Dobson M.,CERMICS |
Dobson M.,University of Minnesota |
Luskin M.,University of Minnesota |
Ortner C.,Mathematical Institute
Archive for Rational Mechanics and Analysis | Year: 2010
Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are popular tools in computational physics. For example, the force-based quasicontinuum (QCF) approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the QCF approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact that the linearized QCF operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the W1,p-W1,q "duality pairing" with 1/p + 1/q = 1, if 1 ≤ p < ∞. However, we were able to establish an inf-sup stability condition for a discrete version of the W1,∞-W1,1 "duality pairing" which leads to optimal order error estimates in a discrete W1,∞-norm. © 2009 Springer-Verlag.
Oberhuber T.,Czech Technical University |
Suzuki A.,CERMICS |
Zabka V.,Czech Technical University
Kybernetika | Year: 2011
We study the use of a GPU for the numerical approximation of the curvature dependent flows of graphs - the mean-curvature flow and the Willmore flow. Both problems are often applied in image processing where fast solvers are required. We approximate these problems using the complementary finite volume method combined with the method of lines. We obtain a system of ordinary differential equations which we solve by the Runge-Kutta-Merson solver. It is a robust solver with an automatic choice of the integration time step. We implement this solver on CPU but also on GPU using the CUDA toolkit. We demonstrate that the mean-curvature flow can be successfully approximated in single precision arithmetic with the speed-up almost 17 on the Nvidia GeForce GTX 280 card compared to Intel Core 2 Quad CPU. On the same card, we obtain the speed-up 7 in double precision arithmetic which is necessary for the fourth order problem - the Willmore flow of graphs. Both speed-ups were achieved without affecting the accuracy of the approximation. The article is structured in such way that the reader interested only in the implementation of the Runge-Kutta-Merson solver on the GPU can skip the sections containing the mathematical formulation of the problems.
Blanc X.,University Pierre and Marie Curie |
Le Bris C.,CERMICS |
Le Bris C.,French Institute for Research in Computer Science and Automation
Networks and Heterogeneous Media | Year: 2010
In quasi-periodic homogenization of elliptic equations or nonlinear periodic homogenization of systems, the cell problem must be in general set on the whole space. Numerically computing the homogenization coefficient therefore implies a truncation error, due to the fact that the problem is approximated on a bounded, large domain. We present here an approach that improves the rate of convergence of this approximation. © American Institute of Mathematical Sciences.
Lelievre T.,CERMICS |
Lelievre T.,French Institute for Research in Computer Science and Automation |
Minoukadeh K.,CERMICS |
Minoukadeh K.,French Institute for Research in Computer Science and Automation
Archive for Rational Mechanics and Analysis | Year: 2011
We present convergence results for an adaptive algorithm to compute free energies, namely the adaptive biasing force (ABF) method (Darve and Pohorille in J Chem Phys 115(20):9169-9183, 2001; Hénin and Chipot in J Chem Phys 121:2904, 2004). The free energy is the effective potential associated to a so-called reaction coordinate ξ(q), where q = (q1, ..., q3N) is the position vector of an N-particle system. Computing free energy differences remains an important challenge in molecular dynamics due to the presence of metastable regions in the potential energy surface. The ABF method uses an on-the-fly estimate of the free energy to bias dynamics and overcome metastability. Using entropy arguments and logarithmic Sobolev inequalities, previous results have shown that the rate of convergence of the ABF method is limited by the metastable features of the canonical measures conditioned to being at fixed values of ξ (Lelièvre et al. in Nonlinearity 21(6):1155-1181, 2008). In this paper, we present an improvement on the existing results in the presence of such metastabilities, which is a generic case encountered in practice. More precisely, we study the so-called bi-channel case, where two channels along the reaction coordinate direction exist between an initial and final state, the channels being separated from each other by a region of very low probability. With hypotheses made on 'channel-dependent' conditional measures, we show on a bi-channel model, which we introduce, that the convergence of the ABF method is, in fact, not limited by metastabilities in directions orthogonal to ξ under two crucial assumptions: (i) exchange between the two channels is possible for some values of ξ and (ii) the free energy is a good bias in each channel. This theoretical result supports recent numerical experiments (Minoukadeh et al. in J Chem Theory Comput 6:1008-1017, 2010), where the efficiency of the ABF approach is demonstrated for such a multiple-channel situation. © 2011 Springer-Verlag.
Allaire G.,Ecole Polytechnique - Palaiseau |
Cances E.,French Institute for Research in Computer Science and Automation |
Structural and Multidisciplinary Optimization | Year: 2016
In this paper we introduce a new variant of shape differentiation which is adapted to the deformation of shapes along their normal direction. This is typically the case in the level-set method for shape optimization where the shape evolves with a normal velocity. As all other variants of the original Hadamard method of shape differentiation, our approach yields the same first order derivative. However, the Hessian or second-order derivative is different and somehow simpler since only normal movements are allowed. The applications of this new Hessian formula are twofold. First, it leads to a novel extension method for the normal velocity, used in the Hamilton-Jacobi equation of front propagation. Second, as could be expected, it is at the basis of a Newton optimization algorithm which is conceptually simpler since no tangential displacements have to be considered. Numerical examples are given to illustrate the potentiality of these two applications. The key technical tool for our approach is the method of bicharacteristics for solving Hamilton-Jacobi equations. Our new idea is to differentiate the shape along these bicharacteristics (a system of two ordinary differential equations). © 2016 Springer-Verlag Berlin Heidelberg