Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations

In this work, we develop methods for the numerical approximation of higher-dimensional functions, given as solutions of linear elliptic operator equations or as eigenfunctions of such operators. The approximations of these functions are generated by iterative schemes, where iterates are represented in a multiplicatively nonlinear tensor decomposition of their wavelet coefficients. As a concrete application problem, we consider the stationary electronic Schrödinger equation, the fundamental equation of quantum chemistry, in model problems with one and two electrons, corresponding to three and six space dimensions, respectively. Besides the dimensionality of these problems, the characteristic singular behaviour of eigenfunctions at the singularities of the potential terms is of particular interest. We first compare the convergence rates that can be achieved for two-electron model systems by different types of wavelet approximation: on the one hand, direct wavelet approximation with linearly parametrized wavelets, using uniform or adaptive refinement, and on the other hand, wavelet approximation with low-rank representation of coefficients, which yields the mentioned nonlinear parametrization of wavelets. In each case, the improvement of possible convergence rates by a combination with a problem-adapted, so-called explicitly correlated ansatz is studied as well. The proposed adaptive methods for computing approximate solutions given by low-rank representations of wavelet coefficients have a structure analogous to known adaptive wavelet methods, based on a perturbed application of an iterative scheme to the wavelet representation of the underlying infinite-dimensional problem. As common for wavelet methods, the transformation to a wavelet representation on `2 involves a diagonal rescaling of the corresponding representation matrices of operators. For finite-dimensional subproblems, this rescaling corresponds to a preconditioning. We consider, in particular, the difficulties that arise at this point in the context of low-rank representations, which leads us to a modification of the tensor representations of coefficients by an appropriate partitioning of wavelet index sets. On this basis, algorithmic building blocks are developed that enable the realization of the basic operations required for adaptive wavelet methods also for suitable low-rank tensor representations. This concerns, in particular, the approximate evaluation of residuals, and the approximation of given iterates with lower representation complexity. We prove the convergence of the resulting methods for suitable step sizes and error tolerances. Here, all parameters controlling the iteration depend only on the infinite-dimensional problem and the wavelet basis, but not on a concrete discretization. The adaptive eigenvalue solver is tested numerically for the electronic Schrödinger equation in the cases of hydrogen and helium, as well as for further model problems. In all those cases, a suitable low-rank format for the approximation of solution coefficients is the Tucker tensor format; however, attention is paid to the applicability of the methods in combination with recently discovered tensor formats that are suitable for very high dimensions. In the realization of the method, the approximation of operators plays a central role. For the particular Coulomb potential terms arising in the Schrödinger equations, we construct approximations by a combination of separable expansions by exponential sums and wavelet matrix compression, and also study a specialized integration scheme for the required integrals.