Unconditionally Stable Finite Difference,Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations

We present an unconditionally energy stable and solvable finite difference scheme for the Cahn-Hilliard-Hele-Shaw (CHHS) equations, which arise in models for spinodal decomposition of a binary fluid in a Hele-Shaw cell, tumor growth and cell sorting, and two phase flows in porous media. We show that the CHHS system is a specialized conserved gradient-flow with respect to the usual Cahn-Hilliard (CH) energy, and thus techniques for bistable gradient equations are applicable. In particular, the scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step-size. Owing to energy stability, we show that the scheme is stable in the L_s^￥(0,T;H_h¹)L_{s}^{\infty}(0,T;H_{h}^{1}) norm, and, assuming two spatial dimensions, we show in an appendix that the scheme is also stable in the L_s²(0,T;H_h²)L_{s}^{2}(0,T;H_{h}^{2}) norm. We demonstrate an efficient, practical nonlinear multigrid method for solving the equations. In particular, we provide evidence that the solver has nearly optimal complexity. We also include a convergence test that suggests that the global error is of first order in time and of second order in space.     

A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems

Abstract We derive a priori error estimates in the finite element method for nonselfadjoint elliptic and parabolic interface problems in a two-dimensional convex polygonal domain. Optimal H ¹-norm and sub-optimal L ²-norm error estimates are obtained for elliptic interface problems. For parabolic interface problems, the continuous-time Galerkin method is analyzed and an optimal order error estimate in the L ²(0,T;H ¹)-norm is established. Further, a discrete-in-time discontinuous Galerkin method is discussed and a related optimal error estimate is obtained. Keywords: Elliptic and parabolic interface problems, finite element method, spatially discrete scheme, discontinuous Galerkin method, error estimates Mathematics Subject Classification (1991): 65N15, 65N20     

Optimal Error Estimates of the Legendre Tau Method for Second-Order Differential Equations

In this paper, we prove that the Legendre tau method has the optimal rate of convergence in L ²-norm, H ¹-norm and H ²-norm for one-dimensional second-order steady differential equations with three kinds of boundary conditions and in C([0,T];L ²(I))-norm for the corresponding evolution equation with the Dirichlet boundary condition. For the generalized Burgers equation, we develop a Legendre tau-Chebyshev collocation method, which can also be optimally convergent in C([0,T];L ²(I))-norm. Finally, we give some numerical examples.     

Boundary control,stabilization and zero-pole dynamics for a non-linear distributed parameter system

In this work we show that the now standard lumped non-linear enhancement of root-locus design still persists for a non-linear distributed parameter boundary control system governed by a scalar viscous Burgers' equation. Namely, we construct a proportional error boundary feedback control law and show that closed-loop trajectories tend to trajectories of the open-loop zero dynamics as the gain parameters are increased to infinity. We also prove a robust version of this result, valid for perturbations by an unknown disturbance with arbitrary L² norm. For the controlled Burgers' equation forced by a disturbance we prove that, for all initial data in L²(0, 1), the closed-loop trajectories converge in L²(0, 1), uniformly in t∈[0, T] and in H¹(0, 1), uniformly in t∈[t₀, T] for any t₀>0, to the trajectories of the corresponding perturbed zero dynamics. We have also extended these results to include the case when additional boundary controls are included in the closed-loop system. This provides a proof of convergence of trajectories in case the zero dynamics is replaced by a non-homogeneous Dirichlet boundary controlled Burgers' equation. As an application of our convergence of trajectories results, we demonstrate that our boundary feedback control scheme provides a semiglobal exponential stabilizing feedback law in L², H¹ and L^∞ for the open-loop system consisting of Burgers' equation with Neumann boundary conditions and zero forcing term. We also show that this result is robust in the sense that if the open-loop system is perturbed by a sufficiently small non-zero disturbance then the resulting closed-loop system is ‘practically semiglobally stabilizable’ in L²-norm. Copyright © 1999 John Wiley & Sons, Ltd.     

