A new parallel solver suited for arbitrary semilinear parabolic partial differential equations based on generalized random trees

A probabilistic representation for initial value semilinear parabolic problems based on generalized random trees has been derived. Two different strategies have been proposed, both requiring generating suitable random trees combined with a Pade approximant for approximating accurately a given divergent series. Such series are obtained by summing the partial contribution to the solution coming from trees with arbitrary number of branches. The new representation greatly expands the class of problems amenable to be solved probabilistically, and was used successfully to develop a generalized probabilistic domain decomposition method. Such a method has been shown to be suited for massively parallel computers, enjoying full scalability and fault tolerance. Finally, a few numerical examples are given to illustrate the remarkable performance of the algorithm, comparing the results with those obtained with a classical method.     

Probabilistically induced domain decomposition methods for elliptic boundary-value problems

Monte Carlo as well as quasi-Monte Carlo methods are used to generate only few interfacial values in two-dimensional domains where boundary-value elliptic problems are formulated. This allows for a domain decomposition of the domain. A continuous approximation of the solution is obtained interpolating on such interfaces, and then used as boundary data to split the original problem into fully decoupled subproblems. The numerical treatment can then be continued, implementing any deterministic algorithm on each subdomain. Both, Monte Carlo (or quasi-Monte Carlo) simulations and the domain decomposition strategy allow for exploiting parallel architectures. Scalability and natural fault tolerance are peculiarities of the present algorithm. Examples concern Helmholtz and Poisson equations, whose probabilistic treatment presents additional complications with respect to the case of homogeneous elliptic problems without any potential term and source.     

Solution of time-convolutionary Maxwell’s equations using parameter-dependent Krylov subspace reduction

We suggest a new algorithm for the solution of the time domain Maxwell equations in dispersive media. After spacial discretization we obtain a large system of time-convolution equations. Then this system is projected onto a small subspace consisting of the Laplace domain solutions for a preselected set of Laplace parameters. This approach is a generalization of the rational Krylov subspace approach for the solution of non-dispersive Maxwell’s systems. We show that the projected system preserves such properties of the initial system as stability and passivity. As an example we consider the 3D quasistationary induced polarization problem with the Cole–Cole conductivity model important for geophysical oil exploration. Our numerical experiments show that the introduction of the induced polarization does not have significant effect on convergence.     

Application of Padé via Lanczos approximations for efficient multifrequency solution of Helmholtz problems

This paper addresses the efficient solution of acoustic problems in which the primary interest is obtaining the solution only on restricted portions of the domain but over a wide range of frequencies. The exterior acoustics boundary value problem is approximated using the finite element method in combination with the Dirichlet-to-Neumann (DtN) map. The restriction domain problem is formally posed in transfer function form based on the finite element solution. In order to obtain the solution over a range of frequencies, a matrix-valued Padé approximation of the transfer function is employed, using a two-sided block Lanczos algorithm. This approach provides a stable and efficient representation of the Padé approximation. In order to apply the algorithm, it is necessary to reformulate the transfer function due to the frequency dependency in the nonreflecting boundary condition. This is illustrated for the case of the DtN boundary condition, but there is no restriction on the approach which can also be applied to other radiation boundary conditions. Numerical tests confirm that the approach offers significant computational speed-up.     

概率神经网络在胃镜样品红外光谱检测中的应用

将人工神经网络方法应用于人体胃镜样品红外光谱检测,以克服常规线性判别分析方法的局限性,从而提高了胃镜样品判别的准确率。概率神经网络是一种适用模式分类的径向基神经网络,采用样本的先验概率和最优判定原则对新的样本进行分类,具有识别率高、训练速度快、不会陷入局部极值等优点。文章采用概率神经网络进行胃镜样品红外光谱模式识别,将预处理后的胃镜样品光谱进行主成分分析,将得分值作为输入,建立概率神经网络判别模型。文中选取118例胃镜离体样品进行红外光谱判别分析,其中正常胃组织19例,胃炎组织64例,胃癌35例,选取其中59例样品建立概率神经网络校正模型,其余样品作为预测集来检验模型。实验结果表明,正常、炎症及癌症胃镜样品检测的总体准确率达到81.4%,对胃镜样品的判别取得了较好的结果。     

