Poisson Solution Research Articles

Maximal inequalities and some applications

A maximal inequality is an inequality which involves the (absolute) supremum sups⩽t|Xs| or the running maximum sups⩽tXs of a stochastic process (Xt)t⩾0. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, Lévy processes, Lévy-type – including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Lévy process –, strong Markov processes and Gaussian processes. Using the Burkholder–Davis–Gundy inequalities we also discuss some relations between maximal estimates in probability and the Hardy–Littlewood maximal functions from analysis.

A Polynomial Approximation to Self Consistent Solution for Schrödinger–Poisson Equations in Superlattice Structures

The paper deals with a new approach to iterative solving the Schrödinger and Poisson equations in the first type of semiconductor superlattice. Assumptions of the transfer matrix method are incorporated into the approach, which allows to take into account the potential varying within each single layer of bias voltage superlattice. The key process of the method is to approximate the charge density and wave functions with polynomials. It allows to obtain semi-analytical solutions for the Schrödinger and Poisson equations, which in turn have significant impact on the accuracy and speed of superlattice simulations. The presented procedure is also suifihue for finding eigenstates extended over relatively large superlattice area, and it can be used as an effective pro-gram module for a superlattice finite model. The obtained quantum states are very similar to the Wannier-Stark functions, and they can serve as the base under non-equilibrium Green’s function formalism (NEGF). Exemplary results for Schrödinger and Poisson solutions for superlattices based on the GaAs/AlGaAs heterostructure are presented to prove all the above.

Electronic Transport in 2D‐Based Printed FETs from a Multiscale Perspective

AbstractAs 2D materials (2DMs) gain the research limelight as a technological option for obtaining on‐demand printed low‐cost integrated circuits with reduced environmental impact, theoretical methods able to provide the necessary fabrication guidelines acquire fundamental importance. Here, a multiscale modeling technique is exploited to study electronic transport in devices consisting of a printed 2DM network of flakes. The approach implements a Monte Carlo scheme to generate the flake distribution. By means of ab initio density functional theory calculations together with non equilibrium Green's functions formalism, detailed physical insights on flake‐to‐flake transport mechanisms are provided. This later feeds a 3D drift‐diffusion and Poisson solution to compute self‐consistently transport and electrostatics in the device. The method is applied to MoS2 and graphene‐based dielectrically gated FETs, highlighting the impact of the structure density and variability on the mobility and sheet resistance. The prediction capability of the proposed approach is validated against electrical measurements of in‐house printed graphene conductive lines as a function of film thickness, demonstrating its strong potential as a guide for future experimental activity in the field.

Active Contour Directed by the Poisson Gradient Vector Field and Edge Tracking

This paper develops a new active contour (AC) model capable of multiple complex objects segmentation in the presence of heavy noise. The model segments images in the framework of two types of partial differential equations (PDEs): the Euler–Lagrange and Poisson PDEs. The former is used to build an evolution algorithm, while the Poisson solution gradient vector field (PGVF) directs the evolution toward the boundaries of all image objects. The AC halts on boundaries and PGVF separatrices, splits on the latter, and leaves at least one segment (called label) on every boundary. Each label tracks its boundary until the corresponding object is enveloped. The advantages of the new method are validated on a number of skin lesions, road, and aircraft images of varying sizes and in the presence of Gaussian noise. The obtained results are compared against results by contemporary and established active contours and neural networks.

Early history of nonlinear acoustics: Waveform distortion, disaster, and redemption

The first (although slightly incorrect) wave equations for finite-amplitude sound in lossless fluids were obtained independently by Euler in 1759 and Lagrange in 1760. Poisson (1808) provided the first major breakthrough with his exact solution for progressive waves of finite amplitude in a lossless gas. Although a far-reaching result, the progressive waveform distortion (and disastrous consequences) implied by his solution went unrecognized for 40 years. Challis (1848) showed that the Poisson solution is not single valued but did not understand why. Stokes (1848) provided the why. He saw that the Poisson waveform distorts as the wave travels, eventually threatening to become multivalued. He postulated that a discontinuity (shock) develops to avoid waveform overturning. He also saw that viscosity (not accounted for by Poisson) would prevent true discontinuities. Earnshaw (1860) and Riemann (1860) cleaned up plane waves in lossless gases. However, how to predict propagation after shocks form? Rankine (1870) and Hugoniot (1887, 1889) provided the first solutions for propagation when dissipation is included. These were the first steps toward redemption. The curtain rang down on this era of nonlinear acoustics with the excellent papers in 1910 by Rayleigh and Taylor on steady shocks in a thermoviscous fluid.

