Liberating the Dimension for Function Approximation and Integration

Abstract

We discuss recent results on the complexity and tractability of problems dealing with ∞-variate functions. Such problems, especially path integrals, arise in many areas including mathematical finance, quantum physics and chemistry, and stochastic differential equations. It is possible to replace the ∞-variate problem by one that has only d variables since the difference between the two problems diminishes with d approaching infinity. Therefore, one could use algorithms obtained in the Information-Based Complexity study, where problems with arbitrarily large but fixed d have been analyzed. However, to get the optimal results, the choice of a specific value of d should be a part of an efficient algorithm. This is why the approach discussed in the present paper is called liberating the dimension. Such a choice should depend on the cost of sampling d-variate functions and on the error demand $\epsilon$. Actually, as recently observed for a specific class of problems, optimal algorithms are from a family of changing dimension algorithms which approximate ∞-variate functions by a combination of special functions, each depending on a different set of variables. Moreover, each such set contains no more than $d(\epsilon ) = \mathcal{O}(\ln (1/\epsilon )/\ln (\ln (1/\epsilon )))$ variables. This is why the new algorithms have the total cost polynomial in $1/\epsilon$ even if the cost of sampling a d-variate function is exponential in d.