We study regularity properties of solutions to operator equations on patchwise smooth manifolds ∂Ω such as, e.g., boundaries of polyhedral domains Ω ⊂ R 3 . Using suitable biorthogonal wavelet bases Ψ, we introduce a new class of Besov-type spaces B α Ψ,q (L p (∂Ω)) of functions u : ∂Ω → C. Special attention is paid on the rate of convergence for best n-term wavelet approximation to functions in these scales since this determines the performance of adaptive numerical schemes. We show embeddings of (weighted) Sobolev spaces on ∂Ω into B α Ψ,τ (L τ (∂Ω)), 1/τ = α/2 + 1/2, which lead us to regularity assertions for the equations under consideration. Finally, we apply our results to a boundary integral equation of the second kind which arises from the double layer ansatz for Dirichlet problems for Laplace's equation in Ω.

We study linear problems S d defined on tensor products of Hilbert spaces with an additional (anti-) symmetry property. We construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its worst case error in terms of the eigenvalues λ of the operator W 1 = S † 1 S 1 of the univariate problem. Moreover, we show that this algorithm is optimal with respect to a wide class of algorithms and investigate its complexity. We clarify the influence of different (anti-) symmetry conditions on the complexity, compared to the classical unrestricted problem. In particular, for symmetric problems with λ 1 ≤ 1 we give characterizations for polynomial tractability and strong polynomial tractability in terms of λ and the amount of the assumed symmetry. Finally, we apply our results to the approximation problem of solutions of the electronic Schrödinger equation.

We study multivariate integration of functions that are invariant under permutations (of subsets) of their arguments. We find an upper bound for the nth minimal worst case error and show that under certain conditions, it can be bounded independent of the number of dimensions. In particular, we study the application of unshifted and randomly shifted rank-1 lattice rules in such a problem setting. We derive conditions under which multivariate integration is polynomially or strongly polynomially tractable with the Monte Carlo rate of convergence O(n −1/2 ). Furthermore, we prove that those tractability results can be achieved with shifted lattice rules and that the shifts are indeed necessary. Finally, we show the existence of rank-1 lattice rules whose worst case error on the permutation-and shiftinvariant spaces converge with (almost) optimal rate. That is, we derive error bounds of the form O(n −λ/2 ) for all 1 ≤ λ < 2α, where α denotes the smoothness of the spaces.

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