Computing the Distance between Two Finite Element Solutions Defined on Different 3D Meshes on a GPU

This article introduces a new method to efficiently compute the distance (i.e., $L^p$ norm of the difference) between two functions supported by two different meshes of the same 3D domain. The functions that we consider are typically finite element solutions discretized in different function spaces supported by meshes that are potentially completely unrelated. Our method computes an approximation of the distance by resampling both fields over a set of parallel 2D regular grids. By leveraging the parallel horse power of computer graphics hardware (graphics processing unit (GPU)), our method can efficiently compute distances between meshes with multimillion elements in seconds. We demonstrate our method applied to different problems (distance between known functions, Poisson solutions, and linear elasticity solutions) using different function spaces (Lagrange polynomials from order one to seven) and different meshes (tetrahedral and hexahedral, with linear or quadratic geometry).

Simple and efficient model for slotless eccentric surface‐mounted PM machines

In this study, the equivalent boundary method is applied for modelling of the air gap flux density in the eccentric surface-mounted permanent magnet (PM) machines. In the presented model, neither conformal transformations nor perturbation analysis methods are applied. In this approach, a two-dimensional Poisson solution for the eccentric PM machines is developed. Therefore, the proposed model is simple, precise and efficient. To make the solution of the Poisson equation in the eccentric air gap easy, the concept of the equivalent boundary method is exploited to find an equivalent concentric geometry for the eccentric problem. All the claimed theorems are mathematically approved. The model is valid for static and dynamic eccentricities at no-load and on-load conditions. Using the analytically predicted air gap flux density, electromagnetic torque, back-EMF and the resulted radial force are computed. The correctness of the predicted results is validated by means of finite-element analysis.

Engineering Approach to Calculating QoS of Server with Self-Similar Incoming Traffic Based on Recursive Scalable Poisson Model

To date the prospects for using the accumulated over many years mathematical and software for the modeling of telecommunications with the Poisson input flow are under a big question. The matter is that a new fractal queuing theory is already on the threshhold. This article formulates and solves the problem of application of a queuing system model with a Poisson incoming flow for the purposes of server modeling described by QS with self-similar incoming traffic of the "fractal Brownian motion" type (according to Norros). Based on the results of the morphological analysis, the Norros model was decomposed into Poisson components connected by a scalable recurrence scheme. The variance of the number of packets in the server, raised to the power determined by the Hurst parameter acts as the similarity coefficient of fractal and Poisson QSs. The method for rescaling Poisson solutions into fractal solutions was constructed on the basis of the similarity coefficient. According to this method in order to find the fractal delay of access, the Poisson delay should be multiplied by the similarity coefficient, and to estimate the probability of packet loss, it is necessary to extract a root of degree equal to the similarity coefficient from classical exponential losses. The scope of the re-scaling method focuses on the pre-project stages of creating telecommunications, where there is no need for high accuracy of simulation results.

Nonlocal Poisson-Fermi model for ionic solvent.

We propose a nonlocal Poisson-Fermi model for ionic solvent that includes ion size effects and polarization correlations among water molecules in the calculation of electrostatic potential. It includes the previous Poisson-Fermi models as special cases, and its solution is the convolution of a solution of the corresponding nonlocal Poisson dielectric model with a Yukawa-like kernel function. The Fermi distribution is shown to be a set of optimal ionic concentration functions in the sense of minimizing an electrostatic potential free energy. Numerical results are reported to show the difference between a Poisson-Fermi solution and a corresponding Poisson solution.

