Finding Saddle Points Research Articles

Improved RGF method to find saddle points.

The predictor-corrector method for following a reduced gradient (RGF) to determine saddle points [Quapp, W. et al., J Comput Chem 1998, 19, 1087] is further accelerated by a modification allowing an implied corrector step per predictor but almost without additional costs. The stability and robustness of the RGF method are improved, and the new version in addition reduces the number of gradient and Hessian calculations.

The Step and Slide method for finding saddle points on multidimensional potential surfaces

We present the Step and Slide method for finding saddle points between two potential-energy minima. The method is applicable when both initial and final states are known. The potential-energy surface is probed by two replicas of the system that converge to the saddle point by following isoenergetic surfaces. The value of the transition-state potential is bracketed in the process, such that a convergence criterion based on the potential can be used. We applied the method to study diffusion mechanisms of a small Ag cluster on a Ag(111) surface using an embedded-atom method potential. Our approach is comparable in efficiency to other commonly used methods.

Characterization of the Activation-Relaxation Technique: Recent Results on Models of Amorphous Silicon

ABSTRACTThe activation-relaxation technique (ART) is a method for finding saddle points in high-dimen- sional energy landscapes. ART has already been applied to a wide range of materials including amorphous semiconductors, Lennard-Jones glasses, and proteins. In spite of its successes, a number of fundamental questions remain to be answered regarding the biases associated with its sampling of the saddle points. We present here results of a detailed analysis of the biases in the simulation of amorphous silicon. We focus in particular on the biases of the method in sampling saddle points, the completeness of the sampling and the sensitivity of these quantities to variations of the different parameters.

Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points

An improved way of estimating the local tangent in the nudged elastic band method for finding minimum energy paths is presented. In systems where the force along the minimum energy path is large compared to the restoring force perpendicular to the path and when many images of the system are included in the elastic band, kinks can develop and prevent the band from converging to the minimum energy path. We show how the kinks arise and present an improved way of estimating the local tangent which solves the problem. The task of finding an accurate energy and configuration for the saddle point is also discussed and examples given where a complementary method, the dimer method, is used to efficiently converge to the saddle point. Both methods only require the first derivative of the energy and can, therefore, easily be applied in plane wave based density-functional theory calculations. Examples are given from studies of the exchange diffusion mechanism in a Si crystal, Al addimer formation on the Al(100) surface, and dissociative adsorption of CH4 on an Ir(111) surface.

A climbing image nudged elastic band method for finding saddle points and minimum energy paths

A modification of the nudged elastic band method for finding minimum energy paths is presented. One of the images is made to climb up along the elastic band to converge rigorously on the highest saddle point. Also, variable spring constants are used to increase the density of images near the top of the energy barrier to get an improved estimate of the reaction coordinate near the saddle point. Applications to CH4 dissociative adsorption on Ir(111) and H2 on Si(100) using plane wave based density functional theory are presented.

EMPHASIZING SADDLE POINTS THROUGH GAME THEORY: A CLASSROOM ACTIVITY

ABSTRACT In optimization problems, often students find a single extremum of a function that is assumed to be a maximum or a minimum. At best, saddle points are discarded when checking second order conditions en route to maxima or minima. Game theory provides a setting where saddle points are the solution concept. We introduce the necessary game-theoretic background and explain how game-theoretic experiments of the Matching Pennies game can be used as a classroom activity to develop intuition about saddle points. Paralleling how students learn to find extrema, we first consider finding saddle points on the interior of a set and then consider saddle-like points that appear on the boundary of compact sets. We conclude with a couple of examples from the game theory literature.

A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives

The problem of determining which activated (and slow) transitions can occur from a given initial state at a finite temperature is addressed. In the harmonic approximation to transition state theory this problem reduces to finding the set of low lying saddle points at the boundary of the potential energy basin associated with the initial state, as well as the relevant vibrational frequencies. Also, when full transition state theory calculations are carried out, it can be useful to know the location of the saddle points on the potential energy surface. A method for finding saddle points without knowledge of the final state of the transition is described. The method only makes use of first derivatives of the potential energy and is, therefore, applicable in situations where second derivatives are too costly or too tedious to evaluate, for example, in plane wave based density functional theory calculations. It is also designed to scale efficiently with the dimensionality of the system and can be applied to very large systems when empirical or semiempirical methods are used to obtain the atomic forces. The method can be started from the potential minimum representing the initial state, or from an initial guess closer to the saddle point. An application to Al adatom diffusion on an Al(100) surface described by an embedded atom method potential is presented. A large number of saddle points were found for adatom diffusion and dimer/vacancy formation. A surprisingly low energy four atom exchange process was found as well as processes indicative of local hex reconstruction of the surface layer.

Saddle Points in Random Matrices: Analysis of Knuth Search Algorithms

We present an analysis of algorithms for finding saddle points in a random matrix, presented by Donald E. Knuth as Exercise 1.3.2-12 in The Art of Computer Programming . We estimate the average computing costs of three saddle point search algorithms. Amusingly, the asymptotic results in this analysis about matrix saddle points uses the same approach that leads to the celebrated saddle point method in complex analysis.

