High-Dimensional Operator Equations - Error Control and Complexity
Conventional numerical concepts, applied to problems posed in high spatial dimensions, suffer from the "curse of dimensionality" paraphrasing an often prohibitive exponential dependence of computational cost on the number of variables. Partial differential equations (PDEs) in high dimensional phase space, such as Fokker-Planck and Schrödinger equations, or Uncertainty Quantification for families of PDEs, depending on a large (even infinite) number of parameters, are important examples of such problems. For both scenarios we discuss intrinsic obstructions, characteristic approximability properties and adaptive solution strategies that realize a given target accuracy while avoiding the curse of dimensionality. Conceptual ingredients are low-rank and tensor methods or reduced basis techniques.