We present a regularized finite difference method for the logarithmic Schr ̈odingerequation (LogSE) and establish its error bound. Due to the blowup of the logarithmic nonlinearity, i.e., ln ρ→ −\infty when ρ→0+v with ρ = |u|^2 being the density and u being the complex-valuedwave function or order parameter, there are significant difficulties in designing numerical methodsand establishing their error bounds for the LogSE. In order to suppress the roundoff error and toavoid blowup, a regularized LogSE (RLogSE) is proposed with a small regularization parameter 0 < ε << 1 and linear convergence is established between the solutions of RLogSE and LogSE interm of ε. Then a semi-implicit finite difference method is presented for discretizing the RLogSEand error estimates are established in terms of the mesh sizehand time stepτas well as the smallregularization parameterε. Finally numerical results are reported to illustrate our error bounds.

In the paper, we consider a continuation approach for the binary quadratic program (BQP) based on a class of NCP-functions. More specifically, we recast the BQP as an equivalent minimization and then seeks its global minimizer via a global continuation method. Such approach had been considered in [11] which is based on the FischerBurmeister function. We investigate this continuation approach again by using a more general function, called the generalized FischerBurmeister function. However, the theoretical background for such extension can not be easily carried over. Indeed, it needs some subtle analysis.

A new residual-based stabilized finite element method is analyzed for solving the Stokes equations in terms of velocity and pressure, where the $H^{−1}$ norm is introduced in the measurement of the residuals to obtain a symmetric positive definite (SPD) method. The $H^{−1}$ norm is computable and can be always easily realized offline by the continuous linear finite element solution or the preconditioned counterpart of the Poisson Dirichlet problem. Although the $H^{−1}$ norm is computed in the linear element space, no matter what the finite element spaces for the velocity and the pressure are, optimal error bounds can be established when using continuous finite element pairs $R_l$−$R_m$ for velocity and pressure for any $l, m\ge 1$. Numerical experiments are performed to confirm the theoretical results obtained.

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The accurate control of process parameters is particularly important for the growth of high quality structures during laser cladding. Melt pool size and cross sectional area are key process characteristics that can be controlled to allow the precise deposition of complex features. A feasible mathematical model is established to describe the interaction between laser and powder particles and to predict and control the cross sectional area. Also, a vision-based sensing system is proposed to detect the transient shape and size of the melt pool. The system focuses on the measuring of the geometric parameters of the melt pool by using a digital image processing method and thus giving researchers the parameters on-line, which gives us a necessary tool to gain insight into the cladding process and to optimize the material quality

The problem of compressive-sensing (CS) L2-L1-TV reconstruc- tion of magnetic resonance (MR) scans from undersampled k-space data has been addressed in numerous studies. However, the regularization parameters in models of CS L2-L1-TV reconstruction are rarely studied. Once the regu- larization parameters are given, the solution for an MR reconstruction model is xed and is less eective in the case of strong noise. To overcome this shortcoming, we present a new alternating formulation to replace the standard L2-L1-TV reconstruction model. A weighted-average alternating minimization method is proposed based on this new formulation and a convergence analysis of the method is carried out. The advantages of and the motivation for the pro- posed alternating formulation are explained. Experimental results demonstrate that the proposed formulation yields better reconstruction results in the case of strong noise and can improve image reconstruction via exible parameter slection.

Local discontinuous Galerkin (LDG) methods are popular for convection-diffusion equations. In LDG methods, we introduce an auxiliary variable $p$ to represent the derivative of the primary variable $u$, and solve them on the same mesh. In this paper, we will introduce a new LDG method, and solve $u$ and $p$ on different meshes. The stability and error estimates will be investigated. The new algorithm is more flexible and flux-free for pure diffusion equations without introducing additional computational cost compared with the original LDG methods, since it is not necessary to solve each equation twice. Moreover, it is possible to construct third-order maximum-principle-preserving schemes based on the new algorithm. However, one cannot anticipate optimal accuracy in some special cases. In this paper, we will find out the reason for accuracy degeneration which further leads to several alternatives to obtain optimal convergence rates. Finally, several numerical experiments will be given to demonstrate the stability and optimal accuracy of the new algorithm.

Let $\Omega\subset\mathbb R^d$, $1\leq d\leq 3$, be a bounded $d$-polytope. Consider the parabolic equation on $\Omega$ with the Dirac delta function on the right hand side. We study the well-posedness, regularity, and the interior error estimate of semidiscrete finite element approximations of the equation. In particular, we derive that the interior error is bounded by the best local approximation error, the negative norms of the error, and the negative norms of the time derivative of the error. This result implies different convergence rates for the numerical solution in different interior regions, especially when the region is close to the singular point. Numerical test results are reported to support the theoretical prediction.

