We consider the piecewise linear approximation of saddle functions of the form (f(x,y)=ax^2-by^2 ) under the (L_ ) error norm. We show that interpolating approximations are not optimal. One can get slightly smaller errors by allowing the vertices of the approximation to move away from the graph of the function.
Bertram, Barnes, Hamann, Joy, Pottmann, and Wushour @cite_1 have extended this approach to an arbitrary bivariate function @math , by taking optimal local approximations on suitably defined patches and stitching'' them together at the patch boundaries. (The setting of this paper actually somewhat different: the bivariate function @math is given as a set of scattered data points.)
@cite_1
2086684904 2291950750 2738326163 2592062427
Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case, the least-squares method is quasi-optimal in expectation in [A. Cohen, M A. Davenport and D. Leviatan. Found. Comput. Math. 13 (2013) 819–834] and in probability in [G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, Found. Comput. Math. 14 (2014) 419–456], under suitable conditions that relate the number of samples with respect to the dimension of the polynomial space. Here “quasi-optimal” means that the accuracy of the least-squares approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the quasi-optimality of the polynomial least-squares method in arbitrary dimension. Our analysis applies to any arbitrary multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space, independently of the anisotropic shape and of the number of variables. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss “inclusion type” elliptic PDE models, and derive an exponential convergence estimate for the least-squares method. Numerical results confirm our estimate, yet pointing out a gap between the condition necessary to achieve optimality in the theory, and the condition that in practice yields the optimal convergence rate. In this paper we propose a distributed version of a randomized block-coordinate descent method for minimizing the sum of a partially separable smooth convex function and a fully separable non-smooth convex function. Under the assumption of block Lipschitz continuity of the gradient of the smooth function, this method is shown to have a sublinear convergence rate. Linear convergence rate of the method is obtained for the newly introduced class of generalized error bound functions. We prove that the new class of generalized error bound functions encompasses both global local error bound functions and smooth strongly convex functions. We also show that the theoretical estimates on the convergence rate depend on the number of blocks chosen randomly and a natural measure of separability of the objective function. We investigate a family of regression problems in a semi-supervised setting. The task is to assign real-valued labels to a set of @math sample points, provided a small training subset of @math labeled points. A goal of semi-supervised learning is to take advantage of the (geometric) structure provided by the large number of unlabeled data when assigning labels. We consider random geometric graphs, with connection radius @math , to represent the geometry of the data set. Functionals which model the task reward the regularity of the estimator function and impose or reward the agreement with the training data. Here we consider the discrete @math -Laplacian regularization. We investigate asymptotic behavior when the number of unlabeled points increases, while the number of training points remains fixed. We uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. We rigorously obtain almost optimal ranges on the scaling of @math for the asymptotic consistency to hold. We prove that the minimizers of the discrete functionals in random setting converge uniformly to the desired continuum limit. Furthermore we discover that for the standard model used there is a restrictive upper bound on how quickly @math must converge to zero as @math . We introduce a new model which is as simple as the original model, but overcomes this restriction. In this paper we employ a parallel version of a randomized (block) coordinate descent method for minimizing the sum of a partially separable smooth convex function and a fully separable nonsmooth convex function. Under the assumption of Lipschitz continuity of the gradient of the smooth function, this method has a sublinear convergence rate. Linear convergence rate of the method is obtained for the newly introduced class of generalized error bound functions. We prove that the new class of generalized error bound functions encompasses both global local error bound functions and smooth strongly convex functions. We also show that the theoretical estimates on the convergence rate depend on the number of blocks chosen randomly and a natural measure of separability of the smooth component of the objective function.