Optimal Algorithms for Computing Average Temperatures

G L Mullendore,G Petrova,S Foucart,M Hielsberg,P Wojtaszczyk

doi:10.1515/mcwf-2019-0003

Abstract

AbstractA numerical algorithm is presented for computing average global temperature (or other quantities of interest such as average precipitation) from measurements taken at speci_ed locations and times. The algorithm is proven to be in a certain sense optimal. The analysis of the optimal algorithm provides a sharp a priori bound on the error between the computed value and the true average global temperature. This a priori bound involves a computable compatibility constant which assesses the quality of the measurements for the chosen model. The optimal algorithm is constructed by solving a convex minimization problem and involves a set of functions selected a priori in relation to the model. It is shown that the solution promotes sparsity and hence utilizes a smaller number of well-chosen data sites than those provided. The algorithm is then applied to canonical data sets and mathematically generic models for the computation of average temperature and average precipitation over given regions and given time intervals. A comparison is provided between the proposed algorithms and existing methods.

Highlights

Computing average temperatures is a particular instance of a common task in data processing, namely that of exploiting measurements made on a function f to estimate a quantity Q(f ) that depends on f, referred to as Quantity of Interest (QoI) below
A numerical algorithm is presented for computing average global temperature from measurements taken at speci ed locations and times
Recall that once the solution a* = [a*, . . . , a*m] ∈ Rm to (11), where m is the number of data sites, has been computed o ine, the nal step for approximating the quantity of interest consists in outputting for the given data values w, . . . , wm

Summary

Introduction

Computing average temperatures is a particular instance of a common task in data processing, namely that of exploiting measurements made on a function f to estimate a quantity Q(f ) that depends on f , referred to as Quantity of Interest (QoI) below. The analysis of the optimal algorithm provides a sharp a priori bound on the error between the computed value and the true average global temperature.

Results

Conclusion