Convergence and Cycling in Walker-type Saddle Search Algorithms

International audienceAlgorithms for computing local minima of smooth objective functions enjoy a mature theory as well as robust and efficient implementations. By comparison, the theory and practice of saddle search is destitute. In this paper we present results for idealized versions of the dimer and gentlest ascent (GAD) saddle search algorithms that show-case the limitations of what is theoretically achievable within the current class of saddle search algorithms: (1) we present an improved estimate on the region of attraction of saddles; and (2) we construct quasi-periodic solutions which indicate that it is impossible to obtain globally convergent variants of dimer and GAD type algorithms

A dimer-type saddle search algorithm with preconditioning and linesearch

The dimer method is a Hessian-free algorithm for computing saddle points. We augment the method with a linesearch mechanism for automatic step size selection as well as preconditioning capabilities. We prove local linear convergence. A series of numerical tests demonstrate significant performance gains

Gentlest ascent dynamics on manifolds defined by adaptively sampled point-clouds

Finding saddle points of dynamical systems is an important problem in practical applications such as the study of rare events of molecular systems. Gentlest ascent dynamics (GAD) is one of a number of algorithms in existence that attempt to find saddle points in dynamical systems. It works by deriving a new dynamical system in which saddle points of the original system become stable equilibria. GAD has been recently generalized to the study of dynamical systems on manifolds (differential algebraic equations) described by equality constraints and given in an extrinsic formulation. In this paper, we present an extension of GAD to manifolds defined by point-clouds, formulated using the intrinsic viewpoint. These point-clouds are adaptively sampled during an iterative process that drives the system from the initial conformation (typically in the neighborhood of a stable equilibrium) to a saddle point. Our method requires the reactant (initial conformation), does not require the explicit constraint equations to be specified, and is purely data-driven

Local movement: agent-based models of pedestrian flows

Modelling movement within the built environment has hitherto been focused on rather coarse spatial scales where the emphasis has been upon simulating flows of traffic between origins and destinations. Models of pedestrian movement have been sporadic, based largely on finding statistical relationships between volumes and the accessibility of streets, with no sustained efforts at improving such theories. The development of object-orientated computing and agent-based models which have followed in this wake, promise to change this picture radically. It is now possible to develop models simulating the geometric motion of individual agents in small-scale environments using theories of traffic flow to underpin their logic. In this paper, we outline such a model which we adapt to simulate flows of pedestrians between fixed points of entry - gateways - into complex environments such as city centres, and points of attraction based on the location of retail and leisure facilities which represent the focus of such movements. The model simulates the movement of each individual in terms of five components; these are based on motion in the direction of the most attractive locations, forward movement, the avoidance of local geometric obstacles, thresholds which constrain congestion, and movement which is influenced by those already moving towards various locations. The model has elements which enable walkers to self-organise as well as learn from their geometric experiences so far. We first outline the structure of the model, present a computable form, and illustrate how it can be programmed as a variant of cellular automata. We illustrate it using three examples: its application to an idealised mall where we show how two key components - local navigation of obstacles and movement towards points of global locational attraction - can be parameterised, an application to the more complex town centre of Wolverhampton (in the UK West Midlands) where the paths of individual walkers are used to explore the veracity of the model, and finally it application to the Tate Gallery complex in central London where the focus is on calibrating the model by letting individual agents learn from their experience of walking within the environment

