: Markov process, molecular dynamics, rate matrix
This article deals with an efficient sampling of the stationary distribution
of dynamical systems in the presence of metastabilities. For such
systems, standard sampling schemes suffer from trapping problems and
critical slowing down. Starting multiple trajectories in different regions of
the sampling space is a promising way out. The different samplings represent
the stationary distribution locally very well, but are still far away
from ergodicity or from the global stationary distribution. We will show
how these samplings can be joined together in order to get one global
sampling of the stationary distribution.
For the treatment of equilibrated molecular systems in a heat bath we
propose a transition state theory that is based on conformation dynamics.
In general, a set-based discretization of a Markov operator P does not
preserve the Markov property. In this article, we propose a discretization
method which is based on a Galerkin approach. This discretization
method preserves the Markov property of the operator and can be interpreted
as a decomposition of the state space into (fuzzy) sets. The
conformation-based transition state theory presented here can be seen as
a first step in conformation dynamics towards the computation of essential
dynamical properties of molecular systems without time-consuming
molecular dynamics simulations.
For an analysis of a molecular system from a computational statistical thermodynamics point of view, extensive
molecular dynamics simulations are very inefficient. During this procedure, at lot of redundant data is generated. Whereas
the algorithms spend most of the computing time for a sampling of configurations within the basins of the potential energy
landscape of the molecular system, the important information about the long-time behaviour of the molecules is given by
transition regions and barriers between the basins, which are sampled rarely only. Thinking of molecular dynamics trajectories,
researchers try to figure out which kind of dynamical model is suitable for an efficient simulation. This article suggests to
change the point of view from extensive simulation of molecular dynamics trajectories to more efficient sampling strategies
of the conformation dynamics approach.
Deformable surface models are often represented as triangular meshes in image segmentation applications. For a fast and easily regularized deformation onto the target object boundary, the vertices of the mesh are commonly moved along line segments (typically surface normals). However, in case of high mesh curvature, these lines may intersect with the target boundary at “non-corresponding” positions, or may not intersect at all. Consequently, certain deformations cannot be achieved. We propose omnidirectional displacements for deformable surfaces (ODDS) to overcome this limitation. ODDS allow each vertex to move not only along a line segment but within a surrounding sphere, and achieve globally optimal deformations subject to local regularization con-
straints. However, allowing a ball-shaped instead of a linear range of motion per vertex significantly increases runtime and memory. To alleviate this drawback, we propose a hybrid approach, fastODDS, with improved runtime and reduced memory requirements. Furthermore, fastODDS can also cope with simultaneous segmentation of multiple objects. We show the theoretical benefits of ODDS with experiments on synthetic data, and evaluate ODDS and fastODDS quantitatively on clinical image data of the mandible and the hip bones. There, we assess both the global segmentation accuracy as well as local accuracy in high curvature regions, such as the tip-shaped mandibular coronoid processes and the ridge-shaped acetabular rims of
the hip bones.
The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces
(2012)
The paper continues the study of the regularity of electronic wave functions in Hilbert spaces of mixed derivatives. It is shown that the eigenfunctions of electronic Schr\"odinger operators and their
exponentially weighted counterparts possess, roughly speaking, square integrable mixed weak derivatives of fractional order $\vartheta$ for $\vartheta<3/4$. The bound $3/4$ is best possible and can neither be reached nor surpassed. Such results are important for the study
of sparse grid-like expansions of the wave functions and show that their asymptotic convergence rate measured in terms of the number of ansatz functions involved does not deteriorate with the number of electrons.
We study the expansion of the eigenfunctions of Schrödinger operators with smooth confinement potentials in Hermite functions; confinement potentials are potentials that become unbounded at infinity. The key result is that such eigenfunctions and all their derivatives decay more rapidly than any exponential function under some mild growth
conditions to the potential and its derivatives. Their expansion in Hermite functions converges therefore very fast, super-algebraically.
Particle methods have become indispensible in conformation dynamics to compute transition rates in protein folding, binding processes and molecular design, to mention a few. Conformation dynamics requires at a decomposition of a molecule's position space into metastable conformations. In this paper, we show how this decomposition can be obtained via the design of either ``soft'' or ``hard'' molecular conformations. We show, that the soft approach results in a larger metastabilitiy of the decomposition and is thus more advantegous. This is illustrated by a simulation of Alanine Dipeptide.
Convergence Analysis of Smoothing Methods for Optimal Control of Stationary Variational Inequalities
(2012)
In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise
control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to
the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained. These rates coincide with
rates obtained by numerical experiments, which are included in the paper.
We derive a-priori estimates on the length of the primal-dual path that results from a Moreau-Yosida approximation of the feasible set for state constrained optimal control problems. These bounds depend on the regularity of the state and the dimension of the problem. Comparison with numerical results indicates that these bounds are sharp and are attained for the case of a single active point.
We present a unified computational framework for matching 3d geometric objects (points, lines, surfaces, volumes) of highly varying shape. Our approach is based on the Large Deformation Diffeomorphic Metric Mapping (LDDMM) method acting on $m$-currents. After stating an optimization algorithm in the function space of admissible morph generating velocity fields, two innovative aspects in this framework are presented: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Secondly, we directly compute the temporal evolution of discrete $m$-current attributes. Several numerical experiments demonstrate the effectiveness of this approach.
Genome-scale metabolic networks are useful tools for achieving a system-level understanding of metabolism. However, due to their large size, analysis of such networks may be difficult and algorithms can be very slow. Therefore, some authors have suggested to analyze subsystems instead of the original genome-scale models.
Flux coupling analysis (FCA) is a well-known method for detecting functionally related reactions in metabolic networks.
In this paper, we study how flux coupling relations may change if we analyze a subsystem instead of the original network. We show mathematically that a pair of fully, partially or directionally coupled reactions may be detected as uncoupled in certain subsystems. Interestingly, this behavior is the opposite of the flux coupling changes that may occur due to missing reactions, or equivalently, deletion of reactions.
Computational experiments suggest that the analysis of plastid (but not mitochondrial) subsystems may significantly influence the results of FCA. Therefore, the results of FCA for subsystems, especially plastid subsystems, should be interpreted with care.