Truncation Dimension for Linear Problems on Multivariate Function Spaces
Provides theoretical bounds on the truncation dimension for linear problems on weighted multivariate function spaces, relevant for high-dimensional approximation.
The paper investigates when approximating a multivariate function by setting all but the first k variables to zero yields small error, and determines the truncation dimension k(ε) needed to achieve error ε. It finds that k(ε) can be very small even for weights with modest decay.
The paper considers linear problems on weighted spaces of multivariate functions of many variables. The main questions addressed are: When is it possible to approximate the solution for the original function of very many variables by the solution for the same function; however with all but the first $k$ variables set to zero, so that the corresponding error is small? What is the truncation dimension, i.e., the smallest number $k=k(\varepsilon)$ such that the corresponding error is bounded by a given error demand $\varepsilon$? Surprisingly, $k(\varepsilon)$ could be very small even for weights with a modest speed of convergence to zero.