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The challenge of mastering computational tasks of enormous size tends tofrequently override questioning the quality of the numerical outcome in termsof accuracy. By this we do not mean the accuracy within the discrete setting,which itself may also be far from evident for ill-conditioned problems or wheniterative solvers are involved. By accuracy-controlled computation we mean thedeviation of the numerical approximation from the exact solution of anunderlying continuous problem in a relevant metric, which has been theinitiating interest in the first place. Can the accuracy of a numerical resultbe rigorously certified - a question that is particularly important in thecontext of uncertainty quantification, when many possible sources ofuncertainties interact. This is the guiding question throughout this article,which reviews recent developments of low-rank approximation methods forproblems in high spatial dimensions. In particular, we highlight the role ofadaptivity when dealing with such strongly nonlinear methods that integrate ina natural way issues of discrete and continuous accuracy.

Adaptive Low-Rank Approximations for Operator Equations: Accuracy Control and Computational Complexity