光脉冲传输数值模拟的分步小波方法

从信号的多尺度小波分解和正交小波变换出发，将描述光学介质中脉冲传输的非线性薛定谔 方程（NLSE）表示为小波域中的分步算符形式，给出了分步小波算法的迭代公式，导出了线 性算符在小波域中的具体表式，并讨论微分算符的矩阵结构.作为一个例子，用分步小波方 法（SSWM）解NLSE，给出了超短高斯脉冲在光纤中线性和非线性传输的波形演化，并与解析 解和分步傅里叶方法的结果作了比较.结果表明，分步小波方法是研究脉冲在光学介质中传 输的一种有效的数值计算方法. 关键词： 分步小波方法 光脉冲传输 非线性薛定谔方程 多尺度小波分解     

Phase Transitions for the Cavity Approach to the Clique Problem on Random Graphs

We give a rigorous proof of two phase transitions for a disordered statistical mechanics system used to define an algorithm to find large cliques inside Erdös random graphs. Such a system is a conservative probabilistic cellular automaton inspired by the cavity method originally introduced in spin glass theory.     

A second-order unsplit Godunov scheme for cell-centered MHD: The CTU-GLM scheme

We assess the validity of a single step Godunov scheme for the solution of the magnetohydrodynamics equations in more than one dimension. The scheme is second-order accurate and the temporal discretization is based on the dimensionally unsplit Corner Transport Upwind (CTU) method of Colella. The proposed scheme employs a cell-centered representation of the primary fluid variables (including magnetic field) and conserves mass, momentum, magnetic induction and energy. A variant of the scheme, which breaks momentum and energy conservation, is also considered. Divergence errors are transported out of the domain and damped using the mixed hyperbolic/parabolic divergence cleaning technique by Dedner et al. (2002) [11]. The strength and accuracy of the scheme are verified by a direct comparison with the eight-wave formulation (also employing a cell-centered representation) and with the popular constrained transport method, where magnetic field components retain a staggered collocation inside the computational cell. Results obtained from two- and three-dimensional test problems indicate that the newly proposed scheme is robust, accurate and competitive with recent implementations of the constrained transport method while being considerably easier to implement in existing hydro codes.     

Fast transform from an adaptive multi-wavelet representation to a partial Fourier representation

We present a fast algorithm to compute the partial transformation of a function represented in an adaptive pseudo-spectral multi-wavelet representation to a partial Fourier representation. Such fast transformations are useful in many contexts in physics and engineering, where changes of representation from a piece wise polynomial basis to a Fourier basis. The algorithm is demonstrated for a Gaussian in one and in three dimensions. For 2D, we apply this approach to a Gaussian in a periodic domain. The accuracy and the performance of this method is compared with direct summation.     

Natural convection of CuO–water nanofluid filled in a partially heated corrugated cavity:KKL model approach

In this article, flow and heat transfer inside a corrugated cavity is analyzed for natural convection with a heated inner obstacle. Thermal performance is analyzed for Cu O–water inside a partially heated domain by defining the constraint along the boundaries. For nanofluid analysis, the Koo and Kleinstreuer Li(KKL) model is implemented to deal with the effective thermal conductivity and viscosity. A heated thin rod is placed inside the corrugated cavity and the bottom portion of the corrugated cavity is partially heated. The dimensionless form of nonlinear partial differential equations are obtained through the compatible transformation along with the boundary constraint. The finite element method is executed to acquire the numerical solution of the obtained dimensional system. Streamlines, isotherms and heat transfers are analyzed for the flow field and temperature distribution. The Nusselt number is calculated at the surface of the partially heated domain for various numerical values of emerging parameters by considering the inner obstacle at cold, adiabatic and heated conditions. The computational simulation was performed by introducing various numerical values of emerging parameters. Important and significant results have been attained for temperature and velocities(in both x-and y-directions) at the vertically and horizontally mean positions of the corrugated duct.     

