We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown coefficients from boundary measurements by varying initial conditions. Based on observability inequalities and a special choice of initial conditions, we provide uniqueness and stability estimates for the recovery of volume and boundary lower order coefficients in wave and heat equations. Some of the results presented here are slightly improved from their original versions.

Overweight is a social disease, which is transmitted through social networks. A mathematical model is proposed to simulate the dynamics of social obesity, where the structures of individual heterogeneity and overeating behaviors are incorporated. The basic reproduction number of the disease is calculated and is shown to be a threshold for disease invasion. Sufficient conditions for the global stability of an endemic equilibrium is established by Lyapunov functions. Numerical simulations are provided to reveal how interventions through treatment to eating behaviors and education to susceptible individuals suppress the progression of the disease.

A good statistical model of speckle formation is useful to design a good speckle reduction model for clinical ultrasound images. We propose a new general distribution to describe the distribution of speckle in clinical ultrasound images according to a log-compression algorithm of clinical ultrasound imaging. A new variational model is designed to remove the speckle noise with the proposed general distribution. The efficiency of the proposed model is confirmed by experiments on synthetic images and real ultrasound images. Compared with previous variational methods which assign a designated distribution, the proposed method is adaptive to remove different kinds of speckle noise by estimating parameters to find suitable distribution. The experiments show that the proposed method can adaptively remove different types of speckle noise.

The epidemic of foot-and-mouth disease (FMD) in cattle remains particular concern in many countries or areas. The epidemic can spread by direct contact with the carrier and symptomatic animals, as well as indirect contact with the contaminated environment. The outbreak of FMD indicates that the infection initially spreads through the farm before spreading between farms. In this paper, considering the cattle population, we establish a dynamical model of FMD with two patches: within-farm and outside-farm, and give the formulae of the basic reproduction number $R$₀. By constructing the Lyapunov function, we prove the disease-free equilibrium is globally asymptotically stable when $R$₀<1, and that of the unique endemic equilibrium when $R$₀>1. By numerical simulations, we confirm the global stability of equilibria. In addition, by carrying out the sensitivity analysis of the basic reproduction number on some parameters, we reach the conclusion that vaccination, quarantining or removing of the carrier and disinfection are the useful control measures for FMD at the large-scale cattle farm.

Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point problems, numerical partial differential equations and optimality conditions of equality constrained optimization. For the unconstrained continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019) carefully considered the very basic question: what is a proper definition of local optima of a minimax optimization problem, and proposed a proper definition of local optimality called local minimax. We shall extend the definition of local minimax point to constrained nonconvex-nonconcave minimax optimization problems. By analyzing Jacobian uniqueness conditions for the lower-level maximization problem and the strong regularity of Karush-Kuhn-Tucker conditions of the maximization problem, we provide both necessary optimality conditions and sufficient optimality conditions for the local minimax points of constrained minimax optimization problems.

Consider the scattering of a time-harmonic electromagnetic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper concerns the numerical solutions of the open cavity scattering problems in both transverse magnetic and transverse electric polarizations. Based on the Dirichlet-to-Neumann (DtN) map for each polarization, a transparent boundary condition is imposed to reduce the scattering problem equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed. The estimate consists of the finite element approximation error and the truncation error of the DtN operator, which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented for both polarizations to illustrate the competitive behavior of the adaptive method.

In this work, we introduce a fast numerical algorithm to solve the time-dependent radiative transport equation (RTE). Our method uses the integral formulation of RTE and applies the treecode algorithm to reduce the computational complexity from $\mathcal{O}$($M$^2+1/$d$) to $\mathcal{O}$($M$^1+1/$d$log$M$), where $M$ is the number of points in the physical domain. The error analysis is presented and numerical experiments are performed to validate our algorithm.