The complexity of linear tensor product problems in (anti)symmetric Hilbert spaces

Abstract

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.

Construction of Quasi-Monte Carlo Rules for Multivariate Integration in Spaces of Permutation-Invariant Functions

“…The CBC construction tries to distinguish between otherwise identical dimensions; so no CBC construction of lattice rules for the integration of permutation-invariant functions can satisfy bounds like (14), in our opinion. Indeed, (16) in Proposition 4.4 above already indicates the complicated influence of already selected components z * 1 , . .…”

“…, d} is some fixed subset of coordinates. As discussed in [16,17] and [10] these functions satisfy the constraint that they are invariant under all permutations of the variables with indices in I d ; i.e., f (x) = f (P (x)) for all x ∈ [0, 1] d and all P ∈ S d ,…”

“…. , k d ) ∈ Z d k i1 ≤ k i2 ≤ · · · ≤ k i #I d , constitutes an orthonormal basis of S I d (F d (r α,β )); see [16] for more details. Here, accounts for the repetitions in the multiindex k. Since the subspace S I d (F d (r α,β )) is equipped with the same norm as the entire space F d (r α,β ), it is again a RKHS.…”

“…, d, according to (22) implies (17) (iv) Observe that (in general) a lattice rule based on a generating vector z * constructed as above is unfortunately not extensible in the dimension. The reason is that with increasing d the factors c u,I d in the definition of B I d ,ℓ (see (16)) will change such that the contributions of the subsets u ⊆ {1, . .…”

“…the exchange of electrons having the same spin, as described by Pauli's exclusion principle which is well-known in theoretical physics. In [16], general linear problems defined on spaces equipped with this type of (anti-) symmetry constraints were shown to be (strongly) polynomially tractable (under certain conditions). That is, the complexity of solving such problems depends at most weakly on the dimension.…”