Control of quantum phenomena: Past, present, and future

Quantum control is concerned with active manipulation of physical and chemical processes on the atomic and molecular scale. This work presents a perspective of progress in the field of control over quantum phenomena, tracing the evolution of theoretical concepts and experimental methods from early developments to the most recent advances. The current experimental successes would be impossible without the development of intense femtosecond laser sources and pulse shapers. The two most critical theoretical insights were (1) realizing that ultrafast atomic and molecular dynamics can be controlled via manipulation of quantum interferences and (2) understanding that optimally shaped ultrafast laser pulses are the most effective means for producing the desired quantum interference patterns in the controlled system. Finally, these theoretical and experimental advances were brought together by the crucial concept of adaptive feedback control, which is a laboratory procedure employing measurement-driven, closed-loop optimization to identify the best shapes of femtosecond laser control pulses for steering quantum dynamics towards the desired objective. Optimization in adaptive feedback control experiments is guided by a learning algorithm, with stochastic methods proving to be especially effective. Adaptive feedback control of quantum phenomena has found numerous applications in many areas of the physical and chemical sciences, and this paper reviews the extensive experiments. Other subjects discussed include quantum optimal control theory, quantum control landscapes, the role of theoretical control designs in experimental realizations, and real-time quantum feedback control. The paper concludes with a prospective of open research directions that are likely to attract significant attention in the future.Comment: Review article, final version (significantly updated), 76 pages, accepted for publication in New J. Phys. (Focus issue: Quantum control

Studies of Classical and Quantum Annealing

A summary of the results of recent applications of PIMC-QA on different optimization problems is given in Chapter 1. In order to gain understanding on these problems, we have moved one step back, and concentrated attention on the simplest textbook problems where the energy landscape is well under control: essentially, one-dimensional potentials, starting from a double-well potential, the simplest form of barrier. On these well controlled landscapes we have carried out a detailed and exhaustive comparison between quantum adiabatic Schr\uf6dinger evolution, both in real and in imaginary time, and its classical deterministic counterpart, i.e., Fokker-Planck evolution [17]. This work will be illustrated in Chapter 2. On the same double well-potential, we have also studied the performance of different stochastic annealing approaches, both classical Monte Carlo annealing and PIMCQA. The CA work is illustrated in Chapter 3, where we analyze the different annealing behaviors of three possible types of Monte Carlo moves (with Box, Gaussian, and Lorentzian distributions) in a numerical and analytical way. The PIMC-QA work is illustrated in Chapter 4, were we show the difficulties that a state-of-the-art PIMCQA algorithm can encounter in describing tunneling even in a simple landscape, and we also investigate the role of the kinetic energy choice, by comparing the standard non-relativistic dispersion, Hkin = Tau(t)p^2, with a relativistic one, Hkin = Tau(t)|p|, which turns out to be definitely more effective. In view of the difficulties encountered by PIMC-QA even in a simple double-well potential, we finally explored the capabilities of another well established QMC technique, the Green's Function Monte Carlo (GFMC), as a base for a QA algorithm. This time, we concentrated our attention on a very studied and challenging optimization problem, the random Ising model ground state search, for which both CA and PIMC-QA data are available [10, 11]. A more detailed summary of the results and achievements described in this Thesis, and a discussion of open issues, is contained in the final section `Conclusions and Perspectives'. Finally, in order to keep this Thesis as self-contained as possible, we include in the appendices a large amount of supplemental material

An Examination of the Strengths and Weaknesses of Newton\u27s Method for Nonlinear Optimization

This thesis begins with the history of operations research and introduces two of its major branches, linear and nonlinear optimization. While other methods are mentioned, the focus is on analytical methods used to solve nonlinear optimization problems. We briefly look at some of the most effective constrained methods for nonlinear optimization and then show how unconstrained methods often play a role in developing effective constrained optimization algorithms. In particular we examine Newton and steepest descent methods, focusing primarily on Newton/quasi-Newton methods. Because Newton\u27s method is primarily viewed as a root-finding method, we start with the basic root-finding algorithm for single variable functions and show its progression into a useful, and often efficient, multivariable optimization algorithm. Comparisons are made between a pure Newton algorithm and a modified Newton algorithm as well as between a pure steepest descent algorithm and a modified steepest descent algorithm. In examining nonlinear functions of varying complexity, we note some of the considerations that must be made when choosing an optimization program as well as some of the difficulties that arise when using Newton\u27s method or steepest descent methods for the optimization of a nonlinear function