We propose the Wishart planted ensemble, a class of zero-field Ising models with tunable algorithmic hardness and specifiable (or planted) ground state. The problem class arises from a simple procedure for generating a family of random integer programming problems with specific statistical symmetry properties but turns out to have intimate connections to a sign-inverted variant of the Hopfield model. The Hamiltonian contains only 2-spin interactions, with the coupler matrix following a type of Wishart distribution. The class exhibits a classical first-order phase transition in temperature. For some parameter settings the model has a locally stable paramagnetic state, a feature which correlates strongly with difficulty in finding the ground state and suggests an extremely rugged energy landscape. We analytically probe the ensemble thermodynamic properties by deriving the Thouless-Anderson-Palmer equations and free energy and corroborate the results with a replica and annealed approximation analysis; extensive Monte Carlo simulations confirm our predictions of the first-order transition temperature. The class exhibits a wide variation in algorithmic hardness as a generation parameter is varied, with a pronounced easy-hard-easy profile and peak in solution time towering many orders of magnitude over that of the easy regimes. By deriving the ensemble-averaged energy distribution and taking into account finite-precision representation, we propose an analytical expression for the location of the hardness peak and show that at fixed precision, the number of constraints in the integer program must increase with system size to yield truly hard problems. The Wishart planted ensemble is interesting for its peculiar physical properties and provides a useful and analytically transparent set of problems for benchmarking optimization algorithms.
We consider classical and semi-classical dynamical systems that start from a given ensemble of configurations and evolve in time until the systems reach a certain fixed stopping criterion, with the mean first-passage time (MFPT) being the quantity of interest. We present a method, projective dynamics, which maps the dynamics of the system onto an arbitrary discrete set of states {ζk}, subject to the constraint that the states ζk are chosen in such a way that only transitions not further than to the neighboring states ζk ± 1 occur. We show that with this imposed condition there exists a master equation with nearest-neighbor coupling with the same MFPT and residence times as the original dynamical system. We show applications of the method for the diffusion process of particles in one- and two-dimensional potential energy landscapes and the folding process of a small biopolymer. We compare results for the MFPT and the mean folding time obtained with the projective dynamics method with those obtained by a direct measurement, and where possible with a semi-analytical solution.
A computational method is presented which is capable to obtain low lying energy structures of topological amorphous systems. The method merges a differential mutation genetic algorithm with simulated annealing. This is done by incorporating a thermal selection criterion, which makes it possible to reliably obtain low lying minima with just a small population size and is suitable for multimodal structural optimization. The method is tested on the structural optimization of amorphous graphene from unbiased atomic starting configurations. With just a population size of six systems, energetically very low structures are obtained. While each of the structures represents a distinctly different arrangement of the atoms, their properties, such as energy, distribution of rings, radial distribution function, coordination number, and distribution of bond angles, are very similar.
The dynamics of a multidimensional system is projected onto a discrete state master equation using the transition rates W(k → k′; t, t + dt) between a set of states {k} represented by the regions {ζk} in phase or discrete state space. Depending on the dynamics Γi(t) of the original process and the choice of ζk, the discretized process can be Markovian or non-Markovian. For absorption processes, it is shown that irrespective of these properties of the projection, a master equation with time-independent transition rates can be obtained, which conserves the total occupation time of the partitions of the phase or discrete state space of the original process. An expression for the transition probabilities is derived based on either time-discrete measurements {ti} with variable time stepping Δ(i + 1)i = ti + 1 − ti or the theoretical knowledge at continuous times t. This allows computational methods of absorbing Markov chains to be used to obtain the mean first passage time (MFPT) of the system. To illustrate this approach, the procedure is applied to obtain the MFPT for the overdamped Brownian motion of particles subject to a system with dichotomous noise and the escape from an entropic barrier. The high accuracy of the simulation results confirms with the theory.
The thermally driven differential mutation algorithm is an evolutionary algorithm dealing with the structural optimization of large amorphous systems represented by empirical potentials. It is a hybrid algorithm that combines a differential mutation evolutionary algorithm with a metropolis selection criterion and a cooling schedule inspired by simulated annealing. In this manuscript, the influence of the cooling rate on the quality of obtained amorphous graphene structures is discussed.
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