Quantum Transport in AlGaSb/InAs TFETs With Gate Field In-Line With Tunneling Direction

Quantum transport simulations are performed in tunneling FETs (TFETs) with the gate electric field in-line with the tunneling junction direction (in-line TFETs). Charge self-consistency and thermalization effects are included in a semiclassical Poisson solution to compute the electrostatic potential. The obtained potential is then used for current calculation with the ballistic nonequilibrium Green’s function method (NEGF) in the tight binding basis. It is shown that the NEGF method predicts a higher subthreshold swing than the often-used dynamic nonlocal (DNL) path band-to-band method. The NEGF method accounts for the direct source–drain tunneling, which is underestimated in the DNL path approach in the studied geometries. Undercut is shown to be essential to obtain switching slope below 60 mV/decade in the in-line TFETs.

Eigen-disfigurement model for simulating plausible facial disfigurement after reconstructive surgery.

BackgroundPatients with facial cancers can experience disfigurement as they may undergo considerable appearance changes from their illness and its treatment. Individuals with difficulties adjusting to facial cancer are concerned about how others perceive and evaluate their appearance. Therefore, it is important to understand how humans perceive disfigured faces. We describe a new strategy that allows simulation of surgically plausible facial disfigurement on a novel face for elucidating the human perception on facial disfigurement.MethodLongitudinal 3D facial images of patients (N = 17) with facial disfigurement due to cancer treatment were replicated using a facial mannequin model, by applying Thin-Plate Spline (TPS) warping and linear interpolation on the facial mannequin model in polar coordinates. Principal Component Analysis (PCA) was used to capture longitudinal structural and textural variations found within each patient with facial disfigurement arising from the treatment. We treated such variations as disfigurement. Each disfigurement was smoothly stitched on a healthy face by seeking a Poisson solution to guided interpolation using the gradient of the learned disfigurement as the guidance field vector. The modeling technique was quantitatively evaluated. In addition, panel ratings of experienced medical professionals on the plausibility of simulation were used to evaluate the proposed disfigurement model.ResultsThe algorithm reproduced the given face effectively using a facial mannequin model with less than 4.4 mm maximum error for the validation fiducial points that were not used for the processing. Panel ratings of experienced medical professionals on the plausibility of simulation showed that the disfigurement model (especially for peripheral disfigurement) yielded predictions comparable to the real disfigurements.ConclusionsThe modeling technique of this study is able to capture facial disfigurements and its simulation represents plausible outcomes of reconstructive surgery for facial cancers. Thus, our technique can be used to study human perception on facial disfigurement.Electronic supplementary materialThe online version of this article (doi:10.1186/s12880-015-0050-7) contains supplementary material, which is available to authorized users.

A two-dimensional fully analytical model with polarization effect for off-state channel potential and electric field distributions of GaN-based field-plated high electron mobility transistor

In this paper, we present a two-dimensional (2D) fully analytical model with consideration of polarization effect for the channel potential and electric field distributions of the gate field-plated high electron mobility transistor (FP-HEMT) on the basis of 2D Poisson's solution. The dependences of the channel potential and electric field distributions on drain bias, polarization charge density, FP structure parameters, AlGaN/GaN material parameters, etc. are investigated. A simple and convenient approach to designing high breakdown voltage FP-HEMTs is also proposed. The validity of this model is demonstrated by comparison with the numerical simulations with Silvaco—Atlas. The method in this paper can be extended to the development of other analytical models for different device structures, such as MIS-HEMTs, multiple-FP HETMs, slant-FP HEMTs, etc.

