Numerical optimization algorithms for wavefront phase retrieval from multiple measurements
Ji Li Tie Zhou

Wavefront phase retrieval from a set of intensity measurements can be formulated as an optimization problem. Two nonconvex models (MLP and its variant LS) based on maximum likelihood estimation are investigated in this paper. We derive numerical optimization algorithms for real-valued function of complex variables and apply them to solve the wavefront phase retrieval problem efficiently. Numerical simulation is given with application to three test examples. The LS model shows better numerical performance than that of the MLP model. An explanation for this is that the distribution of the eigenvalues of Hessian matrix of the LS model is more clustered than that of the MLP model. We find that the LBFGS method shows more robust performance and takes fewer calculations than other line search methods for this problem.

keywords: Wavefront phase retrieval C-R calculus LBFGS
Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions
Hong Lu Ji Li Joseph Shackelford Jeremy Vorenberg Mingji Zhang

A quasi-one-dimensional steady-state Poisson-Nernst-Planck model with Bikerman's local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel is analyzed. Of particular interest is the qualitative properties of ionic flows in terms of individual fluxes without the assumption of electroneutrality conditions, which is more realistic to study ionic flow properties of interest. This is the novelty of this work. Our result shows that ⅰ) boundary concentrations and relative size of ion species play critical roles in characterizing ion size effects on individual fluxes; ⅱ) the first order approximation $\mathcal{J}_{k1} = D_kJ_{k1}$ in ion volume of individual fluxes $\mathcal{ J}_k = D_kJ_k$ is linear in boundary potential, furthermore, the signs of $\partial_V \mathcal{ J}_{k1}$ and $\partial^2_{Vλ} \mathcal{J}_{k1}$, which play key roles in characterizing ion size effects on ionic flows can be both negative depending further on boundary concentrations while they are always positive and independent of boundary concentrations under electroneutrality conditions (see Corollaries 3.2-3.3, Theorems 3.4-3.5 and Proposition 3.7). Numerical simulations are performed to identify some critical potentials defined in (2). We believe our results will provide useful insights for numerical and even experimental studies of ionic flows through membrane channels.