This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass-conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that cannot be localized, in general. This is built on our previous work on the generalized multiscale finite element method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high

We show that the gradient mapping of the squared norm of FischerBurmeister function is globally Lipschitz continuous and semismooth, which provides a theoretical basis for solving nonlinear second-order cone complementarity problems via the conjugate gradient method and the semismooth Newtons method.

This is the second of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of the present paper is to explain why the accuracy of the velocity approximation is not affected by (1) the numerical boundary layers in the approximation of pressure and the intermediate velocity field and (2) the noncommutativity of the projection operator and the laplacian. This is done by using a Godunov–Ryabenki type of analysis in a rigorous fashion. By doing so, we hope to be able to convey the message that normal mode analysis is basically sufficient for understanding the stability and accuracy of a finite-difference method for the Navier–Stokes equation even in the presence of boundaries. As an example, we analyze the second-order projection method based on pressure increment formulations used by van Kan and Bell, Colella, and Glaz. The leading order error term in this case is of $O(\Delta t)$ and behaves as high frequency oscillations over the whole domain, compared with the $O(\Delta t^{{1 / 2}} )$ numerical boundary layers found in the second-order Kim–Momn method.

Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical structure of the manifold, it is difficult to get efficient numerical method to solve PDE on manifold. In the paper, we propose a method called point integral method (PIM) to solve the Poisson-type equations from point clouds. Among different kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM, the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function. This feature makes the integral equation easy to be discretized from point cloud. In the paper, we explain the derivation of the integral equations, describe the point integral method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.

Laser cladding has been widely used in direct laser fabrication and laser refabrication for metal parts. It is a high precision and complicated process due to multiple process parameters involved. These parameters are strongly tied to each other and their effects are still far from being completely understood. In order to make the optimal choice, a feasible mathematical model is established to describe the interaction between laser and powder particles and to predict and control the cross sectional area. Additionally, analysis concentrating on the effect of the various parameters on the cross sectional area is made. Results show that laser power, powder feed rate, scan speed and radius of focused powder stream are the main factors. Correlations between main process parameters and cross sectional area of an individual clad track have been found. The model also provides predictions of range of process parameters for melting powder to form specific cross sectional area, which will guide the experiment and greatly reduce the cost of the experimental investigations.

In this paper, we study the ill posed Perona-Malik equation of image processing [14], and the regularized P-M model i.e. C-model proposed by Catte et al. [4]. The authors present the convex compound of these two models in the form of the system of partial differential equations. The weak solution for the equations is proved in detail. The additive operator splitting (AOS) algorithm for the proposed model is also given. Finally, we show some numeric experiment results on images.

Systematically generalizing planar geometric algorithms to manifold domains is of fundamental importance in computer aided design field. This paper proposes a novel theoretic framework, <i>geometric structure</i>, to conquer this problem. In order to discover the intrinsic geometric structures of general surfaces, we developed a theoretic rigorous and practical efficient method, <i>Discrete Variational Ricci flow</i>.

In this paper, we propose an unfitted Nitsche's method to compute the band structures of phononic crystal with impurities of general geometry. The proposed method does not require the background mesh to fit the interfaces of impurities, and thus avoids the expensive cost of generating body-fitted meshes and simplifies the inclusion of interface conditions in the formulation. The quasi-periodic boundary conditions are handled by the Floquet-Bloch transform, which converts the computation of band structures into an eigenvalue problem with periodic boundary conditions. More importantly, we show the well-posedness of the proposed method using a delicate argument based on the trace inequality, and further prove the convergence by the Babuška-Osborn theory. We achieve the optimal convergence rate at the presence of the impurities of general geometry. We confirm the theoretical results by two numerical examples, and show the capability of the proposed methods for computing the band structures without fitting the interfaces of impurities.

Atmospheric temperature is one of the most important climate variables. This observational study presents detailed descriptions of the temperature variability imprinted in the 9-year brightness temperature data acquired by the Advanced Microwave Sounding Unit- Instrument A (AMSU-A) aboard Aqua since September 2002 over tropical oceans. A non-linear, adaptive method called the Ensemble Joint Multiple Extraction has been employed to extract the principal modes of variability in the AMSU-A/Aqua data. The semi-annual, annual, quasibiennial oscillation (QBO) modes and QBO–annual beat in the troposphere and the stratosphere have been successfully recovered. The modulation by the El Nin˜o/Southern oscillation (ENSO) in the troposphere was found and correlates well with the Multivariate ENSO Index. The longterm variations during 2002–2011 reveal a cooling trend (-0.5 K/decade at 10 hPa) in the tropical stratosphere; the trend below the tropical tropopause is not statistically significant due to the length of our data. A new tropospheric near-annual mode (period *1.6 years) was also revealed in the troposphere, whose existence was confirmed using National Centers for Environmental Prediction Reanalysis air temperature data. The near-annual mode in the troposphere is found to prevail in the eastern Pacific region and is coherent with a near-annual mode in the observed sea surface temperature over the Warm Pool region that has previously been reported. It remains a challenge for climate models to simulate the trends and principal modes of natural variability reported in this work.

