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    Nonparametric change point detection in regression
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Avanesov, Valeriy
    This paper considers the prominent problem of change-point detection in regression. The study suggests a novel testing procedure featuring a fully data-driven calibration scheme. The method is essentially a black box, requiring no tuning from the practitioner. The approach is investigated from both theoretical and practical points of view. The theoretical study demonstrates proper control of first-type error rate under H0 and power approaching 1 under H1. The experiments conducted on synthetic data fully support the theoretical claims. In conclusion, the method is applied to financial data, where it detects sensible change-points. Techniques for change-point localization are also suggested and investigated
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    How to gamble with non-stationary X-armed bandits and have no regrets
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Avanesov, Valeriy
    In X-armed bandit problem an agent sequentially interacts with environment which yields a reward based on the vector input the agent provides. The agent's goal is to maximise the sum of these rewards across some number of time steps. The problem and its variations have been a subject of numerous studies, suggesting sub-linear and sometimes optimal strategies. The given paper introduces a new variation of the problem. We consider an environment, which can abruptly change its behaviour an unknown number of times. To that end we propose a novel strategy and prove it attains sub-linear cumulative regret. Moreover, the obtained regret bound matches the best known bound for GP-UCB for a stationary case, and approaches the minimax lower bound in case of highly smooth relation between an action and the corresponding reward. The theoretical result is supported by experimental study.
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    Simulation of composite materials by a Network FEM with error control
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Eigel, Martin; Peterseim, Daniel
    A novel Finite Element Method (FEM) for the computational simulation in particle reinforced composite materials with many inclusions is presented. It is based on a specially designed mesh consisting of triangles and channel-like connections between inclusions which form a network structure. The total number of elements and, hence, the number of degrees of freedom are proportional to the number of inclusions. The error of the method is independent of the possibly tiny distances of neighbouring inclusions. We present algorithmic details for the generation of the problem adapted mesh and derive an efficient residual a posteriori error estimator which enables to compute reliable upper and lower error bounds. Several numerical examples illustrate the performance of the method and the error estimator. In particular, it is demonstrated that the (common) assumption of a lattice structure of inclusions can easily lead to incorrect predictions about material properties.