Robustness of Solutions of Almost Every System of Equations
Sana Jahedi, TIMOTHY D. SAUER, James A. Yorke · SIAM Journal on Applied Mathematics · 2022
Abstract. In mathematical modeling, it is common to have an equation [Formula: see text], where the exact form of [Formula: see text] is not known. This article shows that there are large classes of [Formula: see text] where almost all [Formula: see text] share the same properties. The classes we investigate are vector spaces [Formula: see text] of [Formula: see text] functions [Formula: see text] that satisfy the following condition: [Formula: see text] has “almost constant rank" (ACR) if there is a constant integer [Formula: see text] such that rank[Formula: see text] for “almost every" [Formula: see text] and almost every [Formula: see text]. If the vector space [Formula: see text] is finite dimensional, then “almost every" is with respect to the Lebesgue measure on [Formula: see text], and otherwise, it means almost every in the sense of prevalence, as described herein. Most function spaces commonly used for modeling purposes are ACR. In particular, we show that if all of the functions in [Formula: see text] are linear or polynomial or real analytic, or if [Formula: see text] is the set of all functions in a “structured system", then [Formula: see text] is ACR. For each [Formula: see text] and [Formula: see text], the solution set of [Formula: see text] is SolSet[Formula: see text]. A solution set of [Formula: see text] is called robust if it persists despite small changes in [Formula: see text] and [Formula: see text]. The following two global results are proved for almost every [Formula: see text] in an ACR vector space [Formula: see text]: (1) Either the solution set SolSet[Formula: see text] is robust for almost every [Formula: see text], or none of the solution sets are robust. (2) The solution set SolSet[Formula: see text] is a [Formula: see text]-manifold of dimension [Formula: see text]. In particular, [Formula: see text] is the same for almost every [Formula: see text].