Particle filtering with path sampling and an application to a bimodal ocean current model

This paper introduces a recursive particle filtering algorithm designed to filter high dimensional systems with complicated non-linear and non-Gaussian effects. The method incorporates a parallel marginalization (PMMC) step in conjunction with the hybrid Monte Carlo (HMC) scheme to improve samples generated by standard particle filters. Parallel marginalization is an efficient Markov chain Monte Carlo (MCMC) strategy that uses lower dimensional approximate marginal distributions of the target distribution to accelerate equilibration. As a validation the algorithm is tested on a 2516 dimensional, bimodal, stochastic model motivated by the Kuroshio current that runs along the Japanese coast. The results of this test indicate that the method is an attractive alternative for problems that require the generality of a particle filter but have been inaccessible due to the limitations of standard particle filtering strategies.     

Implicit spectrally-accurate method for moving boundary problems using immersed boundary conditions concept

A fully implicit, spectral algorithm for the analysis of moving boundary problem is described. The algorithm is based on the concept of immersed boundary conditions (IBC), i.e., the computational domain is fixed while the time dependent physical domain is submerged inside the computational domain, and is described in the context of the diffusion-type problems. The physical conditions along the edges of the physical domain are treated as internal constraints. The method eliminates the need for adaptive grid generation that follows evolution of the physical domain and provides sharp resolution of the location of the boundary. Various tests confirm the spectral accuracy in space and the first- and second-order accuracy in time. The computational cost advantage of the IBC method as compared with the more traditional algorithm based on the mapping concept is demonstrated.     

The moving boundary node method: A level set-based,finite volume algorithm with applications to cell motility

Eukaryotic cell crawling is a highly complex biophysical and biochemical process, where deformation and motion of a cell are driven by internal, biochemical regulation of a poroelastic cytoskeleton. One challenge to built quantitative models that describe crawling cells is solving the reaction–diffusion–advection dynamics for the biochemical and cytoskeletal components of the cell inside its moving and deforming geometry. Here we develop an algorithm that uses the level set method to move the cell boundary and uses information stored in the distance map to construct a finite volume representation of the cell. Our method preserves Cartesian connectivity of nodes in the finite volume representation while resolving the distorted cell geometry. Derivatives approximated using a Taylor series expansion at finite volume interfaces lead to second order accuracy even on highly distorted quadrilateral elements. A modified, Laplacian-based interpolation scheme is developed that conserves mass while interpolating values onto nodes that join the cell interior as the boundary moves. An implicit time stepping algorithm is used to maintain stability. We use the algorithm to simulate two simple models for cellular crawling. The first model uses depolymerization of the cytoskeleton to drive cell motility and suggests that the shape of a steady crawling cell is strongly dependent on the adhesion between the cell and the substrate. In the second model, we use a model for chemical signalling during chemotaxis to determine the shape of a crawling cell in a constant gradient and to show cellular response upon gradient reversal.     

GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model

The compute unified device architecture (CUDA) is a programming approach for performing scientific calculations on a graphics processing unit (GPU) as a data-parallel computing device. The programming interface allows to implement algorithms using extensions to standard C language. With continuously increased number of cores in combination with a high memory bandwidth, a recent GPU offers incredible resources for general purpose computing. First, we apply this new technology to Monte Carlo simulations of the two dimensional ferromagnetic square lattice Ising model. By implementing a variant of the checkerboard algorithm, results are obtained up to 60 times faster on the GPU than on a current CPU core. An implementation of the three dimensional ferromagnetic cubic lattice Ising model on a GPU is able to generate results up to 35 times faster than on a current CPU core. As proof of concept we calculate the critical temperature of the 2D and 3D Ising model using finite size scaling techniques. Theoretical results for the 2D Ising model and previous simulation results for the 3D Ising model can be reproduced.     

