Transform Ranking: a New Method of Fitness Scaling in Genetic Algorithms

The first systematic evaluation of the effects of six existing forms of fitness scaling in genetic algorithms is presented alongside a new method called transform ranking. Each method has been applied to stochastic universal sampling (SUS) over a fixed number of generations. The test functions chosen were the two-dimensional Schwefel and Griewank functions. The quality of the solution was improved by applying sigma scaling, linear rank scaling, nonlinear rank scaling, probabilistic nonlinear rank scaling, and transform ranking. However, this benefit was always at a computational cost. Generic linear scaling and Boltzmann scaling were each of benefit in one fitness landscape but not the other. A new fitness scaling function, transform ranking, progresses from linear to nonlinear rank scaling during the evolution process according to a transform schedule. This new form of fitness scaling was found to be one of the two methods offering the greatest improvements in the quality of search. It provided the best improvement in the quality of search for the Griewank function, and was second only to probabilistic nonlinear rank scaling for the Schwefel function. Tournament selection, by comparison, was always the computationally cheapest option but did not necessarily find the best solutions

Algorithmic differentiation and the calculation of forces by quantum Monte Carlo

We describe an efficient algorithm to compute forces in quantum Monte Carlo using adjoint algorithmic differentiation. This allows us to apply the space warp coordinate transformation in differential form, and compute all the 3M force components of a system with M atoms with a computational effort comparable with the one to obtain the total energy. Few examples illustrating the method for an electronic system containing several water molecules are presented. With the present technique, the calculation of finite-temperature thermodynamic properties of materials with quantum Monte Carlo will be feasible in the near future.Comment: 32 pages, 4 figure, to appear in The Journal of Chemical Physic

Cheap Newton steps for optimal control problems: automatic differentiation and Pantoja's algorithm

Original article can be found at: http://www.informaworld.com/smpp/title~content=t713645924~db=all Copyright Taylor and Francis / Informa.In this paper we discuss Pantoja's construction of the Newton direction for discrete time optimal control problems. We show that automatic differentiation (AD) techniques can be used to calculate the Newton direction accurately, without requiring extensive re-writing of user code, and at a surprisingly low computational cost: for an N-step problem with p control variables and q state variables at each step, the worst case cost is 6(p + q + 1) times the computational cost of a single target function evaluation, independent of N, together with at most p3/3 + p2(q + 1) + 2p(q + 1)2 + (q + l)3, i.e. less than (p + q + l)3, floating point multiply-and-add operations per time step. These costs may be considerably reduced if there is significant structural sparsity in the problem dynamics. The systematic use of checkpointing roughly doubles the operation counts, but reduces the total space cost to the order of 4pN floating point stores. A naive approach to finding the Newton step would require the solution of an Np Np system of equations together with a number of function evaluations proportional to Np, so this approach to Pantoja's construction is extremely attractive, especially if q is very small relative to N. Straightforward modifications of the AD algorithms proposed here can be used to implement other discrete time optimal control solution techniques, such as differential dynamic programming (DDP), which use state-control feedback. The same techniques also can be used to determine with certainty, at the cost of a single Newton direction calculation, whether or not the Hessian of the target function is sufficiently positive definite at a point of interest. This allows computationally cheap post-hoc verification that a second-order minimum has been reached to a given accuracy, regardless of what method has been used to obtain it.Peer reviewe

Algorithmic Differentiation Through Automatic Graph Elimination Ordering (ADTAGEO)

Algorithmic Differentiation Through Automatic Graph Elimination Ordering (ADTAGEO) is based on the principle of Instant Elimination: At runtime we dynamically maintain a DAG representing only active variables that are alive at any time. Whenever an active variable is deallocated or its value is overwritten the corresponding vertex in the Live-DAG will be eliminated immediately by the well known vertex elimination rule [1]. Consequently, the total memory requirement is equal to that of the sparse forward mode. Assuming that local variables are destructed in the opposite order of their construction (as in C++), a single assignment code is in effect differentiated in reverse mode. If compiler-generated temporaries are destroyed in reverse order too, then Instant Elimination yields the statement level reverse mode of ADIFOR [2] naturally. The user determines the elimination order intentionally (or unintentionally) by the order in which he declares variables, which makes hybrid modes of AD possible by combining forward and reverse differentiated parts. By annotating the Live-DAG with local Hessians and applying second order elimination rules, Hessian-vector products can be computed efficiently since the annotated Live-DAG stores one half of the symmetric Hessian graph only (as suggested in [1]). Nested automatic differentiation is done easily by subsequent propagations, since sensitivities between variables alive can be obtained at any point in time within the Live-DAG. The concept of maintaining a Live-DAG fits optimally into the strategy of overloaded operators for classes, it is a very natural example of Object Oriented Programming. A proof-of-concept implementation in C++ is available (contact the first author). References 1. Griewank, A.: Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation. SIAM (2000) 2.Bischof, C.H., Carle, A., Khademi, P., Mauer, A.: ADIFOR 2.0: Automatic differentiation of Fortran 77 programs. IEEE Computational Science & Engineering 3 (1996) 18-3

Applicability of Quasi-Monte Carlo for lattice systems

This project investigates the applicability of quasi-Monte Carlo methods to Euclidean lattice systems in order to improve the asymptotic error scaling of observables for such theories. The error of an observable calculated by averaging over random observations generated from ordinary Monte Carlo simulations scales like $N^{-1/2}$, where $N$ is the number of observations. By means of quasi-Monte Carlo methods it is possible to improve this scaling for certain problems to $N^{-1}$, or even further if the problems are regular enough. We adapted and applied this approach to simple systems like the quantum harmonic and anharmonic oscillator and verified an improved error scaling of all investigated observables in both cases.Comment: on occasion of the 31st International Symposium on Lattice Field Theory - LATTICE 2013, July 29 - August 3, 2013, Mainz, Germany, 7 Pages, 4 figure

QCD critical region and higher moments for three flavor models

One of the distinctive feature of the QCD phase diagram is the possible emergence of a critical endpoint. The critical region around the critical point and the path dependency of the critical exponents is investigated within effective chiral (2+1)-flavor models with and without Polyakov-loops. Results obtained in no-sea mean-field approximations where a divergent vacuum part in the fermion-loop contribution is neglected, are confronted to the renormalized ones. Furthermore, the modifications caused by the back-reaction of the matter fluctuations on the pure Yang-Mills system are discussed. Higher order, non-Gaussian moments of event-by-event distributions of various particle multiplicities are enhanced near the critical point and could serve as a probe to determine its location in the phase diagram. By means of a novel derivative technique higher order generalized quark-number susceptibilities are calculated and their sign structure in the phase diagram is analyzed.Comment: 12 pages, 11 figures. Final PRD version (references and one more equation added