Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

Q-learning with linear function approximation

Abstract: In this paper, we analyze the convergence of Q-learning with linear function approximation. We identify a set of conditions that implies the convergence of this method with probability 1, when a fixed learning policy is used. We discuss the differences and similarities between our results and those obtained in several related works. We also discuss the applicability of this method when a changing policy is used. Finally, we describe the applicability of this approximate method in partially observable scenarios.

A fast and accurate approximation for planar pose graph optimization

Abstract: This work investigates the pose graph optimization problem, which arises in maximum likelihood approaches to simultaneous localization and mapping SLAM. State-of-the-art approaches have been demonstrated to be very efficient in medium- and large-sized scenarios; however, their convergence to the maximum likelihood estimate heavily relies on the quality of the initial guess. We show that, in planar scenarios, pose graph optimization has a very peculiar structure. The problem of estimating robot orientations from relative orientation measurements is a quadratic optimization problem after computing suitable regularization terms; moreover, given robot orientations, the overall optimization problem becomes quadratic. We exploit these observations to design an approximation of the maximum likelihood estimate, which does not require the availability of an initial guess. The approximation, named LAGO Linear Approximation for pose Graph Optimization, can be used as a stand-alone tool or can bootstrap state-of-the-art techniques, reducing the risk of being trapped in local minima. We provide analytical results on existence and sub-optimality of LAGO, and we discuss the factors influencing its quality. Experimental results demonstrate that LAGO is accurate in common SLAM problems. Moreover, it is remarkably faster than state-of-the-art techniques, and is able to solve very large-scale problems in a few seconds.

Optimal power flow based on successive linear approximation of power flow equations

Abstract: In this study, the authors introduce a new solution to the alternating current optimal power flow problem based on successive linear approximation of power flow equations. The polar coordination of the power flow equations is used to take advantage of the quasi-linear P– θ relationship. A mathematical transformation for the crossing term of voltage magnitude is used, which guarantees the accuracy of the linear approximation when the quality of initial points regarding voltage magnitude is relatively low in the first few iterations. As a result, the accuracy of the proposed approximation becomes very high in very few iterations. Linearisation method for the quadratic apparent branch flow limits is provided. Methods to recover the AC feasibility from the obtained optimal power flow solution and correct possible constraint violations are introduced. The proposed method is tested in several IEEE and Polish benchmark systems. The difference in objective functions relative to MATPOWER benchmark results is generally <0.1% when the algorithm terminates.

A Novel Probabilistic Optimal Power Flow Model With Uncertain Wind Power Generation Described by Customized Gaussian Mixture Model

Abstract: A novel probabilistic optimal power flow (P-OPF) model with chance constraints that considers the uncertainties of wind power generation (WPG) and load is proposed in this paper. An affine generation dispatch strategy is adopted to balance the system power uncertainty by several conventional generators, and thus the linear approximation of the cost function with respect to the power uncertainty is proposed to compute the quantile (which is also recognized as the value-at-risk) corresponding to a given probability value. The proposed model applies this quantile as the objective function and minimizes it to meet distinct probabilistic cost regulation purposes via properly selecting the given probability. In particular, the hedging effect due to the used affine generation dispatch is also thoroughly investigated. In addition, an analytical method to calculate probabilistic load flow (PLF) is developed with the probability density function of WPG, which is proposed to be approximated by a customized Gaussian mixture model whose parameters are easily obtained. Accordingly, it is successful to analytically compute the chance constraints on the transmission line power and the power outputs of conventional units. Numerical studies of two benchmark systems show the satisfactory accuracy of the PLF method, and the effectiveness of the proposed P-OPF model.

Chemical association in simple models of molecular and ionic fluids. III: The cavity function

Abstract: Exact equations which relate the cavity function to excess solvation free energies and equilibrium association constants are rederived by using a thermodynamic cycle. A zeroth‐order approximation, derived previously by us as a simple interpolation scheme, is found to be very accurate if the associative bonding occurs on or near the surface of the repulsive core of the interaction potential. If the bonding radius is substantially less than the core radius, the approximation overestimates the association degree and the association constant. For binary association, the zeroth‐order approximation is equivalent to the first‐order thermodynamic perturbation theory (TPT) of Wertheim. For n‐particle association, the combination of the zeroth‐order approximation with a ‘‘linear’’ approximation (for n‐particle distribution functions in terms of the two‐particle function) yields the first‐order TPT result. Using our exact equations to go beyond TPT, near‐exact analytic results for binary hard‐sphere association are obtained. Solvent effects on binary hard‐sphere association and ionic association are also investigated. A new rule which generalizes Le Chatelier’s principle is used to describe the three distinct forms of behaviors involving solvent effects that we find. The replacement of the dielectric‐continuum solvent model by a dipolar hard‐sphere model leads to improved agreement with an experimental observation. Finally, equation of state for an n‐particle flexible linear‐chain fluid is derived on the basis of a one‐parameter approximation that interpolates between the generalized Kirkwood superposition approximation and the linear approximation. A value of the parameter that appears to be near optimal in the context of this application is obtained from comparison with computer‐simulation data.