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##### Department of Mathematics, University of California San Diego

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## The Locator Problem

##### Abstract:

The locator problem models the following physical situation. Suppose one lands an unmanned space craft on an unobservable terrain (e.g., under the clouds of Venus or on the backside of the moon) and wishes to determine the location of the landing site. Suppose that the craft can sample the altitude at the landing site and at several other spots (say at 100 meters east and at 100 meters west). However, the craft does not have a map of the altitudes of the terrain (i.e., $a(x,y)$), but only a single function, $p(x,y)$ that approximates the altitude. The locator problem is to find a function p from a family of functions $P$ that minimizes the error between the actual location of the craft and the computed location of the craft using the approximation $p$. The error is to be minimized over all possible locations that is we seek the $p$ in $P$ to minimize $$\Arrowvert (x,y)- p^{(-1)}(a(x,y))\Arrowvert$$ This is equivalent to classical approximation questions about existence and uniqueness of best approximations from this (non-linear) family of inverse functions. The question is most interesting when the elements in the setting are the most fundamental and basic: for example, when $P$ is the polynomials of degree n and the norm is the uniform norm or the $L_1$ norm. Although this is a rich theoretical setting with five fundamental elements to define (various metrics, data collections and families of approximating functions) and potentially has useful applications, we have the only known solutions. These are for $P$ the increasing polynomials of degree $n$; the domain and range being the unit interval and the norm being either the uniform norm or the $L_1$ norm. In this setting there exist best locator functions and they are unique.

### AP&M 412B

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