The Activation-Relaxation Technique : ART nouveau and kinetic ART

The evolution of many systems is dominated by rare activated events that occur on timescale ranging from nanoseconds to the hour or more. For such systems, simulations must leave aside the full thermal description to focus specifically on mechanisms that generate a configurational change. We present here the activation relaxation technique (ART), an open-ended saddle point search algorithm, and a series of recent improvements to ART nouveau and kinetic ART, an ART-based on-the-fly off-lattice self-learning kinetic Monte Carlo method

On composite systems of dilute and dense couplings

Composite systems, where couplings are of two types, a combination of strong dilute and weak dense couplings of Ising spins, are examined through the replica method. The dilute and dense parts are considered to have independent canonical disordered or uniform bond distributions; mixing the models by variation of a parameter $\gamma$ alongside inverse temperature $\beta$ we analyse the respective thermodynamic solutions. We describe the variation in high temperature transitions as mixing occurs; in the vicinity of these transitions we exactly analyse the competing effects of the dense and sparse models. By using the replica symmetric ansatz and population dynamics we described the low temperature behaviour of mixed systems.Comment: 35 pages, 9 figures, submitted to JPhys

Asymptotic Level Density of the Elastic Net Self-Organizing Feature Map

Whileas the Kohonen Self Organizing Map shows an asymptotic level density following a power law with a magnification exponent 2/3, it would be desired to have an exponent 1 in order to provide optimal mapping in the sense of information theory. In this paper, we study analytically and numerically the magnification behaviour of the Elastic Net algorithm as a model for self-organizing feature maps. In contrast to the Kohonen map the Elastic Net shows no power law, but for onedimensional maps nevertheless the density follows an universal magnification law, i.e. depends on the local stimulus density only and is independent on position and decouples from the stimulus density at other positions.Comment: 8 pages, 10 figures. Link to publisher under http://link.springer.de/link/service/series/0558/bibs/2415/24150939.ht

Composite CDMA - A statistical mechanics analysis

Code Division Multiple Access (CDMA) in which the spreading code assignment to users contains a random element has recently become a cornerstone of CDMA research. The random element in the construction is particular attractive as it provides robustness and flexibility in utilising multi-access channels, whilst not making significant sacrifices in terms of transmission power. Random codes are generated from some ensemble, here we consider the possibility of combining two standard paradigms, sparsely and densely spread codes, in a single composite code ensemble. The composite code analysis includes a replica symmetric calculation of performance in the large system limit, and investigation of finite systems through a composite belief propagation algorithm. A variety of codes are examined with a focus on the high multi-access interference regime. In both the large size limit and finite systems we demonstrate scenarios in which the composite code has typical performance exceeding sparse and dense codes at equivalent signal to noise ratio.Comment: 23 pages, 11 figures, Sigma Phi 2008 conference submission - submitted to J.Stat.Mec

Simultaneous solution approaches for large optimization problems

AbstractIn this paper, efficient simultaneous strategies are presented for the optimization of practical problems involving PDE-models. In particular, reduced sequential quadratic programming methods for problems with only few influence variables and simultaneous quadratic programming iterations are discussed. As a result we obtain algorithms whose overall computational complexity is reduced considerably in comparison to a black-box approach

Shape Optimization by Constrained First-Order Least Mean Approximation

In this work, the problem of shape optimization, subject to PDE constraints, is reformulated as an $L^p$ best approximation problem under divergence constraints to the shape tensor introduced in Laurain and Sturm: ESAIM Math. Model. Numer. Anal. 50 (2016). More precisely, the main result of this paper states that the $L^p$ distance of the above approximation problem is equal to the dual norm of the shape derivative considered as a functional on $W^{1,p^\ast}$ (where $1/p + 1/p^\ast = 1$). This implies that for any given shape, one can evaluate its distance from being a stationary one with respect to the shape derivative by simply solving the associated $L^p$-type least mean approximation problem. Moreover, the Lagrange multiplier for the divergence constraint turns out to be the shape deformation of steepest descent. This provides a way, as an alternative to the approach by Deckelnick, Herbert and Hinze: ESAIM Control Optim. Calc. Var. 28 (2022), for computing shape gradients in $W^{1,p^\ast}$ for $p^\ast \in ( 2 , \infty )$. The discretization of the least mean approximation problem is done with (lowest-order) matrix-valued Raviart-Thomas finite element spaces leading to piecewise constant approximations of the shape deformation acting as Lagrange multiplier. Admissible deformations in $W^{1,p^\ast}$ to be used in a shape gradient iteration are reconstructed locally. Our computational results confirm that the $L^p$ distance of the best approximation does indeed measure the distance of the considered shape to optimality. Also confirmed by our computational tests are the observations that choosing $p^\ast$ (much) larger than 2 (which means that $p$ must be close to 1 in our best approximation problem) decreases the chance of encountering mesh degeneracy during the shape gradient iteration.Comment: 20 pages, 8 figure

Discussion of "Geodesic Monte Carlo on Embedded Manifolds"

Contributed discussion and rejoinder to "Geodesic Monte Carlo on Embedded Manifolds" (arXiv:1301.6064)Comment: Discussion of arXiv:1301.6064. To appear in the Scandinavian Journal of Statistics. 18 page

Calculations of Excited Electronic States by Converging on Saddle Points Using Generalized Mode Following

Variational calculations of excited electronic states are carried out by finding saddle points on the surface that describes how the energy of the system varies as a function of the electronic degrees of freedom. This approach has several advantages over commonly used methods especially in the context of density functional calculations, as collapse to the ground state is avoided and yet, the orbitals are variationally optimized for the excited state. This optimization makes it possible to describe excitations with large charge transfer where calculations based on ground state orbitals are problematic, as in linear response time-dependent density functional theory. A generalized mode following method is presented where an $n^{\text{th}}$-order saddle point is found by inverting the components of the gradient in the direction of the eigenvectors of the $n$ lowest eigenvalues of the electronic Hessian matrix. This approach has the distinct advantage of following a chosen excited state through atomic configurations where the symmetry of the single determinant wave function is broken, as demonstrated in calculations of potential energy curves for nuclear motion in the ethylene and dihydrogen molecules. The method is implemented using a generalized Davidson algorithm and an exponential transformation for updating the orbitals within a generalized gradient approximation of the energy functional. Convergence is found to be more robust than for a direct optimization approach previously shown to outperform standard self-consistent field approaches, as illustrated here for charge transfer excitations in nitrobenzene and N-phenylpyrrole, involving calculations of $4^{\text{th}}$- and $6^{\text{th}}$-order saddle points, respectively. Finally, calculations of a diplatinum and silver complex are presented, illustrating the applicability of the method to excited state energy curves of large molecules.Comment: 57 pages, 12 figures, submitted to the Journal of Chemical Theory and Computatio

The topography of multivariate normal mixtures

Multivariate normal mixtures provide a flexible method of fitting high-dimensional data. It is shown that their topography, in the sense of their key features as a density, can be analyzed rigorously in lower dimensions by use of a ridgeline manifold that contains all critical points, as well as the ridges of the density. A plot of the elevations on the ridgeline shows the key features of the mixed density. In addition, by use of the ridgeline, we uncover a function that determines the number of modes of the mixed density when there are two components being mixed. A followup analysis then gives a curvature function that can be used to prove a set of modality theorems.Comment: Published at http://dx.doi.org/10.1214/009053605000000417 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org