Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases

We investigate the optimality for model selection of the so-called slope heuristics, $V$-fold cross-validation and $V$-fold penalization in a heteroscedastic with random design regression context. We consider a new class of linear models that we call strongly localized bases and that generalize histograms, piecewise polynomials and compactly supported wavelets. We derive sharp oracle inequalities that prove the asymptotic optimality of the slope heuristics---when the optimal penalty shape is known---and $V$ -fold penalization. Furthermore, $V$-fold cross-validation seems to be suboptimal for a fixed value of $V$ since it recovers asymptotically the oracle learned from a sample size equal to $1-V^{-1}$ of the original amount of data. Our results are based on genuine concentration inequalities for the true and empirical excess risks that are of independent interest. We show in our experiments the good behavior of the slope heuristics for the selection of linear wavelet models. Furthermore, $V$-fold cross-validation and $V$-fold penalization have comparable efficiency

Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method

In this paper we propose a new class of preconditioners for the isogeometric discretization of the Stokes system. Their application involves the solution of a Sylvester-like equation, which can be done efficiently thanks to the Fast Diagonalization method. These preconditioners are robust with respect to both the spline degree and mesh size. By incorporating information on the geometry parametrization and equation coefficients, we maintain efficiency on non-trivial computational domains and for variable kinematic viscosity. In our numerical tests we compare to a standard approach, showing that the overall iterative solver based on our preconditioners is significantly faster.Comment: 31 pages, 4 figure

Robust control of robots via linear estimated state feedback

In this note we propose a robust tracking controller for robots that requires only position measurements. The controller consists of two parts: a linear observer part that generates an estimated error state from the error on the joint position and a linear feedback part that utilizes this estimated state. It is shown that this computationally efficient controller yields semi-global uniform ultimate boundedness of the tracking error. An interesting feature of the controller is that it straightforwardly extends results on robust control of robots by linear state feedback to linear estimated-state feedbac

On the spot-futures no-arbitrage relations in commodity markets

In commodity markets the convergence of futures towards spot prices, at the expiration of the contract, is usually justified by no-arbitrage arguments. In this article, we propose an alternative approach that relies on the expected profit maximization problem of an agent, producing and storing a commodity while trading in the associated futures contracts. In this framework, the relation between the spot and the futures prices holds through the well-posedness of the maximization problem. We show that the futures price can still be seen as the risk-neutral expectation of the spot price at maturity and we propose an explicit formula for the forward volatility. Moreover, we provide an heuristic analysis of the optimal solution for the production/storage/trading problem, in a Markovian setting. This approach is particularly interesting in the case of energy commodities, like electricity: this framework indeed remains suitable for commodities characterized by storability constraints, when standard no-arbitrage arguments cannot be safely applied