Brain MRI tissue classification based on local Markov random fields

A new method for tissue classification of brain magnetic resonance images (MRI) of the brain is proposed. The method is based on local image models where each models the image content in a subset of the image domain. With this local modeling approach, the assumption that tissue types have the same characteristics over the brain needs not to be evoked. This is important because tissue type characteristics, such as T1 and T2 relaxation times and proton density, vary across the individual brain and the proposed method offers improved protection against intensity non-uniformity artifacts that can hamper automatic tissue classification methods in brain MRI. A framework in which local models for tissue intensities and Markov Random Field (MRF) priors are combined into a global probabilistic image model is introduced. This global model will be an inhomogeneous MRF and it can be solved by standard algorithms such as iterative conditional modes. The division of the whole image domain into local brain regions possibly having different intensity statistics is realized via sub-volume probabilistic atlases. Finally, the parameters for the local intensity models are obtained without supervision by maximizing the weighted likelihood of a certain finite mixture model. For the maximization task, a novel genetic algorithm almost free of initialization dependency is applied. The algorithm is tested on both simulated and real brain MR images. The experiments confirm that the new method offers a useful improvement of the tissue classification accuracy when the basic tissue characteristics vary across the brain and the noise level of the images is reasonable. The method also offers better protection against intensity non-uniformity artifact than the corresponding method based on a global (whole image) modeling scheme.     

多金属物体CT图像的金属伪影校正

针对多金属伪影的校正问题,本文通过仿真实验分析了多金属伪影的成因,并提出了一种基于投影校正的多金属伪影校正方法.该方法首先直接从投影域分割出金属区域,然后建立对金属区域投影值的校正模型,最后通过调整模型参数达到校正目的.模型以重建图像的灰度熵为目标函数,采用单纯形法迭代求解使熵最小时的校正参数.仿真和实际数据的实验结果表明,本文算法对多金属伪影的校正起到了良好的效果,且校正后的图像质量优于插值校正法.     

Spectrally-accurate algorithm for moving boundary problems for the Navier–Stokes equations

A spectral algorithm based on the immersed boundary conditions (IBC) concept is developed for simulations of viscous flows with moving boundaries. The algorithm uses a fixed computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization uses Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. It has been demonstrated that the algorithm delivers the theoretically predicted accuracy in both time and space. Performances of various linear solvers employed in the solution process have been evaluated and a new class of solver that takes advantage of the structure of the coefficient matrix has been proposed. The new solver results in a significant acceleration of computations as well as in a substantial reduction in memory requirements.     

Computation of demagnetizing fields and particle distribution in magnetic fluid with inhomogeneous density

A new algorithm for calculating magnetic fields in a concentrated magnetic fluid with inhomogeneous density is proposed. Inhomogeneity of the fluid is caused by magnetophoresis. In this case, the diffusion and magnetostatic parts of the problem are tightly linked together and are solved jointly. The dynamic diffusion equation is solved by the finite volume method and, to calculate the magnetic field inside the fluid, an iterative process is performed in parallel. The solution to the problem is sought in Cartesian coordinates, and the computational domain is decomposed into rectangular elements. This technique eliminates the need to solve the related boundary-value problem for magnetic fields, accelerates computations and eliminates the error caused by the finite sizes of the outer region. Formulas describing the contribution of the rectangular element to the field intensity in the case of a plane problem are given. Magnetic and concentration fields inside the magnetic fluid filling a rectangular cavity generated under the action of the uniform external filed are calculated.     

An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner

The projection method is a widely used fractional-step algorithm for solving the incompressible Navier–Stokes equations. Despite numerous improvements to the methodology, however, imposing physical boundary conditions with projection-based fluid solvers remains difficult, and obtaining high-order accuracy may not be possible for some choices of boundary conditions. In this work, we present an unsplit, linearly-implicit discretization of the incompressible Navier–Stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations. Since our scheme is not a fractional-step algorithm, it is straightforward to specify general physical boundary conditions accurately; however, this capability comes at the price of having to solve the time-dependent incompressible Stokes equations at each timestep. To solve this linear system efficiently, we employ a Krylov subspace method preconditioned by the projection method. In our implementation, the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning. The accuracy of the scheme is demonstrated for several problems, including forced and unforced analytic test cases and lid-driven cavity flows. These tests consider a variety of physical boundary conditions with Reynolds numbers ranging from 1 to 30000. The effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the Schur complement for the time-dependent incompressible Stokes operator. The projection method is found to be a more efficient preconditioner in most cases considered in the present work.