Gowri Suryanarayana

Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions

We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection between the worst-case errors of our algorithms in the cosine space and the worst-case errors of some related algorithms in the well-known weighted Korobov space of smooth periodic functions. By exploiting this connection, we are able to obtain constructive worst-case error bounds with good convergence rates for the cosine space.

Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions

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)

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

\mathcal {O}(n^{-1/2})

Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation

. 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 shift-invariant spaces converge with (almost) optimal rate. That is, we derive error bounds of the form 𝓞(n−λ/2)

Reconstruction and Collocation of a Class of Non-periodic Functions by Sampling Along Tent-Transformed Rank-1 Lattices

\mathcal {O}(n^{-\lambda /2})

Approximation in cosine space using tent transformed lattice rules

for all 1≤λ<2α, where α denotes the smoothness of the spaces.

Strang splitting in combination with rank-

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens et al. (Adv Comput Math 42(1):55–84, 2016), the authors derived an upper estimate for the nth minimal worst case error for such problems and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-1 lattice rule that obtains a rate of convergence arbitrarily close to

and rank-

\mathcal {O}(n^{-\alpha })