Tight decomposition functions for mixed monotonicity
Mixed monotonicity is a property of a system’s vector field that says that the vector field admits a decomposition function, in a lifted space, that has some order preserving properties. It is recently shown that this property allows one to efficiently over-approximate the system’s onestep reachable set with a hyperinterval, which is obtained by evaluating the vector field’s decomposition function at two points. Such decomposition functions are usually not unique and some decompositions may not be tight in the sense that the resulting hyperintervals are not the smallest ones that contain the exact one-step reachable set, which leads to conservative over-approximation. In this paper, we show that for a general class of functions, there exists a tight decomposition, which can be implicitly constructed as the solution of certain optimization problems. This implies that any function from Rn to Rm (hence any forward complete system) is mixed-monotone. However, the usefulness of the constructed tight decomposition functions is limited by the fact that it might not be possible to evaluate them efficiently. We show that under certain conditions, the tight decompositions can reduce to a function with explicit expression, which can be directly evaluated. This result suggests that it is not mixed monotonicity itself, more »
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58th IEEE Conference on Decision and Control (CDC)
2. Abstract The classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop families of direct serendipity and direct mixed finite element spaces, which achieve optimal approximation properties and have minimal local dimension. The set of local shape functions for either the serendipity or mixed elements contains the full set of scalar or vector polynomials of degree r , respectively, defined directly on each element (i.e., not mapped from a reference element). Because there are not enough degrees of freedom for global $$H^1$$ H 1 or $$H(\text {div})$$ H ( div ) conformity, exactly two supplemental shape functions must be added to each element when $$r\ge 2$$ r ≥ 2 , and only one when $$r=1$$ r = 1 . The specific choice of supplemental functions gives rise to different families of direct elements. These new spaces are related through a de Rham complex. For index $$r\ge 1$$ r ≥ 1 , the new families of serendipity spaces $${\mathscr {DS}}_{r+1}$$ DS r + 1 are the precursors under the curl operator of our direct mixed finite element spaces, which can be constructed to have reduced or full $$H(\text {div})$$ H (more »
5. A Boolean {\em $k$-monotone} function defined over a finite poset domain ${\cal D}$ alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in ${\cal D}$. Therefore, $k$-monotone functions are natural generalizations of the classical {\em monotone} functions, which are the {\em $1$-monotone} functions. Motivated by the recent interest in $k$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $k$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $k$-monotone (or are close to being $k$-monotone) from functions that are far from being $k$-monotone. Our results include the following: \begin{enumerate} \item We demonstrate a separation between testing $k$-monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$; \item We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=\omega(\log d)$: testing $k$-monotonicity can be performed with $2^{O(\sqrt d \cdot \log d\cdot \log{1/\eps})}$ queries, while learning $k$-monotone functions requires $2^{\Omega(k\cdot \sqrt d\cdot{1/\eps})}$ queries (Blais et al. (RANDOM 2015)). \item We present a tolerant test for functions $f\colon[n]^d\to \{0,1\}$ with complexity independent ofmore »