We consider the problem of solving linear elliptic partial differential equations on a high-dimensional product domain, which we take to be the unit hypercube. These equations can arise, for example, when solving stochastic differential equations. With a standard, piecewise polynomial approximation procedure on the unit hypercube, the rate of the error in H^1 (in terms of the number of unknowns) is at best (d-1)/n, where d is the polynomial order, and n is the space dimension. The fact that this rate decreases with n is known as the curse of dimensionality. Using that the unit hypercube is a product domain, the curse of dimensionality can be circumvented by using a sparse grid basis (Zenger 1991, Bungartz & Griebel 2004) and computing the Galerkin solution. We derive the smoothness that the unknown function being sought has to possess for these methods to work. It is shown that these requirements are not met by the solution of the Poisson equation with a right-hand side term that does not vanish near the boundaries. The key to overcome regularity restrictions is to apply nonlinear approximation. An adaptive method was developed in (Schwab & Stevenson 2008), based on the work of (Cohen, Dahmen, DeVore 2001), that converges with the same rate of the best N-term approximation for a large class of functions, including the solution of the mentioned Poisson problem. In Chapter 5 of this thesis, which will appear as (Dijkema, Schwab & Stevenson 2009), an actual realization of this method is presented. Moreover, it is shown that the constant factor that we may lose is independent of the space dimension. The cost of producing these approximations is proportional to their length with a constant factor that may grow with the space dimension, but only linearly. Most constant factors that are lost in the analysis depend on the Riesz condition number of the stiffness matrix of the wavelet basis. We present the construction of a biorthogonal wavelet basis whose Riesz condition number is generally much smaller than that of existing bases of this kind. We use this basis to construct a tensor product basis for spaces of divergence free functions. In the last chapter, we construct a wavelet basis which gives rise to sparse mass matrices and stiffness matrices. This basis can be used to greatly simplify the realization of adaptive wavelet methods, and improve the quantitative properties of such methods
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