A finite‐volume scheme for the multidimensional quantum drift‐diffusion model for semiconductors

A finite‐volume scheme for the stationary unipolar quantum drift‐diffusion equations for semiconductors in several space dimensions is analyzed. The model consists of a fourth‐order elliptic equation for the electron density, coupled to the Poisson equation for the electrostatic potential, with mixed Dirichlet‐Neumann boundary conditions. The numerical scheme is based on a Scharfetter‐Gummel type reformulation of the equations. The existence of a sequence of solutions to the discrete problem and its numerical convergence to a solution to the continuous model are shown. Moreover, some numerical examples in two space dimensions are presented. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1483–1510, 2011     

Analysis of a Parabolic Cross-Diffusion Semiconductor Model with Electron-Hole Scattering

The global-in-time existence of non-negative solutions to a parabolic strongly coupled system with mixed Dirichlet–Neumann boundary conditions is shown. The system describes the time evolution of the electron and hole densities in a semiconductor when electron-hole scattering is taken into account. The parabolic equations are coupled to the Poisson equation for the electrostatic potential. Written in the quasi-Fermi potential variables, the diffusion matrix of the parabolic system contains strong cross-diffusion terms and is only positive semi-definite such that the problem is formally of degenerate type. The existence proof is based on the study of a fully discretized version of the system, using a backward Euler scheme and a Galerkin method, on estimates for the free energy, and careful weak compactness arguments.     

STEADY-STATE SOLUTIONS FOR A ONE-DIMENSIONAL NONISENTROPIC HYDRODYNAMIC MODEL WITH NON-CONSTANT LATTICE TEMPERATURE

A one-dimensional stationary nonisentropic hydrodynamic model for semiconductor devices with non-constant lattice temperature is studied. This model consists of the equations for the electron density, the electron current density and electron temperature, coupled with the Poisson equation of the electrostatic potential in a bounded interval supplemented with proper boundary conditions. The existence and uniqueness of a strong subsonic steady-state solution with positive particle density and positive temperature is established. The proof is based on the fixed-point arguments, the Stampacchia truncation methods, and the basic energy estimates.     

Existence of Classical Solutions to a Stationary Simplified Quantum Energy-Transport Model in 1-Dimensional Space

The existence of classical solutions to a stationary simplified quantum energytransport model for semiconductor devices in 1-dimensional space is proved.The model consists of a nonlinear elliptic third-order equation for the electron density,including a temperature derivative,an elliptic nonlinear heat equation for the electron temperature,and the Poisson equation for the electric potential.The proof is based on an exponential variable transformation and the Leray-Schauder fixed-point theorem.     

Filtering with Marked Point Process Observations via Poisson Chaos Expansion

We study a general filtering problem with marked point process observations. The motivation comes from modeling financial ultra-high frequency data. First, we rigorously derive the unnormalized filtering equation with marked point process observations under mild assumptions, especially relaxing the bounded condition of stochastic intensity. Then, we derive the Poisson chaos expansion for the unnormalized filter. Based on the chaos expansion, we establish the uniqueness of solutions of the unnormalized filtering equation. Moreover, we derive the Poisson chaos expansion for the unnormalized filter density under additional conditions. To explore the computational advantage, we further construct a new consistent recursive numerical scheme based on the truncation of the chaos density expansion for a simple case. The new algorithm divides the computations into those containing solely system coefficients and those including the observations, and assign the former off-line.     

The zero-electron-mass limit in the hydrodynamic model for plasmas

The limit of the vanishing ratio of the electron mass to the ion mass in the isentropic transient Euler-Poisson equations with periodic boundary conditions is proved. The equations consist of the balance laws for the electron density and current density for a given ion density, coupled to the Poisson equation for the electrostatic potential. The limit is related to the low-Mach-number limit of Klainerman and Majda. In particular, the limit velocity satisfies the incompressible Euler equations with damping. The difference to the zero-Mach-number limit comes from the electrostatic potential which needs to be controlled. This is done by a reformulation of the equations in terms of the enthalpy, higher-order energy estimates and a careful use of the Poisson equation.     

