Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems

This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887–888, 2001; J Comput Phys 219:513–531, 2006; Comput Appl Math 218:79–87, 2008; J Comput Phys 228:2517–2534, 2009; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145–192, 2012; J Comput Appl Math 236:2317–2337, 2012; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915–4946, 2004; Int J Numer Methods Fluids 42:265–295, 2003; Kuzmin and Möller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, 2005; Kuzmin and Turek in J Comput Phys 175:525–558, 2002; J Comput Phys 198:131–158, 2004) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming $P_1$ and $Q_1$ finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix–Raviart element (Crouzeix and Raviart in RAIRO R3 7:33–76, 1973) on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher–Turek elements (Rannacher and Turek in Numer Methods PDEs 8:97–111, 1992). The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection–diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix–Raviart element. Good resolution of smooth and discontinuous profiles is attested to $Q_1^\mathrm{nc}$ approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix–vector multiplications is addressed.     

A Splitting Algorithm for Image Segmentation on Manifolds Represented by the Grid Based Particle Method

We propose a numerical approach to solve variational problems on manifolds represented by the grid based particle method (GBPM) recently developed in Leung et al. (J. Comput. Phys. 230(7):2540–2561, 2011), Leung and Zhao (J. Comput. Phys. 228:7706–7728, 2009a, J. Comput. Phys. 228:2993–3024, 2009b, Commun. Comput. Phys. 8:758–796, 2010). In particular, we propose a splitting algorithm for image segmentation on manifolds represented by unconnected sampling particles. To develop a fast minimization algorithm, we propose a new splitting method by generalizing the augmented Lagrangian method. To efficiently implement the resulting method, we incorporate with the local polynomial approximations of the manifold in the GBPM. The resulting method is flexible for segmentation on various manifolds including closed or open or even surfaces which are not orientable.     

WENO Scheme with Subcell Resolution for Computing Nonconservative Euler Equations with Applications to One-Dimensional Compressible Two-Medium Flows

High order path-conservative schemes have been developed for solving nonconservative hyperbolic systems in (Parés, SIAM J.?Numer. Anal. 44:300?C321, 2006; Castro et al., Math. Comput. 75:1103?C1134, 2006; J.?Sci. Comput. 39:67?C114, 2009). Recently, it has been observed in (Abgrall and Karni, J.?Comput. Phys. 229:2759?C2763, 2010) that this approach may have some computational issues and shortcomings. In this paper, a modification to the high order path-conservative scheme in (Castro et al., Math. Comput. 75:1103?C1134, 2006) is proposed to improve its computational performance and to overcome some of the shortcomings. This modification is based on the high order finite volume WENO scheme with subcell resolution and it uses an exact Riemann solver to catch the right paths at the discontinuities. An application to one-dimensional compressible two-medium flows of nonconservative or primitive Euler equations is carried out to show the effectiveness of this new approach.     

Simulations of Supersonic Astrophysical Jets and Their Environments Using Level Set Methods

Computational fluid dynamics simulations using the WENO method and level set method are applied to high Mach number nonrelativistic astrophysical jets, including the effects of radiative cooling. WENO methods introduced in Liu et al. (J. Comput. Phys., 115:200–212, 1994) have allowed us to simulate HH 1-2 astrophysical jets at Mach number much higher than Mach 80 (Ha et al. in J. Sci. Comput. 24:29–44, 2005). Simulations at high Mach numbers and with radiative cooling are essential for achieving detailed agreement with the astrophysical images. Simulations of interaction between astrophysical jet and environment using level set methods are considered in this paper.     

Hermitian Compact Interpolation on the Cubed-Sphere Grid

The cubed-sphere grid is a spherical grid made of six quasi-cartesian square-like patches. It was originally introduced in Sadourny (Mon Weather Rev 100:136–144, 1972). We extend to this grid the design of high-order finite-difference compact operators (Collatz, The numerical treatment of differential equations. Springer, Berlin, 1960; Lele, J Comput Phys 103:16–42, 1992). The present work is limitated to the design of a fourth-order accurate spherical gradient. The treatment at the interface of the six patches relies on a specific interpolation system which is based on using great circles in an essential way. The main interest of the approach is a fully symmetric treatment of the sphere. We numerically demonstrate the accuracy of the approximate gradient on several test problems, including the cosine-bell test-case of Williamson et al. (J Comput Phys 102:211–224, 1992) and a deformational test-case reported in Nair and Lauritzen (J Comput Phys 229:8868–8887, 2010).     

Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems

The stochastic collocation method (Babu?ka et al. in SIAM J Numer Anal 45(3):1005–1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411–2442, 2008a; SIAM J Numer Anal 46(5):2309–2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118–1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289–294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229–275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41–44):3187–3206, 2009; Arch Comput Methods Eng 17:435–454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $O(1)$ to moderate dimensions $O(10)$ and to high dimensions $O(100)$ . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.     

pth Moment Exponential Stability of Stochastic Recurrent Neural Networks with Markovian Switching

This paper investigates the problem of the pth moment exponential stability for a class of stochastic recurrent neural networks with Markovian jump parameters. With the help of Lyapunov function, stochastic analysis technique, generalized Halanay inequality and Hardy inequality, some novel sufficient conditions on the pth moment exponential stability of the considered system are derived. The results obtained in this paper are completely new and complement and improve some of the previously known results (Liao and Mao, Stoch Anal Appl, 14:165–185, 1996; Wan and Sun, Phys Lett A, 343:306–318, 2005; Hu et al., Chao Solitions Fractals, 27:1006–1010, 2006; Sun and Cao, Nonlinear Anal Real, 8:1171–1185, 2007; Huang et al., Inf Sci, 178:2194–2203, 2008; Wang et al., Phys Lett A, 356:346–352, 2006; Peng and Liu, Neural Comput Appl, 20:543–547, 2011). Moreover, a numerical example is also provided to demonstrate the effectiveness and applicability of the theoretical results.     

Second-Order Accurate Computation of Curvatures in a Level Set Framework Using Novel High-Order Reinitialization Schemes

We present a high-order accurate scheme for the reinitialization equation of Sussman et al.(J. Comput. Phys. 114:146–159, [1994]) that guarantees accurate computation of the interface’s curvatures in the context of level set methods. This scheme is an extension of the work of Russo and Smereka (J. Comput. Phys. 163:51–67, [2000]). We present numerical results in two and three spatial dimensions to demonstrate fourth-order accuracy for the reinitialized level set function, third-order accuracy for the normals and second-order accuracy for the interface’s mean curvature in the L ¹- and L ^∞-norms. We also exploit the work of Min and Gibou (UCLA CAM Report (06-22), [2006]) to show second-order accurate scheme for the computation of the mean curvature on non-graded adaptive grids.     

A linear system form solution to compute the local space average color

In this document, we present an alternative to the method introduced by Ebner (Pattern Recognit 60–67, 2003; J Parallel Distrib Comput 64(1):79–88, 2004; Color constancy using local color shifts, pp 276–287, 2004; Color Constancy, 2007; Mach Vis Appl 20(5):283–301, 2009) for computing the local space average color. We show that when the problem is framed as a linear system and the resulting series is solved, there is a solution based on LU decomposition that reduces the computing time by at least an order of magnitude.     

Some types of filters in residuated lattices

In this paper, inspired by some types of $BL$ -algebra filters (deductive systems) introduced in Haveshki et al. (Soft Comput 10:657–664, 2006), Kondo and Dudek (Soft Comput 12:419–423, 2008) and Turunen (Arch Math Log 40:467–473, 2001), we defined residuated lattice versions of them and study them in connection with Van Gasse et al. (Inf Sci 180(16):3006–3020, 2010), Lianzhen and Kaitai (Inf Sci 177:5725–5738, 2007), Zhu and Xu (Inf Sci 180:3614–3632, 2010). Also we consider some relations between these filters and quotient residuated lattice that are constructed via these filters.     

A High-Order Moving Mesh Kinetic Scheme Based on WENO Reconstruction for Compressible Flows on Unstructured Meshes

In this paper, we present a high-order moving mesh (HMM) kinetic scheme for compressible flow computations on unstructured meshes. To construct the scheme, we employ the frame of the remapping-free ALE-type kinetic method (Ni et al. in J Comput Phys 228:3154–3171, 2009) to get the discretization of compressible system. For the space accuracy, we use the weighted essential non-oscillatory reconstruction on the adaptive moving mesh from Tang and Tang (SIAM J Numer Anal 41:487–515 2003) to achieve time accuracy,we make use of the kinetic flux which includes time accurate integral, and thus obtain a HMM scheme. A number of numerical examples are given, especially an isentropic vortex problem to show the convergence order of the scheme. Numerical results demonstrate the accuracy and robustness of the scheme.     

A Comparison of Iterated Optimal Stopping and Local Policy Iteration for American Options Under Regime Switching

A theoretical analysis tool, iterated optimal stopping, has been used as the basis of a numerical algorithm for American options under regime switching (Le and Wang in SIAM J Control Optim 48(8):5193–5213, 2010). Similar methods have also been proposed for American options under jump diffusion (Bayraktar and Xing in Math Methods Oper Res 70:505–525, 2009) and Asian options under jump diffusion (Bayraktar and Xing in Math Fin 21(1):117–143, 2011). An alternative method, local policy iteration, has been suggested in Huang et al. (SIAM J Sci Comput 33(5):2144–2168, 2011), and Salmi and Toivanen (Appl Numer Math 61:821–831, 2011). Worst case upper bounds on the convergence rates of these two methods suggest that local policy iteration should be preferred over iterated optimal stopping (Huang et al. in SIAM J Sci Comput 33(5):2144–2168, 2011). In this article, numerical tests are presented which indicate that the observed performance of these two methods is consistent with the worst case upper bounds. In addition, while these two methods seem quite different, we show that either one can be converted into the other by a simple rearrangement of two loops.     