Parabolic PDE-based multi-agent formation control on a cylindrical surface

ABSTRACT

This paper considers the modelling and control design of the multi-agent systems in the 3-D space. The communication graph of the agents is a mesh-grid 2-D cylindrical surface. Different from most existing literatures, where the agents are modelled by ordinary differential equations (ODEs), we treat the agents as a continuum in this paper. More specifically, we model the collective dynamics of the agents by two reaction–advection–diffusion 2-D partial differential equations (PDEs). The PDE states represent the agent positions, and the equilibria correspond to possible formation manifolds. These PDEs can be open-loop unstable, and the boundary stabilisation problem of the PDEs on the cylindrical surface is solved using the backstepping method. An all-explicit observer-based output control scheme is constructed, which is distributed in the sense that each agent only needs local information. Closed-loop exponential stability in the L ², H ¹, and H ² spaces is proved for the controller designs. Numerical simulations illustrate the effectiveness of our proposed approach.     

Lattice constants matricial equations for conversion matrices

In this paper we proof a theorem concerning lattice constants and hence three matricial equations for conversion matricesR: if H=Δ^RT we have: i)H ³ =I; ii) H^T Σ H= Σ; iii)(DH) ² =I; where Δ,D, and ε are known when we can enumerate all non-isomorphic graphs withn vertices, we know (for Δ and ε) their edge-number and (for ε) the order of their automorphism group.     

Definite clause programs are canonical (over a suitable domain)

For each first-order languageL with a nonempty Herbrand universe, we construct an algebraC interpreting the function symbols ofL that is a model of the Clark equality theory with languageL and is canonical in the sense that for every definite clause programP in the languageL,T _P ^C is the greatest fixed point ofT _P ^C . The universe of individuals inC is a quotient of the set of terms ofL and is, a fortiori, countable ifL is countable. If contains at least one function symbol of arity at least 2, then the graphs of partial recursive functions onC, suitably defined, are representable in a natural way as individuals inC.Research sponsored in part by U.S. Air Force Contract F30602-85-C-0008.     

Optimal error estimates of the penalty method for the linearized viscoelastic flows

In this article, optimal error estimates of the penalty method for the linearized viscoelastic flows equations arising in the Oldroyd model are derived. Furthermore, error estimates for the backward Euler time discretization scheme in L ² and H ¹-norms are obtained.     

Nonconforming Maxwell Eigensolvers

Three Maxwell eigensolvers are discussed in this paper. Two of them use classical nonconforming finite element approximations, and the other is an interior penalty type discontinuous Galerkin method. A main feature of these solvers is that they are based on the formulation of the Maxwell eigenproblem on the space H ₀(curl;Ω)∩H(div⁰;Ω). These solvers are free of spurious eigenmodes and they do not require choosing penalty parameters. Furthermore, they satisfy optimal order error estimates on properly graded meshes, and their analysis is greatly simplified by the underlying compact embedding of H ₀(curl;Ω)∩H(div⁰;Ω) in L ₂(Ω). The performance and the relative merits of these eigensolvers are demonstrated through numerical experiments.     

Adaptive iterative learning control for a class of non-linearly parameterised systems with input saturations

In this paper, an adaptive iterative learning control scheme is proposed for a class of non-linearly parameterised systems with unknown time-varying parameters and input saturations. By incorporating a saturation function, a new iterative learning control mechanism is presented which includes a feedback term and a parameter updating term. Through the use of parameter separation technique, the non-linear parameters are separated from the non-linear function and then a saturated difference updating law is designed in iteration domain by combining the unknown parametric term of the local Lipschitz continuous function and the unknown time-varying gain into an unknown time-varying function. The analysis of convergence is based on a time-weighted Lyapunov–Krasovskii-like composite energy function which consists of time-weighted input, state and parameter estimation information. The proposed learning control mechanism warrants a L²_{[0, T]} convergence of the tracking error sequence along the iteration axis. Simulation results are provided to illustrate the effectiveness of the adaptive iterative learning control scheme.     

Feedback representations of critical controls for well-posed linear systems

This is the first part in a three part study of the suboptimal full information H^∞ problem for a well-posed linear system with input space U, state space H, and output space Y. We define a cost function Q(x₀,u)=∫〈y(s),Jy(s)〉_Yds, where y∈L²_loc( R ⁺; Y) is the output of the system with initial state x₀∈H and control u∈L²_loc( R ⁺; U), and J is a self-adjoint operator on Y. The cost function Qis quadratic in x₀ and u, and we suppose (in the stable case) that the second derivative of Q(x₀, u) with respect to u is non-singular. This implies that, for each x₀∈H, there is a unique critical control u^crit such that the derivative of Q(x₀, u) with respect to u vanishes at u=u^crit. We show that u^crit can be written in feedback form whenever the input/output map of the system has a coprime factorization with a (J, S)-inner numerator; here S is a particular self-adjoint operator on U. A number of properties of this feedback representation are established, such as the equivalence of the (J, S)-losslessness of the factorization and the positivity of the Riccati operator on the reachable subspace. © 1998 John Wiley & Sons, Ltd.     

