Glosarium Matematika

273 p.; 24 cm

Calculus of the exponent of Kurdyka-{\L}ojasiewicz inequality and its applications to linear convergence of first-order methods

In this paper, we study the Kurdyka-{\L}ojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo-Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is $\frac12$. The Luo-Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function's KL exponent is $\frac12$. This includes the least squares problem with smoothly clipped absolute deviation (SCAD) regularization or minimax concave penalty (MCP) regularization and the logistic regression problem with $\ell_1$ regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step-sizes for some specific models that arise in sparse recovery.Comment: The paper is accepted for publication in Foundations of Computational Mathematics: https://link.springer.com/article/10.1007/s10208-017-9366-8. In this update, we fill in more details to the proof of Theorem 4.1 concerning the nonemptiness of the projection onto the set of stationary point

Parallel software tool for decomposing and meshing of 3d structures

An algorithm for automatic parallel generation of three-dimensional unstructured computational meshes based on geometrical domain decomposition is proposed in this paper. Software package build upon proposed algorithm is described. Several practical examples of mesh generation on multiprocessor computational systems are given. It is shown that developed parallel algorithm enables us to reduce mesh generation time significantly (dozens of times). Moreover, it easily produces meshes with number of elements of order 5 · 107, construction of those on a single CPU is problematic. Questions of time consumption, efficiency of computations and quality of generated meshes are also considered

Maintenance models applied to wind turbines. A comprehensive overview

Producción CientíficaWind power generation has been the fastest-growing energy alternative in recent years, however, it still has to compete with cheaper fossil energy sources. This is one of the motivations to constantly improve the efficiency of wind turbines and develop new Operation and Maintenance (O&M) methodologies. The decisions regarding O&M are based on different types of models, which cover a wide range of scenarios and variables and share the same goal, which is to minimize the Cost of Energy (COE) and maximize the profitability of a wind farm (WF). In this context, this review aims to identify and classify, from a comprehensive perspective, the different types of models used at the strategic, tactical, and operational decision levels of wind turbine maintenance, emphasizing mathematical models (MatMs). The investigation allows the conclusion that even though the evolution of the models and methodologies is ongoing, decision making in all the areas of the wind industry is currently based on artificial intelligence and machine learning models