Superposition of Multi-Valued Solutions in High Frequency Wave Dynamics

The weakly coupled WKB system captures high frequency wave dynamics in many applications. For such a system a level set method framework has been recently developed to compute multi-valued solutions to the Hamilton-Jacobi equation and evaluate position density accordingly. In this paper we propose two approaches for computing multi-valued quantities related to density, momentum as well as energy. Within this level set framework we show that physical observables evaluated in Jin et al. (J. Comput. Phys. 210(2):497–518, [2005]; J. Comput. Phys. 205(1):222–241, [2005]) are simply the superposition of their multi-valued correspondents. A series of numerical tests is performed to compute multi-valued quantities and validate the established superposition properties.     

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see Castro et al. in Math. Comput. 79:1427?C1472, 2010). This Riemann solver is based on a suitable decomposition of a Roe matrix (see Toumi in J. Comput. Phys. 102(2):360?C373, 1992) by means of a parabolic viscosity matrix (see Degond et al. in C. R. Acad. Sci. Paris 1 328:479?C483, 1999) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L ^??-stable, well-balanced, and it doesn??t require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods.     

Comments and further improvements on “pth moment stability of stochastic neural networks with Markov volatilities”

In a very recent paper, Peng and Liu (Neural Comput Appl 20:543–547, 2011) investigated the pth moment stability of the stochastic Grossberg–Hopfield neural networks with Markov volatilities by Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1). We should point out that Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1) investigated the pth moment exponentially stable for a class of stochastic dynamical systems with constant delay; however, this theorem cannot apply to the case of variable time delays. It is also worthy to emphasize that Peng and Liu (Neural Comput Appl 20:543–547, 2011) discussed by Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1) the pth moment exponentially stable for the Grossberg–Hopfield neural networks with variable delays, and therefore, there are some gaps between Peng and Liu (Neural Comput Appl 20:543–547, 2011, Theorem 1) and Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1). In this paper, we fill up this gap. Moreover, a numerical example is also provided to demonstrate the effectiveness and applicability of the theoretical results.     

On sunflowers and matrix multiplication

We present several variants of the sunflower conjecture of Erd?s & Rado (J Lond Math Soc 35:85–90, 1960) and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to the questions of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990) and Cohn et al. (2005) regarding possible approaches for obtaining fast matrix-multiplication algorithms. Specifically, we show that the Erd?s–Rado sunflower conjecture (if true) implies a negative answer to the “no three disjoint equivoluminous subsets” question of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990); we also formulate a “multicolored” sunflower conjecture in ${\mathbb{Z}_3^n}$ and show that (if true) it implies a negative answer to the “strong USP” conjecture of Cohn et al. (2005) (although it does not seem to impact a second conjecture in Cohn et al. (2005) or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith–Winograd conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in ${\mathbb{Z}_3^n}$ is a strengthening of the well-known (ordinary) sunflower conjecture in ${\mathbb{Z}_3^n}$ , and we show via our connection that a construction from Cohn et al. (2005) yields a lower bound of (2.51 . . .) ⁿ on the size of the largest multicolored 3-sunflower-free set, which beats the current best-known lower bound of (2.21 . . . ) ⁿ Edel (2004) on the size of the largest 3-sunflower-free set in ${\mathbb{Z}_3^n}$ .     

An Asymptotic Preserving Method for Linear Systems of Balance Laws Based on Galerkin’s Method

We apply the concept of asymptotic preserving schemes (SIAM J Sci Comput 21:441–454, 1999) to the linearized \(p\) -system and discretize the resulting elliptic equation using standard continuous Finite Elements instead of Finite Differences. The fully discrete method is analyzed with respect to consistency, and we compare it numerically with more traditional methods such as Implicit Euler’s method.     

A Conservative Method for the Simulation of?the?Isothermal Euler System with the van-der-Waals Equation of State

In this article, we are interested in the simulation of phase transition in compressible flows, with the isothermal Euler system, closed by the van-der-Waals model. We formulate the problem as an hyperbolic system, with a source term located at the interface between liquid and vapour. The numerical scheme is based on (Abgrall and Saurel, J. Comput. Phys. 186(2):361?C396, 2003; Le Métayer et al., J. Comput. Phys. 205(2):567?C610, 2005). Compared with previous discretizations of the van-der-Waals system, the novelty of this algorithm is that it is fully conservative. Its Godunov-type formulation allows an easy implementation on multi-dimensional unstructured meshes.