Energy image density property and the lent particle method for Poisson measures

We introduce a new approach to absolute continuity of laws of Poisson functionals. It is based on the energy image density property for Dirichlet forms. The associated gradient is a local operator and gives rise to a nice formula called the lent particle method which consists in adding a particle and taking it back after some calculation.     

A simplified quantum energy-transport model for semiconductors

The existence of global-in-time weak solutions to a quantum energy-transport model for semiconductors is proved. The equations are formally derived from the quantum hydrodynamic model in the large-time and small-velocity regime. They consist of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients; an elliptic nonlinear heat equation for the electron temperature; and the Poisson equation for the electric potential. The equations are solved in a bounded domain with periodic boundary conditions. The existence proof is based on an entropy-type estimate, exponential variable transformations, and a fixed-point argument. Furthermore, we discretize the equations by central finite differences and present some numerical simulations of a one-dimensional ballistic diode.     

On familial longitudinal Poisson mixed models with gamma random effects

Poisson mixed models are used to analyze a wide variety of cluster count data. These models are commonly developed based on the assumption that the random effects have either the log-normal or the gamma distribution. Obtaining consistent as well as efficient estimates for the parameters involved in such Poisson mixed models has, however, proven to be difficult. Further problem gets mounted when the data are collected repeatedly from the individuals of the same cluster or family. In this paper, we introduce a generalized quasilikelihood approach to analyze the repeated familial data based on the familial structure caused by gamma random effects. This approach provides estimates of the regression parameters and the variance component of the random effects after taking the longitudinal correlations of the data into account. The estimators are consistent as well as highly efficient.     

Global convergence of the Euler‐Poisson system for ion dynamics

We consider smooth solutions of the Euler‐Poisson system for ion dynamics in which the electron density is replaced by a Boltzmann relation. The system arises in the modeling of plasmas, where appear two small parameters, the relaxation time and the Debye length. When the initial data are sufficiently close to constant equilibrium states, we prove the convergence of the system for all time, as each of the parameters goes to zero. The limit systems are drift‐diffusion equations and compressible Euler equations. The proof is based on uniform energy estimates and compactness arguments.     

A Charlier–Parseval approach to poisson approximation and its applications

We propose a new approach to Poisson approximation. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. We also give a selected survey on diverse Poisson approximation results.     

IMPLICIT-EXPLICIT MULTISTEP FINITE ELEMENT-MIXED FINITE ELEMENT METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

The transient behavior of a semiconductor device consists of a Poisson equa-tion for the electric potential and of two nonlinear parabolic equations for the electrondensity and hole density.The electric potential equation is discretized by a mixed finiteelement method. The electron and hole density equations are treated by implicit-explicitmultistep finite element methods. The schemes are very efficient. The optimal order errorestimates both in time and space are derived.     

A class of proper priors for Bayesian simultaneous prediction of independent Poisson observables

Simultaneous prediction and parameter inference for the independent Poisson observables model are considered. A class of proper prior distributions for Poisson means is introduced. Bayesian predictive densities and estimators based on priors in the introduced class dominate the Bayesian predictive density and estimator based on the Jeffreys prior under Kullback-Leibler loss.     

Point process diagnostics based on weighted second-order statistics and their asymptotic properties

A new approach for point process diagnostics is presented. The method is based on extending second-order statistics for point processes by weighting each point by the inverse of the conditional intensity function at the point’s location. The result is generalized versions of the spectral density, R/S statistic, correlation integral and K-function, which can be used to test the fit of a complex point process model with an arbitrary conditional intensity function, rather than a stationary Poisson model. Asymptotic properties of these generalized second-order statistics are derived, using an approach based on martingale theory.     

Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application

A method based on higher-order partial differential equation (PDE) numerical scheme are proposed to obtain the transition cumulative distribution function (CDF) of the diffusion process (numerical differentiation of the transition CDF follows the transition probability density function (PDF)), where a transformation is applied to the Kolmogorov PDEs first, then a new type of PDEs with step function initial conditions and 0, 1 boundary conditions can be obtained. The new PDEs are solved by a fourth-order compact difference scheme and a compact difference scheme with extrapolation algorithm. After extrapolation, the compact difference scheme is extended to a scheme with sixth-order accuracy in space, where the convergence is proved. The results of the numerical tests show that the CDF approach based on the compact difference scheme to be more accurate than the other estimation methods considered; however, the CDF approach is not time-consuming. Moreover, the CDF approach is used to fit monthly data of the Federal funds rate between 1983 and 2000 by CKLS model.     

An Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems

A new fast algorithm based on the augmented immersed interface method and a fast Poisson solver is proposed to solve three dimensional elliptic interface problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable which should be chosen such that the original flux jump condition is satisfied. The discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing PDEs in a neighborhood of control points on the interface. The interpolation scheme is the key to the success of the augmented IIM particularly. In this paper, the key new idea is to select interpolation points along the normal direction in line with the flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The number of the GMRES iterations is independent of the mesh size.     

Buckling analysis of circular functionally graded plate under uniform radial compression including shear deformation with linear and quadratic thickness variation on the Pasternak elastic foundation

A numerical scheme for buckling analysis of functionally graded circular plate (FGCP) subjected to uniform radial compression including shear deformation rested on Pasternak elastic foundation is presented. The linear and quadratic thickness variation patterns with various boundary conditions are considered. A modified Euler–Lagrange equation is achieved and then solved by converting differential equation to a nonlinear algebraic system of equations. Also, based on traction–free surface without using shear correction factor, a new approach by considering shear deformation for buckling analysis of FGCP rested on elastic foundation is carried out. The stability equation based on shear stress-free surface is solved by the spectral Ritz method. The spectral Ritz method has good flexibility in the sense of satisfying the boundary conditions. The effects of both linear and quadratic thickness variations and Poisson’s ratio are investigated. By taking small numbers of the basis, the outcomes in literature are improved.     

Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps

The optimization problem of fishing for a stochastic logistic model is studied in this paper. Besides a standard geometric Brownian motion, another two driving processes are taken into account: a stationary Poisson point process and a continuous-time finite-state Markov chain. The classical harvesting problem for this model is a big difficult puzzle since the corresponding Fokker–Planck equations with three types of noise are very difficult to solve. Our main goal of this paper is to work out the optimization problem with respect to stationary probability density. One of the main contributions is to provide a new equivalent approach to overcome this problem. More precisely, an ergodic method is used to show the almost surely equivalency between the time averaging yield and sustainable yield. Results show that the optimal strategy changes with environment. An interesting thing is that the optimal strategy for each state is equivalent to the global optimality.     

An immersed interface method for anisotropic elliptic problems on irregular domains in 2D

This work is a generalization of the immersed interface method for discretization of a nondiagonal anisotropic Laplacian in 2D. This first‐order discretization scheme enforces weakly diagonal dominance of the numerical scheme whenever possible. A necessary and sufficient condition depending on the mesh size h for the existence of this scheme at an interior grid point is found in terms of the anisotropy matrix. A linear programming approach is introduced for finding the weights of the schemes. The method is tested with a parametrized family of anisotropic Poisson equations. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Conservative finite-difference scheme for computer simulation of contrast 3D spatial-temporal structures induced by a laser pulse in a semiconductor

The numerical approach for computer simulation of femtosecond laser pulse interaction with a semiconductor is considered under the formation of 3D contrast time-dependent spatiotemporal structures. The problem is governed by the set of nonlinear partial differential equations describing a semiconductor characteristic evolution and a laser pulse propagation. One of the equations is a Poisson equation concerning electric field potential with Neumann boundary conditions that requires fulfillment of the well-known condition for Neumann problem solvability. The Poisson equation right part depends on free-charged particle concentrations that are governed by nonlinear equations. Therefore, the charge conservation law plays a key role for a finite-difference scheme construction as well as for solvability of the Neumann difference problem. In this connection, the iteration methods for the Poisson equation solution become preferable than using direct methods like the fast Fourier transform. We demonstrate the following: if the finite-difference scheme does not possess the conservatism property, then the problem solvability could be broken, and the numerical solution does not correspond to the differential problem solution. It should be stressed that for providing the computation in a long-time interval, it is crucial to use a numerical method that possessing asymptotic stability property. In this regard, we develop an effective numerical approach—the three-stage iteration process. It has the same economic computing expenses as a widely used split-step method, but, in contrast to the split-step method, our method possesses conservatism and asymptotic stability properties. Computer simulation results are presented.