A Deuflhard-type exponential integrator sine pseudospectral (DEI-SP) method is proposed and analyzed for solving the generalized improved Boussinesq (GIBq) equation. The numerical scheme is based on a second-order exponential integrator for time integration and a sine pseudospectral discretization in space. Rigorous analysis and abundant experiments show that the method converges quadratically and spectrally in time and space, respectively. Finally the DEI-SP method is applied to investigate the complicated and interesting long-time dynamics of the GIBq equation.

In this paper, we generalize the so-called inverse Lax-Wendroff boundary treatment \cite{ilw} for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge-Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems.

In this paper, we propose a numerical method to solve isotropic elliptic equations on point cloud by generalizing the point integral method. The idea of the point integral method is to approximate the differential operators by integral operators and discretize the corresponding integral equation on point cloud. The key step is to get the integral approximation. In this paper, with proper kernel function, we get an integral approximation for the elliptic operators with isotropic coefficients. Moreover, the integral approximation has been proved to keep the coercivity of the original elliptic operator. The convergence of the point integral method is also proved.

This paper proposes a neural network approach to efficiently solve nonlinear convex programs with the second-order cone constraints. The neural network model is designed by the generalized FischerBurmeister function associated with second-order cone. We study the existence and convergence of the trajectory for the considered neural network. Moreover, we also show stability properties for the considered neural network, including the Lyapunov stability, the asymptotic stability and the exponential stability. Illustrative examples give a further demonstration for the effectiveness of the proposed neural network. Numerical performance based on the parameter being perturbed and numerical comparison with other neural network models are also provided. In overall, our model performs better than two comparative methods.

The numerical solution of the one-dimensional KleinGordon equation on an unbounded domain is analyzed in this paper. Two artificial boundary conditions are obtained to reduce the original problem to an initial boundary value problem on a bounded computational domain, which is discretized by an explicit difference scheme. The stability and convergence of the scheme are analyzed by the energy method. A fast algorithm is obtained to reduce the computational cost and a discrete artificial boundary condition (DABC) is derived by the <i>Z</i>-transform approach. Finally, we illustrate the efficiency of the proposed method by several numerical examples.

In this paper, we propose a model reduction method for solving multiscale elliptic PDEs with random coefficients in the multiquery setting using an optimization approach. The optimization approach enables us to construct a set of localized multiscale data-driven stochastic basis functions that give an optimal approximation property of the solution operator. Our method consists of the offline and online stages. In the offline stage, we construct the localized multiscale data-driven stochastic basis functions by solving an optimization problem. In the online stage, using our basis functions, we can efficiently solve multiscale elliptic PDEs with random coefficients with relatively small computational costs. Therefore, our method is very efficient in solving target problems with many different force functions. The convergence analysis of the proposed method is also presented and has been verified by the numerical simulations.

We present two uniformly accurate numerical methods for discretizing the Zakharovsystem (ZS) with a dimensionless parameter 0< ε ≤ 1, which is inversely proportional to theacoustic speed. In the subsonic limit regime, i.e., 0< ε << 1, the solution of ZS propagates waves with O(ε)- andO(1)-wavelengths in time and space, respectively, and/or rapid outgoing initial layerswith speed O(1/ε) in space due to the singular perturbation of the wave operator in ZS and/or theincompatibility of the initial data. By adopting an asymptotic consistent formulation of ZS, wepresent a time-splitting exponential wave integrator (TS-EWI) method via applying a time-splittingtechnique and an exponential wave integrator for temporal derivatives in the nonlinear Schr ̈odingerequation and wave-type equation, respectively. By introducing a multiscale decomposition of ZS, wepropose a time-splitting multiscale time integrator (TS-MTI) method. Both methods are explicitand convergent exponentially in space for all kinds of initial data, which is uniformly for ε ∈ (0,1].The TS-EWI method is simpler to be implemented and it is only uniformly and optimally accuratein time for well-prepared initial data, while the TS-MTI method is uniformly and optimally accuratein time for any kind of initial data. Extensive numerical results are reported to show their efficiencyand accuracy, especially in the subsonic limit regime. Finally, the TS-MTI method is applied tostudy numerically convergence rates of ZS to its limiting models when ε→0+.

We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31, pp. 334-368, 2008] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL condition. This condition tends to a parabolic CFL condition for small mean free paths, and is close to a convection CFL condition for large mean free paths. Ou r analysis is based on very simple energy estimates.