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Various graph clustering methods have been introduced to identify communities in social or biological networks. This paper studies the entropy-based and the Markov chain-based methods in clustering the undirected graph. We examine the performance of two clustering methods with conventional methods based on quality measures of clustering. For the real applications, we collect the mathematical subject classification (MSC) codes of research papers from published mathematical databases and construct the weighted code-to-document matrix for applying graph clustering methods. We pursue to group MSC codes into the same cluster if the corresponding MSC codes appear in many papers simultaneously. We compare the MSC clustering results based on the several assessment measures and conclude that the Markov chain-based method is suitable for clustering the MSC codes.

In this paper, we propose high order and unconditionally energy stable methods for a modified phase field crystal equation by applying the strategy of the energy quadratization Runge–Kutta methods. We transform the original model into an equivalent system with auxiliary variables and quadratic free energy. The modified system preserves the laws of mass conservation and energy dissipation with the associated energy functional. We present rigorous proofs of the mass conservation and energy dissipation properties of the proposed numerical methods and present numerical experiments conducted to demonstrate their accuracy and energy stability. Finally, we compare long-term simulations using an indicator function to characterize the pattern formation.

The phase field crystal (PFC) model was extended to the modified phase field crystal (MPFC) model, which is a sixth-order nonlinear damped wave equation, to include not only diffusive dynamics but also elastic interactions. In this paper, we present temporally first- and second-order accurate methods for the MPFC equation, which are based on an appropriate splitting of the energy for the PFC equation. And we use the Fourier spectral method for the spatial discretization. The first- and second-order methods are shown analytically to be unconditionally stable with respect to the energy and pseudoenergy of the MPFC equation, respectively. Numerical experiments are presented demonstrating the accuracy and energy stability of the proposed methods.

The phase-field crystal (PFC) equation is derived by the gradient flow for the Swift–Hohenberg free energy functional; thus, the numerical method requires the energy of the functional to decrease. Convex Splitting Runge–Kutta (CSRK) methods can be suitably applied to achieve high-order temporal accuracy as well as unconditional energy stability and unique solvability. For the PFC equation, we prove the unconditional energy stability and unique solvability of the CSRK methods and provide one family of parameters of the second-order CSRK methods and possible examples of third-order CSRK methods. We present numerical experiments to demonstrate the accuracy and energy stability of the methods. Specifically, based on the high-order accuracy and energy stability of the CSRK method, we propose an indicator function capable of characterizing the pattern formation of the phase-field crystal model for long-time simulation.

L∞-error estimates for B-spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B-spline Galerkin scheme is studied using appropriate projections. For fully discrete B-spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016

Allen–Cahn (AC) type equations with nonlinear source terms have been applied to a wide range of problems, for example, the vector-valued AC equation for phase separation and the phase-field equation for dendritic crystal growth. In contrast to the well developed first and second order methods for the AC equation, not many second order methods are suggested for the AC type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically. In this paper, we propose a simple and stable second order operator splitting method. A core idea of the method is to decompose the original equation into three subequations with the free-energy evolution term, the heat evolution term, and a nonlinear source term, respectively. It is important to combine these three subequations in proper order to achieve the second order accuracy and stability. We propose a method with a half-time free-energy evolution solver, a half-time heat evolution solver, a full-time midpoint solver for the nonlinear source term, and a half-time heat evolution solver followed by a final half-time free-energy evolution solver. We numerically demonstrate the second order accuracy of the new numerical method through the simulations of the phase separation and the dendritic crystal growth.

In this paper, we present an efficient numerical method for multiphase image segmentation using a multiphase-field model. The method combines the vector-valued Allen- Cahn phase-field equation with initial data fitting terms containing prescribed interface width and fidelity constants. An efficient numerical solution is achieved using the recently developed hybrid operator splitting method for the vector-valued Allen-Cahn phase-field equation. We split the modified vector-valued Allen-Cahn equation into a nonlinear equation and a linear diffusion equation with a source term. The linear diffusion equation is discretized using an implicit scheme and the resulting implicit discrete system of equations is solved by a multigrid method. The nonlinear equation is solved semi-analytically using a closed-form solution. And by treating the source term of the linear diffusion equation explicitly, we solve the modified vector-valued Allen-Cahn equation in a decoupled way. By decoupling the governing equation, we can speed up the segmentation process with multiple phases. We perform some characteristic numerical experiments for multiphase image segmentation.

