arXiv subjects

Han Feng

Publications and source records attributed to Han Feng.

45 records · Page 3Linked to original sources

arXiv2020

Mathematical Modeling of Business Reopening when Facing SARS-CoV-2 Pandemic: Protection, Cost and Risk

The sudden onset of the coronavirus (SARS-CoV-2) pandemic has resulted in tremendous loss of human life and economy in more than 210 countries and territories around the world. While self-protections such as wearing mask, sheltering in place and quarantine polices and strategies are necessary for containing virus transmission, tens of millions people in the U.S. have lost their jobs due to the shutdown of businesses. Therefore, how to reopen the economy safely while the virus is still circulating in population has become a problem of significant concern and importance to elected leaders and business executives. In this study, mathematical modeling is employed to quantify the profit generation and the infection risk simultaneously from the point of view of a business entity. Specifically, an ordinary differential equation model was developed to characterize disease transmission and infection risk. An algebraic equation is proposed to determine the net profit that a business entity can generate after reopening and take into account the costs associated of several protection/quarantine guidelines. All model parameters were calibrated based on various data and information sources. Sensitivity analyses and case studies were performed to illustrate the use of the model in practice.

physics.soc-ph↗

arXiv2019

Escaping Locally Optimal Decentralized Control Polices via Damping

We study the evolution of locally optimal decentralized controllers with the damping of the control system. Empirically it is shown that even for instances with an exponential number of connected components, damping merges all local solutions to the one global solution. We characterize the evolution of locally optimal solutions with the notion of hemi-continuity and further derive asymptotic properties of the objective function and of the locally optimal controllers as the damping becomes large. Especially, we prove that with enough damping, there is no spurious locally optimal controller with favorable control structures. The convoluted behavior of the locally optimal trajectory is illustrated with numerical examples.

math.OC↗

arXiv2019

Aggressive Local Search for Constrained Optimal Control Problems with Many Local Minima

This paper is concerned with numerically finding a global solution of constrained optimal control problems with many local minima. The focus is on the optimal decentralized control (ODC) problem, whose feasible set is recently shown to have an exponential number of connected components and consequently an exponential number of local minima. The rich literature of numerical algorithms for nonlinear optimization suggests that if a local search algorithm is initialized in an arbitrary connected component of the feasible set, it would search only within that component and find a stationary point there. This is based on the fact that numerical algorithms are designed to generate a sequence of points (via searching for descent directions and adjusting the step size), whose corresponding continuous path is trapped in a single connected component. In contrast with this perception rooted in convex optimization, we numerically illustrate that local search methods for non-convex constrained optimization can obliviously jump between different connected components to converge to a global minimum, via an aggressive step size adjustment using backtracking and the Armijio rule. To support the observations, we prove that from almost every arbitrary point in any connected component of the feasible set, it is possible to generate a sequence of points using local search to jump to different components and converge to a global solution. However, due to the NP-hardness of the problem, such fine-tuning of the parameters of a local search algorithm may need prior knowledge or be time consuming. This paper offers the first result on escaping non-global local solutions of constrained optimal control problems with complicated feasible sets.

math.OC↗

arXiv2017

Best polynomial approximation on the triangle

Let $E_n(f)_{α,β,γ}$ denote the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_{α,β,γ})$ on the triangle $\{(x,y): x, y \ge 0, x+y \le 1\}$, where $\varpi_{α,β,γ}(x,y) := x^αy ^β(1-x-y)^γ$ for $α,β,γ> -1$. Our main result gives a sharp estimate of $E_n(f)_{α,β,γ}$ in terms of the error of best approximation for higher order derivatives of $f$ in appropriate Sobolev spaces. The result also leads to a characterization of $E_n(f)_{α,β,γ}$ by a weighted $K$-functional.

math.CA↗

arXiv2017

Chebyshev-type cubature formulas for doubling weights on spheres, balls and simplexes