Drain current model of double-gate MOSFETs considering both electrons and holes

We present a rigorously derived current solution for undoped double-gate (DG) MOSFETs with two carriers, which is based on surface potentials. The third-order Newton–Raphson (NR) method is used to solve the surface-potential equations resulting from the application of the boundary conditions to the general Poisson solution, with an initial guess very close to the true solution. The results demonstrate surface-potential solutions for DG MOSFETs with 2–7 iterations to achieve an accuracy of 10−15. The drain current model for two carriers is presented as a benchmark to test the accuracy of one-carrier current approximation. © 2014 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Extension theory and Kreĭn-type resolvent formulas for nonsmooth boundary value problems

The theory of selfadjoint extensions of symmetric operators, and more generally the theory of extensions of dual pairs, was implemented some years ago for boundary value problems for elliptic operators on smooth bounded domains. Recently, the questions have been taken up again for nonsmooth domains. In the present work we show that pseudodifferential methods can be used to obtain a full characterization, including Kreĭn resolvent formulas, of the realizations of nonselfadjoint second-order operators on C32+ε domains; more precisely, we treat domains with Bp,232-smoothness and operators with Hq1-coefficients, for suitable p>2(n−1) and q>n. The advantage of the pseudodifferential boundary operator calculus is that the operators are represented by a principal part and a lower-order remainder, leading to regularity results; in particular we analyze resolvents, Poisson solution operators and Dirichlet-to-Neumann operators in this way, also in Sobolev spaces of negative order.

MONTE CARLO TRANSPORT AND HEAT GENERATION IN SEMICONDUCTORS

Understanding transport and energy use in nanoscale devices is essential for the design of low-power electronics and efficient thermoelectrics. This chapter examines transport physics and the electron-phonon interaction in the context of Monte Carlo simulations, which treat electrons and phonons with comparable attention to detail. The Monte Carlo method is described in depth, including scattering physics, electron energy band and phonon dispersions, Poisson solution, and contacts within realistic devices. This approach uncovers, for instance, that Joule heating in silicon devices is distributed between the (slow) optical and (fast) acoustic phonon modes by a ratio of two to one. In nanoscale transistors, nonequilibrium transport affects heat generation near strongly peaked electric fields, and Joule heating occurs almost entirely in the drain of short, quasi-ballistic devices. Evidence is also uncovered for thermionic cooling at the source terminal of transistors, and the physics of this phenomenon are described. Although the discussion is often with respect to silicon for specificity, key methods can be broadly applied to many semiconductor devices and structures. Such aspects are only expected to increase in importance as nanoscale devices are reduced to dimensions comparable to or smaller than the electron and phonon mean free paths (10−100 nm).

A small box Fast Fourier Transformation method for fast Poisson solutions in large systems

We present a new divide-and-conquer algorithm to efficiently evaluate the Coulomb interaction in a large system, which is an essential part of self-consistent first-principle calculations. The total Coulomb potential ϕ(r)=1/|r| is divided into a short range part ϕS(r) and a smooth long range part ϕL(r). The system is divided into many cuboids, with a small box defined for each cuboid plus a buffer region. For the short range part, the interaction convolution integral is calculated directly using a Fast Fourier Transformation (FFT) in the small box. For the smooth long range part, the convolution integral is evaluated by a global FFT but on a coarse grid. The conversion between the dense grid and coarse grid values is done using small box FFTs with proper mask functions. Using this small box FFT method, the total Coulomb potentials of test quantum dot systems on 4803 grid and 24003 grid are calculated. For the 24003 grid case, the calculation is carried out by tens of thousands of processors with a computational speed up close to 10 times when compared with direct global FFT calculations using the FFTW package with the maximumly allowed number of processors. The maximum relative error is 4×10−5 while the absolute error is less than 0.1 meV.

Interactive Poisson Photometric Propagation for Facial Composite

In image composition, the inconsistent illumination of the source images is one of the major problems for seamless stitching of separated patches. The Poisson image editing is a sound technique for seamless image composition. In this paper, we have generalized and improved this technique and applied it onto solving the illumination discontinuity problem for facial image composition. Toward stitched image with patches of arbitrary shapes, number, and severe photometric discrepancy, an extended Poisson equation is proposed and formulated into a linear equation problem. To solve this equation efficiently, a layer-based Poisson solution propagation algorithm is designed. Based on it, an interactive photometric alignment system for facial compositing image is built. In the experiments, the photometric propagation effects with respect to the standard Poisson editing and other relevant algorithms are compared. Its time performance is also investigated. The experimental results verified the effectiveness and efficiency of the proposed method.

