Diffusion equations over arbitrary triangulated surfaces for filtering and texture applications

In computer graphics, triangular mesh representations of surfaces have become very popular. Compared with parametric and implicit forms of surfaces, triangular mesh surfaces have many advantages, such as easy to render, convenient to store and the ability to model geometric objects with arbitrary topology. In this paper, we are interested in data processing over triangular mesh surfaces through PDEs (partial differential equations). We study several diffusion equations over triangular mesh surfaces, and present corresponding numerical schemes to solve them. Our methods work for triangular mesh surfaces with arbitrary geometry (the angles of each triangle are arbitrary) and topology (open meshes or closed meshes of arbitrary genus). Besides the flexibility, our methods are efficient due to the implicit/semi-implicit time discretization. We finally apply our methods to several filtering and texture applications such as image processing, texture generating and regularization of harmonic maps over triangular mesh surfaces. The results demonstrate the flexibility and effectiveness of our methods.     

Quantum broadcast scheme and multi-output quantum teleportation via four-qubit cluster state

In this paper, two theoretical schemes of the arbitrary single-qubit states via four-qubit cluster state are proposed. One is three-party quantum broadcast scheme, which realizes the broadcast among three participants. The other is multi-output quantum teleportation. Both allow two distant receivers to simultaneously and deterministically obtain the arbitrary single-qubit states, respectively. Compared with former schemes of an arbitrary single-qubit state, the proposed schemes realize quantum multi-cast communication efficiently, which enables Bob and Charlie to obtain the states simultaneously in the case of just knowing Alice’s measurement results. The proposed schemes play an important role in quantum information, specially in secret sharing and quantum teleportation.     

Second-order-accurate explicit fluctuation splitting schemes for unsteady problems

This paper is concerned with the issue of obtaining explicit fluctuation splitting schemes which achieve second-order accuracy in both space and time on an arbitrary unstructured triangular mesh. A theoretical analysis demonstrates that, for a linear reconstruction of the solution, mass lumping does not diminish the accuracy of the scheme provided that a Galerkin space discretization is employed. Thus, two explicit fluctuation splitting schemes are devised which are second-order accurate in both space and time, namely, the well known Lax-Wendroff scheme and a Lax-Wendroff-type scheme using a three-point-backward discretization of the time derivative. A thorough mesh-refinement study verifies the theoretical order of accuracy of the two schemes on meshes with increasing levels of nonuniformity.     

Open-loop discretization methods for control systems design

In this paper, a unified theory is presented which addresses essential aspects of the open-loop discretization procedure. Initially, factors which affect the generation and propagation of discretization errors are identified by analytical, heuristic and experimental arguments. Following this, a discretization algorithm is presented which takes these factors into account. The fundamental idea of the discretization method is the replacement of the analogue integrators of the prototype continuous-time system by discrete-time approximations. This is done in such a way as to optimize a given cost function with respect to a given input. Unlike many discretization schemes presented in the literature, the procedure developed attempts to first determine the magnitude of the expected discretization errors, and then discretize with an appropriate complexity, giving the designer control over the order of the discrete-time system. This appears to be an effective means of designing filters with low complexity which still retain essential properties of the prototype system. An emphasis of this work is to view the discretization process from a control theory and engineering perspective. This results in a number of new perspectives relating to the discretization process.     

Mimetic discretization of two-dimensional Darcy convection

We consider discretization of the planar convection of the incompressible fluid in a porous medium filling rectangular enclosure. This problem belongs to the class of cosymmetric systems and admits an existence of a continuous family of steady states in the phase space. Mimetic finite-difference schemes for the primitive variables equation are developed. The connection of a derived staggered discretization with a finite-difference approach based on the stream function and temperature equations is established. Computations of continuous cosymmetric families of steady states are presented for the case of uniform and nonuniform grids.     

信息表的闭离散化方案研究

提出对象域U的有序划分概念,讨论一种特殊的离散化方案（闭离散化方案）。给出对象域U的有序划分对应的闭离散化方案获取算法CDA,分析闭离散化方案与对象域U的有序划分之间的关系,证明了闭离散化方案在离散格到划分格的映射f下能保持交并运算。     

Dispersion-bounded numerical integration of the elastodynamic equations with cost-effective staggered schemes

Known procedures for designing numerical schemes for the integration of elastodynamic equations with explicit control over numerical dispersion are reviewed. In the literature, the analysis of such schemes has concentrated on the discrete space differentiators, and has neglected the role played by time discretization in the overall accuracy. In this paper we define a computational cost for a given dispersion error bound which fully includes the effect of temporal differencing. For some representative schemes based on leap-frog time marching, we provide an optimal operating point (time sampling rate and number of grid points per shortest wavelength) which minimizes the computational cost for a given dispersion error threshold. Based on this notion of cost, we introduce new optimal operators for staggered grids. Additionally, we introduce the notion of composite differentiators to design still more cost-effective schemes. The cost of the proposed schemes is shown to be less than that of known finite difference (FD) operators and compares favorably with pseudo-spectral (PS) algorithms. Numerical simulations are presented to illustrate the effectiveness of the new operators.     

