Hilbert numbers of Sobolev spaces with mixed smoothness in the sup-norm

Let Ω i ⊂ R n i \Omega _i\subset \mathbb {R}^{n_i} , i = 1 , … , m i=1,\ldots ,m , be given domains. In this article, we study the low-rank approximation with respect to L 2 ( Ω 1 × ⋯ × Ω m ) L^2(\Omega _1\times \dots \times \Omega _m) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare Griebel and Harbrecht [IMA J. Numer. Anal. 34 (2014), pp. 28–54] and Griebel and Harbrecht [IMA J. Numer. Anal. 39 (2019), pp. 1652–1671], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.

We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by $n^{-1/2}$ times the $L_2$-norm of the Fourier transform outside a region around the origin, where $n$ is the expected number of function evaluations. As corollaries we obtain the optimal order of convergence for functions from the Sobolev spaces $H^s_p$ with isotropic, anisotropic, or mixed smoothness with given compact support for all values of the parameters. If the region of integration is the unit cube, we obtain the same optimal orders for functions without boundary conditions. This proves, in particular, that the optimal order of convergence in the latter case is $n^{-s-1/2}$ for $p\ge2$, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than $n^{-1/2}$ for a whole class of natural function spaces.

Sampling numbers of smoothness classes via ℓ1-minimization

It is well-known that the problem of sampling recovery in the L 2 L_2 -norm on unweighted Korobov spaces (Sobolev spaces with mixed smoothness) as well as classical smoothness classes such as Hölder classes suffers from the curse of dimensionality. We show that the problem is tractable for those classes if they are intersected with the Wiener algebra of functions with summable Fourier coefficients. In fact, this is a relatively simple implication of powerful results from the theory of compressed sensing. Tractability is achieved by the use of non-linear algorithms, while linear algorithms cannot do the job.

Monte Carlo methods for uniform approximation on periodic Sobolev spaces with mixed smoothness

Numerical weighted integration of functions having mixed smoothness

Non-linear Approximations Using Sets of Finite Cardinality or Finite Pseudo-dimension

We study the information-based complexity of approximating integrals of smooth functions at absolute precision e > 0 with confidence level 1 − δ ∈ (0, 1) using function evaluations within randomized algorithms. The probabilistic error criterion is new in the context of integrating smooth functions. In previous research, Monte Carlo integration was studied in terms of the expected error (or the root mean squared error), for which linear methods achieve optimal rates of the error e(n) in terms of the number n of function evaluations. In our context, usually methods that provide optimal confidence properties exhibit non-linear features. The optimal probabilistic error rate e(n,δ) for multivariate functions from classical isotropic Sobolev spaces ${W_{p}^{r}}(G)$ with sufficient smoothness on bounded Lipschitz domains $G \subset {\mathbb R}^{d}$ is determined. It turns out that the integrability index p has an effect on the influence of the uncertainty δ in the complexity. In the limiting case p = 1, we see that deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the additional effort for diminishing the uncertainty. Finally, we add a discussion about this problem for function spaces with mixed smoothness.

Let $$X_n = \{x^j\}_{j=1}^n$$Xn={xj}j=1n be a set of n points in the d-cube $${\mathbb {I}}^d:=[0,1]^d$$Id:=[0,1]d, and $$\Phi _n = \{\varphi _j\}_{j =1}^n$$źn={źj}j=1n a family of n functions on $${\mathbb {I}}^d$$Id. We consider the approximate recovery of functions f on $${{\mathbb {I}}}^d$$Id from the sampled values $$f(x^1), \ldots , f(x^n)$$f(x1),ź,f(xn), by the linear sampling algorithm $$ L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. $$Ln(Xn,źn,f):=źj=1nf(xj)źj. The error of sampling recovery is measured in the norm of the space $$L_q({\mathbb {I}}^d)$$Lq(Id)-norm or the energy quasi-norm of the isotropic Sobolev space $$W^\gamma _q({\mathbb {I}}^d)$$Wqź(Id) for $$1 0. Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces $$B^{\alpha ,\beta }_{p,\theta }$$Bp,źź,β of a hybrid of mixed smoothness $$\alpha > 0$$ź>0 and isotropic smoothness $$\beta \in {\mathbb {R}}$$βźR, and spaces $$B^a_{p,\theta }$$Bp,źa of a nonuniform mixed smoothness $$a \in {\mathbb {R}}^d_+$$aźR+d. We constructed asymptotically optimal linear sampling algorithms $$L_n(X_n^*,\Phi _n^*,\cdot )$$Ln(Xnź,źnź,·) on special sparse grids $$X_n^*$$Xnź and a family $$\Phi _n^*$$źnź of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in $$B^{\alpha ,\beta }_{p,\theta }$$Bp,źź,β and $$B^a_{p,\theta }$$Bp,źa. As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.

