We consider the problem of approximating the regression function f_μ : Ω→ Y from noisy μ -distributed vector-valued data (ω _m,y_m)∈Ω× Y by an online learning algorithm using a reproducing kernel Hilbert space H (RKHS) as prior. In an online algorithm, i.i.d. samples become available one by one via a random process and are successively processed to build approximations to the regression function. Assuming that the regression function essentially belongs to H (soft learning scenario), we provide estimates for the expected squared error in the RKHS norm of the approximations f^(m)∈ H obtained by a standard regularized online approximation algorithm. In particular, we show an order-optimal estimate 𝔼(‖ϵ ^(m)‖ _H^2)≤ C (m+1)^-s/(2+s), m=1,2,… , where ϵ ^(m) denotes the error term after m processed data, the parameter 0

My joint paper (Numerische Mathematik 135:1207-1220, 2017. ) with W. Zhou contains two errors which concern the derivation of some auxiliary norm estimates of the lower triangular projection for positive semi-definite Hermitean matrices in dependence on coordinate permutations. These errors are corrected. The main results of Oswald and Zhou (Numerische Mathematik 135:1207-1220, 2017. ) about the convergence behavior of so-called shuffled and preshuffled SOR iterations are not affected.

My joint paper (Numerische Mathematik 135:1207–1220, 2017. https://doi.org/10.1007/s00211-016-0829-7 ) with W. Zhou contains two errors which concern the derivation of some auxiliary norm estimates of the lower triangular projection for positive semi-definite Hermitean matrices in dependence on coordinate permutations. These errors are corrected. The main results of Oswald and Zhou (Numerische Mathematik 135:1207–1220, 2017. https://doi.org/10.1007/s00211-016-0829-7 ) about the convergence behavior of so-called shuffled and preshuffled SOR iterations are not affected.

We prove the convergence of greedy and randomized versions o f Schwarz iterative methods for solving linear elliptic variational problems b ased on infinite space splittings of a Hilbert space. For the greedy case, we show a squared error de cay rate ofO((m+ 1)−1) for elements of an approximation space A1 related to the underlying splitting. For the randomized case, we show an expected squared error decay rate of O((m+ 1)−1) on a classA π ∞ ⊂ A1 depending on the probability distribution.

Охарактеризованы все случаи, в которых система $d$-мерных всплесков Хаара $H^d$ на единичном кубе $I^d$ образует условный или безусловный базис Шаудера в классических изотропных функциональных пространствах Бесова ${B}_{p,q,1}^s(I^d)$, $0

We determine all cases for which the -dimensional Haar wavelet system on the unit cube is a conditional or unconditional Schauder basis in the classical isotropic Besov function spaces , , , defined in terms of first-order -moduli of smoothness. We obtain similar results for the tensor-product Haar system , and characterize the parameter range for which the dual of is trivial for . Bibliography: 31 titles.

We present convergence results in expectation for stochastic subspace correction schemes and their accelerated versions to solve symmetric positive-definite variational problems, and discuss their potential for achieving fault tolerance in an unreliable compute network. We employ the standard overlapping domain decomposition algorithm for PDE discretizations to discuss the latter aspect.

On the example of the simplest C spline quarklet construction of [1], we demonstrate the possible reduction of the complexity estimates for the approximation of singularity functions on the unit interval given in [3], to come closer to the complexity estimates known for hpmethods. To come up with a simplified argument, we explore the fact that the CDF biorthogonal spline wavelets [2] used in the construction of quarklets can be obtained by the lifting scheme [4]. Similar results are also possible for spline quarklet systems built from smooth B-splines with higher order of vanishing moments.

We reconsider some estimates the paper "M. Griebel, P. Oswald, On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math. 66 (1994), 449-463" concerning the hierarchical basis preconditioner for sparse grid discretizations. The improvement is in three directions: We consider arbitrary space dimensions d>1, give bounds for generalized sparse grid spaces with arbitrary monotone index set, and show that the bounds are sharp up to constants depending only on d, at least for a subclass of generalized sparse grid spaces containing full grid, standard sparse grid spaces, and energy-norm optimized sparse grid spaces.

We show that the d-dimensional Haar system H^d on the unit cube I^d is a Schauder basis in the classical Besov space B_{p,q,1}^s(I^d), 0

We consider an incremental approximation method for solving variational problems in infinite-dimensional separable Hilbert spaces, where in each step a randomly and independently selected subproblem from an infinite collection of subproblems is solved. We show that convergence rates for the expectation of the squared error can be guaranteed under weaker conditions than previously established in Griebel and Oswald (Constr Approx 44(1):121–139, 2016).

We present an approach to defining Hilbert spaces of functions depending on infinitely many variables or parameters, with emphasis on a weighted tensor product construction based on stable space splittings. The construction has been used in an exemplary way for guiding dimension- and scale-adaptive algorithms in application areas such as statistical learning theory, reduced order modeling, and information-based complexity. We prove results on compact embeddings, norm equivalences, and the estimation of ϵ -dimensions. A new condition for the equivalence of weighted ANOVA and anchored norms is also given.