Various Cahn&ndash, Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit&ndash, explicit Runge&ndash, Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme.

A discontinuous Galerkin type nonconforming element method and a local flux matching nonconforming element method for the second order elliptic boundary value problems are presented. Both of these methods enjoy the local flux conservation property. The local flux matching method finds a numerical solution in the same solution space of the DG type nonconforming element method, but it achieves much faster iterative convergence speed by embedding continuity requirement in the approximation functions rather than using constraint equations that are used in the DG type nonconforming element method. The merits of the proposed local flux matching method are as follows: the formulation of the method is simple and the solution satisfies local flux conservation property. Moreover, it can be easily applied to general elliptic equations.

We study the existence of solutions to nonlinear discrete boundary value problems with the discrete p-Laplacian, potential, and nonlinear source terms. Using variational methods, we demonstrate that there exist at least two positive solutions. The existence strongly depends on the smallest positive eigenvalue of Dirichlet eigenvalue problems and the growth conditions of the source terms.

A solution of the conductivity problem with anomalies of piecewise constant conductivities in a homogeneous medium can be represented as a single layer potential. We propose a simple disk reconstruction method based on the Laurent expansion of the single layer potential to estimate anomalies that can be used as an initial guess for an iterative searching algorithm. Using a simple linear relationship between the normal domain perturbation distance and the boundary perturbation, we develop an iterative algorithm to find anomalies within the domain. The performance of the algorithm is illustrated via numerical examples. The iterative searching algorithm works well in many cases using a single boundary measurement. However, some features of the anomalies require that the algorithm utilize multiple measurements. We discuss the limitations of the inverse conductivity problem using a single Cauchy data set and present a modified version of the algorithm for use with double boundary measurements. The improved per...

This paper deals with the problem of reconstructing the electromagnetic parameters and the shape of a target from multi-static response matrix measurements at a single frequency. The target is of characteristic size less than the operating wavelength. Using long-wavelength asymptotic expansions of the measurements of high-order, we show how the electromagnetic parameters and the equivalent ellipse of the target can be reconstructed. The asymptotic expansions of the measurements are written in terms of the new concept of frequency dependent polarization tensors. Moreover, we extend the optimization approach proposed in Part I [9] to image geometric details of an electromagnetic target that are finer than the equivalent ellipse. The equivalent ellipse still provides a good initial guess for the optimization procedure. However, compared to the conductivity case in [9], the cost functional measures the discrepancy between the computed and measured high-order frequency dependent polarization tensors rather than between the generalized polarization tensors. The main reason for such a modification of the cost functional is the fact that the (measured) frequency dependent polarization tensors can be easily obtained from multistatic measurements by solving a linear system while the derivation of the generalized polarization tensors from measurements requires more delicate treatment. The proposed methods are numerically implemented to demonstrate their validity and efficiency. Mathematics Subject Classification (MSC2000): 35R30, 35B30.

We consider a problem of reconstructing a cluster of small elastic inclusions which are located close to each other. We show that the location of the cluster and the elastic moment tensor associated with it can be reconstructed by the measurements of the displacement vectors on the boundary corresponding to the traction applied on the boundary. The detected elastic moment tensor represents the overall (or effective) property of the cluster of inclusions. We implement this idea of reconstruction for the two-dimensional linear isotropic elasticity to demonstrate its viability. We also perform a numerical study on the relation between the elastic moment tensor and the total size of the inclusions of general shape.