Efficient and stable numerical solution of the Heston–Cox–Ingersoll–Ross partial differential equation by alternating direction implicit finite difference schemes

This paper concerns the numerical solution of the three-dimensional Heston–Cox–Ingersoll–Ross partial differential equation for the fair values of European-style financial options. We follow the method-of-lines approach and first semidiscretize on suitable nonuniform Cartesian spatial grids by applying finite difference schemes. The main aim of this paper is to investigate various prominent alternating direction implicit schemes for the effective time discretization of the obtained, very large, semidiscrete systems. For this purpose an extensive numerical study is performed, among others, for arbitrary correlation factors, for time-dependent mean-reversion levels, for short and long maturities, for cases where the Feller condition is satisfied and for cases where it is not. Also, the approximation of the hedging quantities Delta and Gamma is discussed and double barrier knock-out call options are considered.     

On a collocation B-spline method for the solution of the Navier–Stokes equations

A novel B-spline collocation method for the solution of the incompressible Navier–Stokes equations is presented. The discretization employs B-splines of maximum continuity, yielding schemes with high-resolution power. The Navier–Stokes equations are solved by using a fractional step method, where the projection step is considered as a Div–Grad problem, so that no pressure boundary conditions need to be prescribed. Pressure oscillations are prevented by introducing compatible B-spline bases for the velocity and pressure, yielding efficient schemes of arbitrary order of accuracy. The method is applied to two-dimensional benchmark flows, and mass lumping techniques for cost-effective computation of unsteady problems are discussed.     

Lattice-based volumetric global illumination

We describe a novel volumetric global illumination framework based on the Face-Centered Cubic (FCC) lattice. An FCC lattice has important advantages over a Cartesian lattice. It has higher packing density in the frequency domain, which translates to better sampling efficiency. Furthermore, it has the maximal possible kissing number (equivalent to the number of nearest neighbors of each site), which provides optimal 3D angular discretization among all lattices. We employ a new two-pass (illumination and rendering) global illumination scheme on an FCC lattice. This scheme exploits the angular discretization to greatly simplify the computation in multiple scattering and to minimize illumination information storage. The GPU has been utilized to further accelerate the rendering stage. We demonstrate our new framework with participating media and volume rendering with multiple scattering, where both are significantly faster than traditional techniques with comparable quality.     

Faithful quantum broadcast beyond the no-go theorem

The main superiority of the quantum remote preparation over quantum teleportation lies the classical resource saving. This situation may be changed from the following constructions. Our purpose in this paper is to find some special differences between these two quantum tasks besides the classical resource costs. Some novel schemes show that the first one is useful to simultaneously broadcast arbitrary quantum states, while the second one cannot because of the quantum no-cloning theorem. Moreover, these broadcast schemes may be adapted to satisfying the different receivers’ requirements or distributing the classical information, which are important in various quantum applications such as the quantum secret distribution or the quantum network communication.     

Optimal Control of a Conventionally Correct System with Conjugation Conditions

New optimal control problems are constructed and considered for distributed systems whose states are described by Neumann boundary-value problems with conjugation conditions and non-unique solutions. The paper proposes accurate higher-order computation schemes for discretization of optimization problems for the case where a control set coincides with a full Hilbert space.     

Extension of numerical quadrature formulae to cater for end point singular behaviours over finite intervals

The N-point Gaussian quadrature method is generalised to cater for various possible singular behaviours at the end points of the interval of integration at the expense of being algebraically exact for a polynomial of lower order than usual. Weights and abscissae are chosen to exactly integrate an integrand which is the sum of the singular functions and an arbitrary polynomial. This allows us to cater for several different end-point singularities in the same quadrature formula and in this way differs from published quadratures where a singular behaviour is incorporated in a weight function that multiplies an arbitrary polynomial. We present tables of weights and abscissae that cater for (i) logarithmic end point singularities and (ii) logarithmic plus inverse square root singular behaviours. Also a 10-point quadrature is presented that exactly caters for log(x), x^-1/4,x^-1/2,x^-3/4 singular behaviours and is recommended for programmable calculator use. Finally a brief comparison study of the various (10-point) quadratures herein considered is made.     

