A modified homogeneous algorithm for function minimization

AbstractRecently, an algorithm for function minimization was presented, based upon an homogeneous, rather than upon a quadratic, model. Numerical experiments with this algorithm indicated that it rapidly minimizes the standard test functions available in the literature. Although it was proved that the algorithm produces function values which continually descend, no proof of convergence was supplied.In this paper, the homogeneous algorithm is modified primarily by replacing the cubic interpolation routine by Armijo's step size rule. Although not quite as fast as the original version on the standard test functions, this modified form has the advantage that a proof of convergence follows from a general theorem of Polak

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

Entanglement entropy, black holes and holography

We observe that the entanglement entropy resulting from tracing over a subregion of an initially pure state can grow faster than the surface area of the subregion (indeed, proportional to the volume), in contrast to examples studied previously. The pure states with this property have long-range correlations between interior and exterior modes and are constructed by purification of the desired density matrix. We show that imposing a no-gravitational collapse condition on the pure state is sufficient to exclude faster than area law entropy scaling. This observation leads to an interpretation of holography as an upper bound on the realizable entropy (entanglement or von Neumann) of a region, rather than on the dimension of its Hilbert space.Comment: 4 pages, revte

Optimal low-thrust trajectories to asteroids through an algorithm based on differential dynamic programming

In this paper an optimisation algorithm based on Differential Dynamic Programming is applied to the design of rendezvous and fly-by trajectories to near Earth objects. Differential dynamic programming is a successive approximation technique that computes a feedback control law in correspondence of a fixed number of decision times. In this way the high dimensional problem characteristic of low-thrust optimisation is reduced into a series of small dimensional problems. The proposed method exploits the stage-wise approach to incorporate an adaptive refinement of the discretisation mesh within the optimisation process. A particular interpolation technique was used to preserve the feedback nature of the control law, thus improving robustness against some approximation errors introduced during the adaptation process. The algorithm implements global variations of the control law, which ensure a further increase in robustness. The results presented show how the proposed approach is capable of fully exploiting the multi-body dynamics of the problem; in fact, in one of the study cases, a fly-by of the Earth is scheduled, which was not included in the first guess solution

The Basics of Water Waves Theory for Analogue Gravity

This chapter gives an introduction to the connection between the physics of water waves and analogue gravity. Only a basic knowledge of fluid mechanics is assumed as a prerequisite.Comment: 36 pages. Lecture Notes for the IX SIGRAV School on "Analogue Gravity", Como (Italy), May 201

Reduction of quartz to silicon monoxide by methane-hydrogen mixtures

The reduction of quartz was studied isothermally in a fluidized bed reactor using continuously flowing methane-hydrogen gas mixture in the temperature range from 1623 K to 1773 K (1350 °C to 1500 °C). The CO content in the off-gas was measured online using an infrared gas analyzer. The main phases of the reduced samples identified by XRD analysis were quartz and cristobalite. Significant weight loss in the reduction process indicated that the reduction products were SiO and CO. Reduction of SiO2 to SiO by methane starts with adsorption and dissociation of CH4 on the silica surface. The high carbon activity in the CH4-H2 gas mixture provided a strongly reducing condition. At 1623 K (1350 °C), the reduction was very slow. The rate and extent of reduction increased with the increasing temperature to 1723 K (1450 °C). A further increase in temperature to 1773 K (1500 °C) resulted in a decrease in the rate and extent of reduction. An increase in the gas flow rate from 0.4 to 0.8 NL/min and an increase in the methane content in the CH4-H2 gas mixture from 0 to 5 vol pct facilitated the reduction. Methane content in the gas mixture should be maintained at less than 5 vol pct in order to suppress methane cracking

An eddy-correlation measurement of NO2 flux to vegetation and comparison to O3 flux

Eddy-correlation measurements with a newly developed fast-response NOx sensor indicate that the deposition velocity at a height of about 6m above a soybean field has a maximum value near 0.6cms-1 for NOx and is usually about 2/3 ofthat found for ozone. In these studies, over 90% of the NOx is NO2. The corresponding minimum surface resistance for NOx calculated as the quantity remaining after atmospheric resistances are subtracted is about 1.3 s cm-1, which is larger than expected on the basis of leaf stomatal resistance alone. Emission of NO from sites in the plant canopy and soil where NO2 is deposited and reduced to NO or release of NOx as a result of biological activity may have lessened the downward fluxes of NOx as measured. During windy conditions at night, surface resistances are found to have values of about 15scm-1 for NOx (again, greater than 90% NO2) and 1.8scm-1 for O3, corresponding to deposition velocities of 0.05cms-1 and 0.3cms-1, respectively.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24138/1/0000395.pd

A thermodynamic motivation for dark energy

It is argued that the discovery of cosmic acceleration could have been anticipated on thermodynamic grounds, namely, the generalized second law and the approach to equilibrium at large scale factor. Therefore, the existence of dark energy -or equivalently, some modified gravity theory- should have been expected. In general, cosmological models that satisfy the above criteria show compatibility with observational data.Comment: 22 pages, 7 eps figures; Key words: dark energy, thermodynamics, modified gravity. Comments added and arguments sharpene

Adaptive Optimal Feedback Control with Learned Internal Dynamics Models

Optimal Feedback Control (OFC) has been proposed as an attractive movement generation strategy in goal reaching tasks for anthropomorphic manipulator systems. Recent developments, such as the Iterative Linear Quadratic Gaussian (ILQG) algorithm, have focused on the case of non-linear, but still analytically available, dynamics. For realistic control systems, however, the dynamics may often be unknown, difficult to estimate, or subject to frequent systematic changes. In this chapter, we combine the ILQG framework with learning the forward dynamics for simulated arms, which exhibit large redundancies, both, in kinematics and in the actuation. We demonstrate how our approach can compensate for complex dynamic perturbations in an online fashion. The specific adaptive framework introduced lends itself to a computationally more efficient implementation of the ILQG optimisation without sacrificing control accuracy – allowing the method to scale to large DoF systems