Finding saddle points on polyhedra: Solving certain continuous minimax problems

This article reviews procedures for computing saddle points of certain continuous concave-convex functions defined on polyhedra and investigates how certain parameters and payoff functions influence equilibrium solutions. The discussion centers on two widely studied applications : missile defense and market-share attraction games. In both settings, each player allocates a limited resource, called effort, among a finite number of alternatives. Equilibrium solutions to these two-person games are particularly easy to compute under a proportional effectiveness hypothesis, either in closed form or in a finite number of steps. One of the more interesting qualitative properties we establish is the identification of conditions under which the maximizing player can ignore the values of the alternatives in determining allocation decisions.

R-Algorithms and ellipsoids

Transformation (6) smoothing thef(x) level lines explains the effectiveness ofr(α)-algorithms from visual geometrical considerations. It may be regarded as a satisfactory interpretation of space dilation in the direction of the difference of two successive subgradients. On the other hand, it preserves the gradient flavor of the method, in contrast to the classical ellipsoid method [11, 12], which is a successful interpretation of the subgradient method with space dilation in the direction of the subgradient. A sensible combination of ellipsoids of a special kind [5] with the ellipsoids ell(x0,a, b, c) is quite capable of producing, on the basis of a one-dimensional space dilation operator, effective algorithms that solve a broader class of problems than convex programming problems, e.g., the problem to find saddle points of convex-concave functions, particular cases of the problem of solving variational inequalities, and also special classes of linear and nonlinear complementarity problems.

Monotone?-subgradient method for finding saddle points of convex?concave functions

Monotone?-subgradient method for finding saddle points of convex?concave functions

A saddle point finding algorithm for functionals

We present an algorithm which can be used to find saddle points of non-linear functionals or functions with a large number of variables. The saddle points that can be found have only one unstable direction. The algorithm is tested on a “φ 4” functional and yields the well known circular “critical droplet” solution in two dimensions. Saddle points corresponding to two widely separated droplets were also found.

Ridge method for finding saddle points on potential energy surfaces

A new method is proposed for locating saddle points on potential energy surfaces. The method involves walking on the ridge separating reactants’ and products’ valleys toward its minimum, which is a saddle point in coordinate space. Of particular advantage for ab initio calculations, the ridge method does not require evaluation of second derivatives of the potential energy. Another important feature of the method is that no assumptions about the transition state geometry are needed, and it is easy to impose linear constraints on the molecular structure. The ridge method is supplemented by a heuristic detour algorithm, which enables one to deal with unfortunate choices of reactants’ and products’ coordinates. Both algorithms are illustrated by several examples where the complexity of the potential energy surface ranges from a simple analytical formula to a numerical many-body ab initio potential.

Gauge invariant extremization

Recently, Duncan and Mawhinney introduced a method to find saddle points of the action in simulations of non-abelian lattice gauge theory. The idea, called ‘extremization’, is to minimize ∝( δS/ δA μ ) 2 instead of the action S itself as in conventional ‘cooling’. The method was implemented in an explicitly gauge variant way, however, and gauge dependence showed up in the results. Here we present a gauge invariant formulation of extremization on the lattice, applicable to any gauge group and any lattice action. The procedure is worked out in detail for U(1) and SU(N) lattice gauge theory with the plaquette action.

An algorithm for determining dynamically defined reaction paths (DDRP)

A numerically stable and well-parallelizable curve variational algorithm is described for determining tangent curves of vector fields between two given stationary points. In particular, the method is suitable for finding reaction paths and saddle points on potential energy hypersurfaces (PHS). The stability of the procedure is illustrated by an artificial mathematical function, showing phases of following the reaction on the PHS.

A two-level subgradient method for finding saddle points of convex-concave functions

A two-level subgradient method for finding saddle points of convex-concave functions

Gauge invariant extremization on the lattice

Recently, a method was proposed and tested to find saddle points of the action in simulations of non-abelian lattice gauge theory. The idea, called “extremization”, is to minimize ∫( δS δ Aμ ) 2 . The method was implemented in an explicitly gauge variant way, however, and gauge dependence showed up in the results. Here we show how extremization can be formulated in a way that preserves gauge invariance on the lattice. The method applies to any gauge group and any lattice action. The procedure is worked out in detail for the standard plaquette action with gauge groups U(1) and SU( N).

An improved algorithm for finding saddlepoints of two-person zero-sum games

It is well-known that one can sort (order)n real numbers in at mostF 0(n) =nl − 2 l + 1 steps (comparisons), wherel = [log2 n]. We snow how to find the strict saddlepoint or prove its absence in anm byn matrix,m ≤n, in at mostF 0(m)+F 0(m+1)+n+m − 3 + (n−m) [log2(m+1)] steps.

How to find a saddle point

AbstractThis paper explains a method for finding saddle points on a multidimensional surface and shows how it may be used to define saddle‐point seeking curves that have properties similar to well‐known orthogonal trajectories. It is shown that a gradient extremal is a special case of one of these curves, and its chemical significance as the path, defined by local criteria, which starts from a stable structure and leads to a transition state, is discussed briefly in relation to the intrinsic reaction coordinate. It is emphasized that this theory gives a natural method for locating points that have Hessians of similar structure to those of transition states.

Finding saddle points for clusters

The Cerjan–Miller eigenvector-following method is highly successful in finding saddle points for a variety of clusters whose potential energy surfaces are well described using simple analytical pair potentials. Examples are given for argon clusters, including Ar33 and Ar55, potassium chloride clusters and clusters of ions held in an ‘‘ion trap.’’ The method reveals a number of important patterns and opens the way to a more detailed understanding of the relation between energetics, reactivity, and potential energy surfaces for these and other systems.