Two-level stabilized finite element method for the transient Navier–Stokes equations

In this article, a two-level stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations is analysed. This new stabilized method presents attractive features such as being parameter-free, or being defined for nonedge-based data structures. Some new a priori bounds for the stabilized finite element solution are derived. The two-level stabilized method involves solving one small Navier–Stokes problem on a coarse mesh with mesh size 0<H<1, and a large linear Stokes problem on a fine mesh with mesh size 0<h?H. A H ¹-optimal velocity approximation and a L ²-optimal pressure approximation are obtained. If we choose h=O(H ²), the two-level method gives the same order of approximation as the standard stabilized finite element method.     

Inherent robustness of MRAC schemes

A standard model reference adaptive control (MRAC) scheme without modification of the adaptive law is inherently robust with respect to L^∞ ∩ L² disturbances in the sense that all closed-loop signals remain bounded and the tracking error belongs to L^∞ ∩ L². A MRAC scheme with a new adaptive law is inherently robust with respect to the disturbances in L^∞ ∩ L^1+α, 0 < α < ∞, with an L^1+β tracking error, for .     

并行马尔可夫随机场计算变分光流方法

目的 基于马尔可夫随机场（MRF）的变分光流计算是一种较为鲁棒的光流计算方法,但是计算效率很低。置信传播算法（BP） 是一种针对MRF较为高效的全局优化算法。本文提出一种MRF变分光流计算模型并采用并行BP方法实现,极大提高计算效率。方法 提出的MRF变分光流计算模型中的数据项采用了Horn等人根据灰度守恒假设得到的光流基本约束方程,并采用非平方惩罚函数进行调整以平滑边界影响。为在CUDA平台上实现高效并行处理,本文提出了一种优化的基于置信传播的MRF并行光流计算方法。该优化方法在采用置信传播最小化MRF光流能量函数时,采用了一种4层的3维网络结构进行并行计算,每层对应MRF4邻域模型中的一个方向的信息传播,同时在每层中为每个像素分配多个线程采用并行降维法计算所要传递的信息,大大降低单线程计算负荷,大幅度提高计算效率。结果 采用旋转小球图像序列进行实验,计算效率提高314倍;采用旋转小球、Yosemite山谷和RubberWhale 3种不同图像序列,与Horn算法、Weickert算法、Hossen并行Lucas算法、Grauer-Gray并行MRF算法进行对比实验,本文方法得到最低的平均端点误差（AEE）,分别为0.13、0.55和0.34。结论 本文提出了一种新的MRF光流计算模型,并在CUDA平台上实现了并行优化计算。实验结果表明,本文提出的并行计算方法在保持计算精度的同时极大提高了计算效率。本文方法对内存需求巨大,在处理高分辨率图像时,限制了采样点数,难以计算大位移。     

High Order Numerical Discretization for Hamilton–Jacobi Equations on Triangular Meshes

In this paper we construct several numerical approximations for first order Hamilton–Jacobi equations on triangular meshes. We show that, thanks to a filtering procedure, the high order versions are non-oscillatory in the sense of satisfying the maximum principle. The methods are based on the first order Lax–Friedrichs scheme [2] which is improved here adjusting the dissipation term. The resulting first order scheme is -monotonic (we explain the expression in the paper) and converges to the viscosity solution as for the L -norm. The first high order method is directly inspired by the ENO philosophy in the sense where we use the monotonic Lax–Friedrichs Hamiltonian to reconstruct our numerical solutions. The second high order method combines a spatial high order discretization with the classical high order Runge–Kutta algorithm for the time discretization. Numerical experiments are performed for general Hamiltonians and L ¹, L ² and L -errors with convergence rates calculated in one and two space dimensions show the k-th order rate when piecewise polynomial of degree k functions are used, measured in L ¹-norm.     