This paper proves that given a doubling weight $w$ on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$, there exists a positive constant $K_w$ such that for each positive integer $n$ and each integer $N\geq \max_{x\in \mathbb{S}^{d-1}} \frac {K_w} {w(B(x, n^{-1}))}$, there exists a set of $N$ distinct nodes $z_1,\cdots, z_N$ on $\mathbb{S}^{d-1}$ which admits a strict Chebyshev-type cubature formula (CF) of degree $n$ for the measure $w(x) dσ_d(x)$, $$ \frac 1{w(\mathbb{S}^{d-1})} \int_{\mathbb{S}^{d-1}} f(x) w(x)\, dσ_d(x)=\frac 1N \sum_{j=1}^N f(z_j),\ \ \forall f\inΠ_n^d, $$ and which, if in addition $w\in L^\infty(\mathbb{S}^{d-1})$, satisfies $$\min_{1\leq i\neq j\leq N}\mathtt{d}(z_i,z_j)\geq c_{w,d} N^{-\frac1{d-1}}$$ for some positive constant $c_{w,d}$. Here, $dσ_d$ and $\mathtt{d}(\cdot, \cdot)$ denote the surface Lebesgue measure and the geodesic distance on $\mathbb{S}^{d-1}$ respectively, $B(x,r)$ denotes the spherical cap with center $x\in\mathbb{S}^{d-1}$ and radius $r>0$, $w(E)=\int_E w(x) \, dσ_d(x)$ for $E\subset\mathbb{S}^{d-1}$, and $Π_n^d$ denotes the space of all spherical polynomials of degree at most $n$ on $\mathbb{S}^{d-1}$. It is also shown that the minimal number of nodes $\mathcal{N}_{n} (wdσ_d)$ in a strict Chebyshev-type CF of degree $n$ for a doubling weight $w$ on $\mathbb{S}^{d-1}$ satisfies $$\mathcal{N}_n (wdσ_d) \sim \max_{x\in \mathbb{S}^{d-1}} \frac 1 {w(B(x, n^{-1}))},\ \ n=1,2,\cdots.$$ Proofs of these results rely on new convex partitions of $\mathbb{S}^{d-1}$ that are regular with respect to a given weight $w$ and integer $N$. Our results extend the recent results of Bondarenko, Radchenko, and Viazovska on spherical designs ({\it Ann. of Math. (2)} {\bf 178}(2013), no. 2, 443--452,{\it Constr. Approx.} {\bf 41}(2015), no. 1, 93--112).

math.CA↗

arXiv2016

Gambling in contests with random initial law

This paper studies a variant of the contest model introduced in Seel and Strack [J. Econom. Theory 148 (2013) 2033-2048]. In the Seel-Strack contest, each agent or contestant privately observes a Brownian motion, absorbed at zero, and chooses when to stop it. The winner of the contest is the agent who stops at the highest value. The model assumes that all the processes start from a common value $x_0>0$ and the symmetric Nash equilibrium is for each agent to utilise a stopping rule which yields a randomised value for the stopped process. In the two-player contest, this randomised value has a uniform distribution on $[0,2x_0]$. In this paper, we consider a variant of the problem whereby the starting values of the Brownian motions are independent, nonnegative random variables that have a common law $μ$. We consider a two-player contest and prove the existence and uniqueness of a symmetric Nash equilibrium for the problem. The solution is that each agent should aim for the target law $ν$, where $ν$ is greater than or equal to $μ$ in convex order; $ν$ has an atom at zero of the same size as any atom of $μ$ at zero, and otherwise is atom free; on $(0,\infty)$ $ν$ has a decreasing density; and the density of $ν$ only decreases at points where the convex order constraint is binding.

econ.GN↗

arXiv2015

Uncertainty Principles on weighted spheres, balls and simplexes

This paper studies the uncertainty principle for spherical $h$-harmonic expansions on the unit sphere of $\mathbb{R}^d$ associated with a weight function invariant under a general finite reflection group, which is in full analogy with the classical Heisenberg inequality. Our proof is motivated by a new decomposition of the Dunkl-Laplace-Beltrami operator on the weighted sphere.

math.CA↗

arXiv2014

Reverse Hölder's inequality for spherical harmonics

This paper determines the sharp asymptotic order of the following reverse Hölder inequality for spherical harmonics $Y_n$ of degree $n$ on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$ as $n\to \infty$: \[\|Y_n\|_{L^q(\mathbb{S}^{d-1})}\leq C n^{α(p,q)}\|Y_n\|_{L^p(\mathbb{S}^{d-1})},\quad 0<p<q\leq \infty.\] In many cases, these sharp estimates turn out to be significantly better than the corresponding estimates in the Nilkolskii inequality for spherical polynomials. Furthermore, they allow us to improve two recent results on the restriction conjecture and the sharp Pitt inequalities for the Fourier transform on $\mathbb{R}^d$.

math.CA↗

arXiv2013

Gambling in contests with regret

This paper discusses the gambling contest introduced in Seel & Strack (Gambling in contests, Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 375, Mar 2012.) and considers the impact of adding a penalty associated with failure to follow a winning strategy. The Seel & Strack model consists of $n$-agents each of whom privately observes a transient diffusion process and chooses when to stop it. The player with the highest stopped value wins the contest, and each player's objective is to maximise their probability of winning the contest. We give a new derivation of the results of Seel & Strack based on a Lagrangian approach. Moreover, we consider an extension of the problem in which in the case when an agent is penalised when their strategy is suboptimal, in the sense that they do not win the contest, but there existed an alternative strategy which would have resulted in victory.

q-fin.PM↗