We present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous, high contrast materials. By high contrast, we mean that the electrical or thermal conductivity ratios are high. Our approach involves a modification of the standard integral representation using, in electrostatic terms, the polarization charge as an unknown. It is related to the perturbation approach proposed by Tausch and White [J. Tausch, J. White, Capacitance extraction of 3-D conductor systems in dielectric media with high-permittivity ratios, IEEE Trans. Microwave Theory Tech., 47 (1999) 18-26], but based on a simple form of projection rather than on the solution of auxiliary integral equations. Numerical implementation is straightforward in the multiple dielectric case.

The nonequispaced or nonuniform fast Fourier transform (NUFFT) arises in a variety of application areas, including imaging processing and the numerical solution of partial differential equations. In its most general form, it takes as input an irregular sampling of a function and seeks to compute its Fourier transform at a nonuniform sampling of frequency locations. This is sometimes referred to as the NUFFT of type 3. Like the fast Fourier transform, the amount of work required is of the order O(NlogN), where N denotes the number of sampling points in both the physical and spectral domains. In this short note, we present the essential ideas underlying the algorithm in simple terms. We also illustrate its utility with application to problems in magnetic resonance imagin and heat flow.

We consider a reconstruction formula for the internal conductivity and uniqueness of conductivity in magnetic resonance electrical impedance tomography (MREIT) which aims to reconstruct the conductivity distribution using internal current distribution. We provide a counter-example of uniqueness for a single measurement of current density with Neumann boundary data and show that at least two measurements are required unless Dirichlet boundary data are given. We present a reconstruction formula and a non-iterative reconstruction method using two internal current densities, which gives a unique conductivity distribution up to a constant factor even without any boundary measurement. The curl-J method is based on the fact that the distortion of the current density vector is induced by the gradient of conductivity orthogonal to the current flow and the fact that no MREIT method can detect the conductivity gradient parallel to the current flow direction directly. We demonstrate the feasibility of our method with several realistic numerical examples.

Two-dimensional vortex sheet suffers an unstable deformation from the Kelvin–Helmholtz instability and a curvature singularity which develops in a finite time. In this paper, long time computations of the two-dimensional vortex sheet are performed by a robust and efficient numerical method with high accuracy. To handle the rapid and non-uniform stretching of the interface, we adopt the adaptive point insertion and redistribution procedures. Computational results show highly refined structures of a complex and chaotic pattern for the vortex sheet up to very long time.

We consider the problem of identifying unknown inclusions inside an elastic body by means of traction–displacement relations measured on the boundary. Based on the asymptotic formula of Ammari et al (2003 J. Elast. at press), we propose an algorithm to reconstruct unknown inclusions. The algorithm of this paper detects the location and the elastic moment tensors (elastic Polya–Szego tensors) of the inclusion. Since the elastic moment tensors carry information on the size of the inclusion, we can represent the inclusion in two dimensions by a disc of the detected size with the detected centre. We also propose an algorithm to find an ellipse which represents the elastic moment tensors. For this purpose, we explicitly compute elastic moment tensors associated with ellipses. Several results of numerical experiments are presented. We emphasize that only shear stresses are used for the reconstruction of the inclusion: linear traction to detect elastic moment tensors and quadratic polynomials to detect the location.

We consider magnetic resonance electrical impedance tomography, which aims to reconstruct the conductivity distribution using the internal current density furnished by magnetic resonance imaging. We show the uniqueness of the conductivity reconstruction with one measurement imposing the Dirichlet boundary condition. We also propose a fast non-iterative numerical algorithm for the conductivity reconstruction using the internal current vector information. The algorithm is mainly based on efficient numerical construction of equipotential lines. The resulting numerical method is stable in the sense that the error of the computed conductivity is linearly proportional to the input noise level and the introduction of internal current data makes the impedance tomography problem well-posed. We present various numerical examples to show the feasibility of using our method.

A numerical method is constructed for two-dimensional Navier–Stokes flows in a circular domain. Adaptive computational grids are employed to resolve the boundary layer and the boundary vorticity values are obtained in high order of accuracy. Numerical results show a quite good agreement with the asymptotic expansion calculation (Kim, SIAM J. Appl. Math. 58(5) (1998) 1394–1413). Extension to general shape domains is briefly addressed.