Some Ergodic Results on Stochastic Iterative Discrete Events Systems

This paper deals with the asymptotic behavior of the stochastic dynamics of discrete event systems. In this paper we focus on a wide class of models arising in several fields and particularly in computer science. This class of models may be characterized by stochastic recurrence equations in ^K of the form T(n+1) = _n+1(T(n)) where _n is a random operator monotone and 1—linear. We establish that the behaviour of the extremas of the process T(n) are linear. The results are an application of the sub-additive ergodic theorem of Kingman. We also give some stability properties of such sequences and a simple method of estimating the limit points.     

Il metodo di Treffiz-Fichera per i semigruppi. potenziale di tipo potenza

A variation of the Trefftz-Fichera method is presented to compute lower bounds for the eigenvalues of a positive self-adjoint operator with discrete spectrum with grow at least in a logarithmic way as the index diverges. As suggested by Barnes et al. [2] to compute ground state, the semigroupe ^−βH, β>0, is used rather than the iterated resolvent(H+β) ⁻ⁿ,n=1,2,... As an example, the method is applied to the operatorH=−Δ+|x|^γ. inL ²(R), 1≤γ≤4.      

Quadrature based finite element approximations to time dependent parabolic equations with nonsmooth initial data

The purpose of this paper is to study the effect of numerical quadrature in the finite element analysis for a time dependent parabolic equation with nonsmooth initial data. Both semidiscrete and fully discrete schemes are analyzed using standard energy techniques. For the semidiscrete case, optimal order error estimates are derived in the L ² and H ¹-norms and quasi-optimal order in the L ^∞-norm, when the initial function is only in H ₀ ¹. Finally, based on the backward Euler method, a time discretization scheme is discussed and almost optimal rates of convergence in the L ², H ¹ and L ^∞-norms are established. Received: September 1997 / Accepted: October 1997     

On nonlinear H ∞ sliding mode control for a class of nonlinear cascade systems

In this work two main robust control strategies, the sliding mode control (SMC) and nonlinear H ^∞ control, are integrated to function in a complementary manner for tracking control tasks. The SMC handles matched L ^∞[0,∞) type system uncertainties with known bounding functions. H ^∞ control deals with unmatched disturbances of L ₂[0,∞) type where the upper-bound knowledge is not available. The new control method is designed for a class of nonlinear uncertain systems with two cascade subsystems. Nonlinear H ^∞ control is applied to the first subsystem in the presence of unmatched disturbances. Through solving a Hamilton-Jacoby inequality, the nonlinear H ^∞ control law for the first subsystem well defines a nonlinear switching surface. By virtue of nonlinear H ^∞ control, the resulting sliding manifold in the sliding phase possesses the desired L ₂ gain property and to a certain extend the optimality. Associated with the new switching surface, the SMC is applied to the second subsystem to accomplish the tracking task, and ensure the L ₂ gain robustness in the reaching phase. Two illustrative examples are given to show the effectiveness of the proposed robust control scheme.     

Generation of robust root loci for linear systems with parametric uncertainties in an ellipsoid

Given a parametric polynomial family p(s; Q) := {ⁿ _k=0 a_k (q)s^k : q ] Q}, Q R ^m , the robust root locus of p(s; Q) is defined as the two-dimensional zero set _p,Q := {s ] C:p(s; q) = 0 for some q ] Q}. In this paper we are concerned with the problem of generating robust root loci for the parametric polynomial family p(s; E) whose polynomial coefficients depend polynomially on elements of the parameter vector q ] E which lies in an m-dimensional ellipsoid E. More precisely, we present a computational technique for testing the zero inclusion/exclusion of the value set p(z; E) for a fixed point z in C, and then apply an integer-labelled pivoting procedure to generate the boundary of each subregion of the robust root locus _p,E . The proposed zero inclusion/exclusion test algorithm is based on using some simple sufficient conditions for the zero inclusion and exclusion of the value set p(z,E) and subdividing the domain E iteratively. Furthermore, an interval method is incorporated in the algorithm to speed up the process of zero inclusion/exclusion test by reducing the number of zero inclusion test operations. To illustrate the effectiveness of the proposed algorithm for the generation of robust root locus, an example is provided.