Lattice sums arising from the Poisson equation

In recent times, attention has been directed to the problem of solving the Poisson equation, either in engineering scenarios (computational) or in regard to crystal structure (theoretical). Herein we study a class of lattice sums that amount to Poisson solutions, namely the n-dimensional formsφn(r1,⋯,rn)=1π2∑m1,⋯,mn odd eiπ(m1r1+⋯+mnrn)m12+⋯+mn2.?>By virtue of striking connections with Jacobi ϑ-function values, we are able to develop new closed forms for certain values of the coordinates rk, and extend such analysis to similar lattice sums. A primary result is that for rational x, y, the natural potential ϕ2(x, y) is where A is an algebraic number. Various extensions and explicit evaluations are given. Such work is made possible by number-theoretical analysis, symbolic computation and experimental mathematics, including extensive numerical computations using up to 20 000-digit arithmetic.

Comparison of three-dimensional Poisson solution methods for particle-based simulation and inhomogeneous dielectrics

Particle-based simulation represents a powerful approach to modeling physical systems in electronics, molecular biology, and chemical physics. Accounting for the interactions occurring among charged particles requires an accurate and efficient solution of Poisson's equation. For a system of discrete charges with inhomogeneous dielectrics, i.e., a system with discontinuities in the permittivity, the boundary element method (BEM) is frequently adopted. It provides the solution of Poisson's equation, accounting for polarization effects due to the discontinuity in the permittivity by computing the induced charges at the dielectric boundaries. In this framework, the total electrostatic potential is then found by superimposing the elemental contributions from both source and induced charges. In this paper, we present a comparison between two BEMs to solve a boundary-integral formulation of Poisson's equation, with emphasis on the BEMs' suitability for particle-based simulations in terms of solution accuracy and computation speed. The two approaches are the collocation and qualocation methods. Collocation is implemented following the induced-charge computation method of D. Boda et al. [J. Chem. Phys. 125, 034901 (2006)]. The qualocation method is described by J. Tausch et al. [IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 20, 1398 (2001)]. These approaches are studied using both flat and curved surface elements to discretize the dielectric boundary, using two challenging test cases: a dielectric sphere embedded in a different dielectric medium and a toy model of an ion channel. Earlier comparisons of the two BEM approaches did not address curved surface elements or semiatomistic models of ion channels. Our results support the earlier findings that for flat-element calculations, qualocation is always significantly more accurate than collocation. On the other hand, when the dielectric boundary is discretized with curved surface elements, the two methods are essentially equivalent; i.e., they have comparable accuracies for the same number of elements. We find that ions in water--charges embedded in a high-dielectric medium--are harder to compute accurately than charges in a low-dielectric medium.

Piecewise Linearization Technique for Compact Charge Modeling of Independent DG MOSFET

Charge linearization techniques have been used over the years in advanced compact models for bulk and double-gate MOSFETs in order to approximate the position along the channel as a quadratic function of the surface potential (or inversion charge densities) so that the terminal charges can be expressed as a compact closed-form function of source and drain end surface potentials (or inversion charge densities). In this paper, in case of the independent double-gate MOSFETs, we show that the same technique could be used to model the terminal charges quite accurately only when the 1-D Poisson solution along the channel is fully hyperbolic in nature or the effective gate voltages are same. However, for other bias conditions, it leads to significant error in terminal charge computation. We further demonstrate that the amount of nonlinearity that prevails between the surface potentials along the channel actually dictates if the conventional charge linearization technique could be applied for a particular bias condition or not. Taking into account this nonlinearity, we propose a compact charge model, which is based on a novel piecewise linearization technique and shows excellent agreement with numerical and Technology Computer-Aided Design (TCAD) simulations for all bias conditions and also preserves the source/drain symmetry which is essential for Radio Frequency (RF) circuit design. The model is implemented in a professional circuit simulator through Verilog-A, and simulation examples for different circuits verify good model convergence.