On improving mass-conservation properties of the hybrid particle-level-set method

For the computation of multi-phase flows level-set methods are an attractive alternative to volume-of-fluid or front-tracking approaches. For improving their accuracy and efficiency the hybrid particle-level-set modification was proposed by Enright et al. [Enright D, Fedkiw R, Ferziger J, Mitchell I. A hybrid particle-level-set method for improved interface capturing. J Comput Phys 2002;183:83-116]. In actual applications the overall properties of a level-set method, such as mass conservation, are strongly affected by discretization schemes and algorithmic details. In this paper we address these issues with the objective of determining the optimum alternatives for the purpose of direct numerical simulation of dispersed-droplet flows. We evaluate different discretization schemes for curvature and unit normal vector at the interface. Another issue is the particular formulation of the reinitialization of the level-set function which significantly affects the quality of computational results. Different approaches employing higher-order schemes for discretization, supplemented either by a correction step using marker particles (Enright et al., 2002) or by additional constraints [Sussman M, Almgren AS, Bell JB, Colella P, Howell LH, Welcome ML. An adaptive level set approach for incompressible two-phase flows. J Comput Phys 1999;148:81-124] are analyzed. Different parameter choices for the hybrid particle-level-set method are evaluated with the purpose of increasing the efficiency of the method. Aiming at large-scale computations we find that in comparison with pure level-set methods the hybrid particle-level-set method exhibits better mass-conservation properties, especially in the case of marginally resolved interfaces.     

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.     

A Finite Element Method on Convex Polyhedra

We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping. This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.     

An efficient ECC-based mechanism for securing network coding-based P2P content distribution

Network coding has been demonstrated to be able to improve the performance of P2P content distribution. However, it is vulnerable to pollution attacks where malicious peers can flood the network with corrupted blocks easily, leading to substantial performance degradation. Moreover, existing corruption detection schemes for network coding are not well suited to P2P systems. Effective scheme to detect the corruption and identify the attacker is required to thwart such attacks. In this paper, we propose an efficient ECC-based mechanism for securing network coding-based P2P content distribution, namely ESNC, which includes an efficient network coding signature scheme and an identity-based malicious peer identification scheme. The two schemes cooperate to thwart pollution attacks on network coding effectively in P2P networks, not only detecting corrupted blocks on-the-fly efficiently, but also precisely identifying all the malicious peers quickly. ESNC is mainly based on elliptic curve cryptography (ECC) and can provide high level of security. It incurs significantly less computation and communication overheads than other comparable state-of-the-art schemes for P2P systems. ESNC can work with arbitrary topologies, as it is the case in P2P networks. Security analysis demonstrates that ESNC can resist hash collision attacks, signature forgery attacks, and collusion attacks with arbitrary number of colluding malicious peers. Simulation results show that ESNC effectively limits the corruption spread and identifies all the malicious peers in a short time under different practical settings.     

Arbitrary high order finite-volume methods for electromagnetic wave propagation

Problems in electromagnetic wave propagation often require high accuracy approximations with low resolution computational grids. For non-stationary problems such schemes should possess the same approximation order in space and time. In the present article we propose for electromagnetic applications an explicit class of robust finite-volume (FV) schemes for the Maxwell equations. To achieve high accuracy we combine the FV method with the so-called ADER approach resulting in schemes which are arbitrary high order accurate in space and time. Numerical results and convergence investigations are shown for two and three-dimensional test cases on Cartesian grids, where the used FV-ADER schemes are up to 8th order accurate in both space and time.     

GEncode: Geometry-driven compression for General Meshes

Performances of actual mesh compression algorithms vary significantly depending on the type of model it encodes. These methods rely on prior assumptions on the mesh to be efficient, such as regular connectivity, simple topology and similarity between its elements. However, these priors are implicit in usual schemes, harming their suitability for specific models. In particular, connectivity‐driven schemes are difficult to generalize to higher dimensions and to handle topological singularities. GEncode is a new single‐rate, geometry‐driven compression scheme where prior knowledge of the mesh is plugged into the coder in an explicit manner. It encodes meshes of arbitrary dimension without topological restrictions, but can incorporate topological properties, such as manifoldness, to improve the compression ratio. Prior knowledge of the geometry is taken as an input of the algorithm, represented by a function of the local geometry. This suits particularly well for scanned and remeshed models, where exact geometric priors are available. Compression results surfaces and volumes are competitive with existing schemes.     

Splitting meshless deforming objects with explicit surface tracking

We present a novel algorithm for efficiently splitting deformable solids along arbitrary piecewise linear crack surfaces in cutting and fracture simulations. The algorithm combines a meshless discretization of the deformation field with explicit surface tracking using a triangle mesh. We decompose the splitting operation into a first step where we synthesize crack surfaces, and a second step where we use the newly synthesized surfaces to update the meshless discretization of the deformation field. We present a novel visibility graph for facilitating fast update of shape functions in the meshless discretization. The separation of the splitting operation into two steps, along with our novel visibility graph, enables high flexibility and control over the splitting trajectories, provides fast dynamic update of the meshless discretization, and allows for an easy implementation. As a result, our algorithm is scalable, versatile, and suitable for a large range of applications, from computer animation to interactive medical simulation.