Surface Approximation Is Sometimes Easier than Surface Integration

Abstract

The approximation and integration problems consist of finding an approximation to a function $f$ or its integral over some fixed domain $\Sigma$. For the classical version of these problems, we have partial information about the functions $f$ and complete information about the domain $\Sigma$; for example, $\Sigma$ might be a cube or ball in $\reals^d$. When this holds, it is generally the case that integration is not harder than approximation; moreover, integration can be much easier than approximation. What happens if we have partial information about $\Sigma$? This paper studies the surface approximation and surface integration problems, in which $\Sigma=\Sigma_g$ for functions $g$. More specifically, the functions $f$ are $r$ times continuously differentiable scalar functions of $l$ variables, and the functions $g$ are $s$ times continuously differentiable injective functions of $d$ variables with $l$ components. The class of surfaces considered is generated as images of cubes or balls, or as oriented cellulated regions. Error for the surface approximation problem is measured in the $L_q$-sense. These problems are well defined, provided that $d\le l$, $r\ge 0$, and $s\ge 1$. Information consists of function evaluations of $f$ and $g$. We show that the $\e$-complexity of surface approximation is proportional to $(1/\e)^{1/\mu}$ with $\mu=\mrs/d$. We also show that if $s\ge 2$, then the $\e$-complexity of surface integration is proportional to $(1/\e)^{1/\nu}$ with $$ \nu=\min\left\{ \frac{r}{d},\frac{s-\delta_{s,1}(1-\delta_{d,l})}{\min\{d,l-1\}}\right\}. $$ (This bound holds as well for several subcases of $s=1$; we conjecture that it holds for all $r\ge0$, $s\ge1$, and $d\le l$.) Using these results, we determine when surface approximation is easier than, as easy as, or harder than, surface integration; all three possibilities can occur. In particular, we find that if $r=s=1$ and $d<l$, then $\mu=1/d$ and $\nu=0$, so that surface integration is unsolvable and surface approximation is solvable; this is an extreme case for which surface approximation is easier than surface integration.