keywords: Ion channel local hard-sphere potential critical potentials individual fluxes electroneutrality conditions
Invariant foliations for random dynamical systems
Ji Li Kening Lu Peter W. Bates
We prove the existence of invariant foliations of stable and unstable manifolds of a normally hyperbolic random invariant manifold. The normally hyperbolic random invariant manifold referred to here is that which was shown to exist in a previous paper when a deterministic dynamical system having a normally hyperbolic invariant manifold is subjected to a small random perturbation.
keywords: random normally hyperbolic invariant manifolds Random dynamical systems random invariant foliations.
A smooth QP-free algorithm without a penalty function or a filter for mathematical programs with complementarity constraints
Jianling Li Chunting Lu Youfang Zeng
In this paper, a smooth QP-free algorithm without a penalty function or a filter is proposed for a special kind of mathematical programs with complementarity constraints (MPCC for short). Firstly, the investigated problem is transformed into sequential parametric standard nonlinear programs by perturbed techniques and a generalized complementarity function. Then the trial step, at each iteration, is accepted such that either the value of the objective function or the measure of the constraint violation is sufficiently reduced. Finally, it is shown that every limit point of the iterative sequence is feasible, and there exists a limit point that is a KKT point for the problem under mild conditions.
keywords: penalty-function-free global convergence. Mathematical programs complementarity constraints QP-free filter-free
Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method
Jingzhi Li Hongyu Liu Qi Wang
Some effective imaging schemes for inverse scattering problems were recently proposed in [13,14] for locating multiple multiscale electromagnetic (EM) scatterers, namely a combination of components of possible small size and regular size compared to the detecting EM wavelength. In this paper, instead of using a single far-field measurement, we relax the assumption of one fixed frequency to multiple ones, and develop efficient numerical techniques to speed up those imaging schemes by adopting multi-frequency and Multilevel ideas in a two-stage manner. Numerical tests are presented to demonstrate the efficiency and the salient features of the proposed fast imaging scheme.
keywords: single-shot multilevel. sampling method Inverse scattering two-stage
Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands
Jianbin Li Niu Yu Zhixue Liu Lianjie Shu
We examine the pure rebate strategies in a two-echelon supply chain under stochastic demand with multiplicative error. Given exogenous wholesale price and retail price, we characterize the unique Nash equilibrium when both manufacturer and retailer provide rebate policy to consumers under nonlinear and linear price-dependent demand functions, including iso-elastic multiplicative demand function (EMDF) and linear multiplicative demand function (LMDF). Based on a game theoretical framework, we prove that there still exists a unique equilibrium when the price elasticity is rather small with constraint conditions in the former case. We also find that in this case the retailer(manufacturer) may increase its rebate value in reaction to the manufacturer's(retailer's) rebate value in order to stimulate sales, which is contrary to the conventional wisdom that the retailer(manufacturer) will shrink its rebate value to gain an ``extra advantage" unfairly. As a result, both parties share the same profit at equilibrium. Further, we compare the expected profit outcomes at equilibrium among joint-rebate game, single-party rebate game and no-rebate game by using numerical examples. It is shown that the joint-rebate policy is not always dominates the others unless the price elasticity is sufficiently flexible.
keywords: Supply chain equilibrium. newsvendor model rebate Nash game
A mathematical model of HTLV-I infection with two time delays
Xuejuan Lu Lulu Hui Shengqiang Liu Jia Li
In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxic T lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection, where one is the intracellular infection delay and the other is the immune delay to account for a series of immunological events leading to the CTL response. We show that the global dynamics of the model system are determined by two threshold values $R_0$, the corresponding reproductive number of a viral infection, and $R_1$, the corresponding reproductive number of a CTL response, respectively. If $R_0<1$, the infection-free equilibrium is globally asymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-free equilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with no persistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-I infection becomes chronic with a persistent CTL response. Moreover, we show that the immune delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numerical simulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize the HAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, the stability of the HAM/TSP equilibrium may generate rich dynamics combining the ``stabilizing" effects from the intracellular delay with those ``destabilizing" influences from immune delay.
keywords: Hopf bifurcation. Epidemic threshold HTLV-I infection Lyapunov functional time delay
Optimal capacity reservation policy on innovative product
Jianbin Li Ruina Yang Niu Yu
We examine the problem of optimal capacity reservation policy on innovative product in a setting of one supplier and one retailer. The parameters of capacity reservation policy are two dimensional: reservation price and excess capacity that the supplier will have in additional to the reservation amount. The above problem is analyzed using a two-stage Stackelberg game. In the first stage, the supplier announces the capacity reservation policy. The retailer forecasts the future demand and then determines the reservation amount. After receiving the reservation amount, the supplier expands the capacity. In the second stage, the uncertainty in demand is resolved and the retailer places a firm order. The supplier salvages the excess capacity and the associated payments are made.
    In the paper, with exogenous reservation price or exogenous excess capacity level, we study the optimal expansion policy and then investigate the impacts of reservation price or excess capacity level on the optimal strategies. Finally, we characterize Nash Equilibrium and derive the optimal capacity reservation policy, in which the supplier will adopt exact capacity expansion policy.
keywords: Capacity reservation game theory deductible reservation contract.
Imaging acoustic obstacles by singular and hypersingular point sources
Jingzhi Li Hongyu Liu Hongpeng Sun Jun Zou
We investigate a qualitative method for imaging acoustic obstacles in two and three dimensions by boundary measurements corresponding to hypersingular point sources. Rigorous mathematical justification of the imaging method is established, and numerical experiments are presented to illustrate the effectiveness of the proposed imaging scheme.
keywords: hypersingular point sources. Dirichelet-to-Neumann map sampling method indicator function Inverse acoustic scattering boundary measurment
Jibin Li Kening Lu Junping Shi Chongchun Zeng
This special issue of Discrete and Continuous Dynamical Systems-A is dedicated to Peter W. Bates on the occasion of his 60th birthday, and in recognition of his outstanding contributions to infinite dimensional dynamical systems and the mathematical theory of phase transitions.
    Peter Bates was born in Manchester, England on December 27, 1947. He graduated from the University of London in mathematics in 1969 after which he moved to United States with his family. Later, he attended the University of Utah and received his Ph.D. in 1976. Following his graduation, Peter moved to Texas and taught at University of Texas at Pan American and Texas A&M University. He returned to Utah in 1984 and taught at Brigham Young University until 2004. He is currently a professor of mathematics at Michigan State University.

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