We investigate the approximation of d-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $$s>0$$ on the d-dimensional torus, where the approximation error is measured in the $$L_2$$ -norm. In other words, we study the approximation numbers $$a_n$$ of the Sobolev embeddings $$H^s_\mathrm{mix}(\mathbb {T}^d)\hookrightarrow L_2(\mathbb {T}^d)$$ , with particular emphasis on the dependence on the dimension d. For any fixed smoothness $$s>0$$ , we find two-sided estimates for the approximation numbers as a function in n and d. We observe super-exponential decay of the constants in d, if n, the number of linear samples of f, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by approximations using only a few linear samples (small n). We present some surprising results for the so-called “preasymptotic” decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.

We consider the Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity and potential in the body based on the measurement of the boundary voltages on the m m electrodes for a given electrode current. The variational formulation is introduced as a PDE constrained coefficient optimal control problem in Sobolev spaces with dominating mixed smoothness. Electrical conductivity and boundary voltages are control parameters, and the cost functional is the L 2 L_2 -norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. EIT optimal control problem is fully discretized using the method of finite differences. The existence of the optimal control and the convergence of the sequence of finite-dimensional optimal control problems to EIT coefficient optimal control problem is proved both with respect to functional and control in 2- and 3-dimensional domains.

We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness $\alpha\geq 2$ defined over the $s$-dimensional unit cube. We propose a new quasi-Monte Carlo (QMC)-based quadrature rule, named \emph{extrapolated polynomial lattice rule}, which achieves the almost optimal rate of convergence. Extrapolated polynomial lattice rules are constructed in two steps: i) construction of classical polynomial lattice rules over $\mathbb{F}_b$ with $\alpha$ consecutive sizes of nodes, $b^{m-\alpha+1},\ldots,b^{m}$, and ii) recursive application of Richardson extrapolation to a chain of $\alpha$ approximate values of the integral obtained by consecutive polynomial lattice rules. We prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence of the worst-case error in Sobolev spaces with general weights. Then, by restricting to product weights, we show that such good extrapolated polynomial lattice rules can be constructed by the fast component-by-component algorithm under a computable quality criterion. The required total construction cost is of order $(s+\alpha)N\log N$, which improves the currently known result for interlaced polynomial lattice rule, that is of order $s\alpha N\log N$. We also study the dependence of the worst-case error bound on the dimension. A big advantage of our method compared to interlaced polynomial lattice rules is that the fast QMC matrix vector method can be used in this setting, while still achieving the same rate of convergence. Such a method was previously not known. Numerical experiments for test integrands support our theoretical result.

Optimal approximation of multivariate periodic Sobolev functions in the sup-norm

This paper describes an extension of Fourier approximation methods for multivariate functions defined on the torus $\mathbb{T}^d$ to functions in a weighted Hilbert space $L_{2}(\mathbb{R}^d, \omega)$ via a multivariate change of variables $\psi:\left(-\frac{1}{2},\frac{1}{2}\right)^d\to\mathbb{R}^d$. We establish sufficient conditions on $\psi$ and $\omega$ such that the composition of a function in such a weighted Hilbert space with $\psi$ yields a function in the Sobolev space $H_{\mathrm{mix}}^{m}(\mathbb{T}^d)$ of functions on the torus with mixed smoothness of natural order $m \in \mathbb{N}_{0}$. In this approach we adapt algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus $\mathbb{T}^d$ based on single and multiple reconstructing rank-$1$ lattices. Since in applications it may be difficult to choose a related function space, we make use of dimension incremental construction methods for sparse frequency sets. Various numerical tests confirm obtained theoretical results for the transformed methods.

<abstract><p>It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Hölder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the Hölder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $ n $-dimensional rectangle and the inductive application of the Sobolev trace embedding results. The new method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. Counterexamples demonstrate that in terms of minimal weak regularity degree the Sobolev spaces with dominating mixed smoothness present the largest class of weakly differentiable functions with the upgrade of pointwise regularity to continuity. Remarkably, it also presents the largest class of weakly differentiable functions where the generalized Newton-Leibniz formula holds.</p></abstract>