In many applications, as for example physics, economics, finance and computational sciences, high-dimensional integration and approximation are problems which have to be solved numerically. In this thesis we study several aspects of high-dimensional solution algorithms for these problems.
In the first part of the thesis we consider tractability of multivariate continuous problems. This means that we are interested in how much information a numerical algorithm needs to solve the problem with accuracy . We study how fast the number of information evaluations required increases if the number of variables goes to infinity or the error demand tends to zero. We consider the two examples of a weighted Hermite space and of a hybrid function space.
In the second part of the thesis we investigate the problem of constructing point sets in the s-dimensional unit cube, which are used in a certain type of numerical algorithms, so-called quasi-Monte Carlo algorithms, which are widely used to numerically solve high-dimensional integration problems. We present several fast construction methods which provide point sets having certain good properties.