We present a parallel Poisson solver on distributed computing environments. In the solver, the parallel implementation of the Fast Multipole Method (FMM) is designed to minimize amount of data communication and the number of data transfers and synchronizations. The experimental results show linear speedup, good load balancing, and reasonable performance under failure and demonstrate the viability of loosely coupled heterogeneous workstations for large scale scientific computations.

We study the free boundary problem to identify the plasma region in a stationary tokamak machine using the magnetic flux function. Uniqueness has been proved and the free boundary has been explicitly computed in the case where the domain is an n-dimensional ball. We also derive an energy distribution identity which is applicable to identify the location of the free boundary for more general domains in two-dimensional space.

We present a direct, adaptive solver for the Poisson equation which can achieve any prescribed order of accuracy. It is based on a domain decomposition approach using local spectral approximation, as well as potential theory and the fast multipole method. In two space dimensions, the algorithm requiresO(NK) work, whereNis the number of discretization points andKis the desired order of accuracy.

We present a new integral equation method for the calculation of two-dimensional scattering from periodic structures involving triple-points (multiple materials meeting at a single point). The combination of a robust and high-order accurate integral representation and a fast direct solver permits the efficient simulation of scattering from fixed structures at multiple angles of incidence. We demonstrate the performance of the scheme with several numerical examples., Comment: 19 Pages. 8 Figures

In this paper we consider the inverse problem of identifying a linear inclusion inside an elastic body from exterior boundary measurements. Based on the asymptotic formula by Beretta and Francini (2006 SIAM J. Math. Anal. 38 1249–61), we design an effective reconstruction algorithm to find the endpoints and the thickness of a linear inclusion. Numerical experiments show that the algorithm is effective and stable.

A theoretical description of radial energetic‐ion transport induced by electromagnetic waves of ion cyclotron range of frequencies (ICRF) is presented in a tokamak with general flux surface geometry. A known form of quasilinear operator is used to represent the ICRF wave‐induced particle scattering in velocity space. The theory is applied to both nonflowing and flowing groups of energetic ions, represented by alpha particles born from deuterium–tritium reactions, and energetic ions born from unbalanced neutral beam injection, respectively. It is found that ICRF waves with reasonable power can be used for manipulation of the local radial energetic ion flow at practical level. A simple integral representation of the radial fluxes is suggested for implementation into a transport code. A brief discussion on the possibility of the strong negative electric field generation by ICRH at the plasma edge is also presented.

A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an algorithm which computes the convolution of arbitrarily spaced data with the sinc kernel in [math] operations, where [math] denotes the number of data points. We briefly discuss its application to the construction of optimal density compensation weights for Fourier reconstruction and to the iterative approximation of the pseudoinverse of the signal equation in MRI.

In magnetic resonance electrical impedance tomography (MREIT), we try to reconstruct a cross-sectional resistivity (or conductivity) image of a subject. When we inject a current through surface electrodes, it generates a magnetic field. Using a magnetic resonance imaging (MRI) scanner, we can obtain the induced magnetic flux density from MR phase images of the subject. We use recessed electrodes to avoid undesirable artefacts near electrodes in measuring magnetic flux densities. An MREIT image reconstruction algorithm produces cross-sectional resistivity images utilizing the measured internal magnetic flux density in addition to boundary voltage data. In order to develop such an image reconstruction algorithm, we need a three-dimensional forward solver. Given injection currents as boundary conditions, the forward solver described in this paper computes voltage and current density distributions using the finite element method (FEM). Then, it calculates the magnetic flux density within the subject using the Biot-Savart law and FEM. The performance of the forward solver is analysed and found to be enough for use in MREIT for resistivity image reconstructions and also experimental designs and validations. The forward solver may find other applications where one needs to compute voltage, current density and magnetic flux density distributions all within a volume conductor.