Author:

Abstract:

The aim of this research is to develop algorithms to approximate the solutions of problems defined on spaces of d-variate functions, where d can be arbitrarily large. We study quasi-Monte Carlo methods, which in contrast to Monte Carlo methods, use deterministic point sets. These methods converge significantly faster with well chosen points that minimize deterministic error bounds. Additionally, we require that the computational hardness of these methods has only a moderate or no dependence on d. We study this using the information complexity n(d, epsilon), the minimal number of information operations required to achieve an approximation with an error of epsilon or less. If n(d, epsilon) grows exponentially in d, the problem is said to have the 'curse of dimension' and the goal is to vanquish this curse. Our research resulted in several contributions. We studied multivariate integration for a Hilbert space of functions that are invariant under permutations of the variables; this is inspired from the symmetry constraints induced on solutions of the Schrödinger equation for indistinguishable particles. We showed that there exist quasi-Monte Carlo methods that achieve a convergence rate of O(n^−1/2) with the implied constant depending only polynomially on d and under some conditions independent of d (and that shifted rank-1 lattice rules can attain the same). We then provided a construction algorithm for a shifted rank-1 lattice that obtains the (almost) optimal rate of convergence. Next, we extended the use of rank-1 lattice points, traditionally used for the cubature and approximation of periodic functions, to the non-periodic setting. We did this in the cosine space setting, where we extensively studied the use of tent-transformed lattice points for cubature and approximation of nonperiodic functions. We were able to reuse algorithms from the periodic setting by establishing relations between the cosine space and the Korobov space of periodic functions. Finally we studied some special cases of approximation in two dimensions. We derived explicit expressions for various degrees of exactness for the Fibonacci lattice points. We end the thesis with an insight on the Lebesgue constant of trigonometric interpolation using lattice points as interpolation nodes.