Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.4, pp. 383-404, 2012, DOI:10.3970/cmes.2012.084.383

Abstract An iterative algorithm based on the critical descent vector is proposed to solve an ill-posed linear system: Bx = b. We define a future cone in the Minkowski space as an invariant manifold, wherein the discrete dynamics evolves. A critical value α_c in the critical descent vector u = α_cr + B^Tr is derived, which renders the largest convergence rate as to be the globally optimal iterative algorithm (GOIA) among all the numerically iterative algorithms with the descent vector having the form u = αr + B^Tr to solve the ill-posed linear problems. Some numerical examples are used to reveal… More >

Chein-Shan Liu

CMES-Computer Modeling in Engineering & Sciences, Vol.83, No.5, pp. 499-526, 2012, DOI:10.3970/cmes.2012.083.499

Abstract The iterative algorithms based on the concept of best vector are proposed to solve an ill-conditioned linear system: Bx-b=0, which might be a discretization of linear inverse problem. In terms of r:=Bx-b and a monotonically increasing positive function Q(t) of a time-like variable t, we define a future cone in the Minkowski space, wherein the discrete dynamics of the proposed algorithm is evolved. We propose two methods to approximate the best vector B-1r, and obtain three iterative algorithms for solving x, which we label them as the steepest-descent and optimal vectors iterative algorithm (SOVIA), the mixed optimal iterative algorithm (MOIA),… More >

Chein-Shan Liu^1,2, Hong-Hua Dai¹, Satya N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.81, No.3&4, pp. 335-363, 2011, DOI:10.3970/cmes.2011.081.335

Abstract To solve an ill- (or well-) conditioned system of Nonlinear Algebraic Equations (NAEs): F(x) = 0, we define a scalar hyper-surface h(x,t) = 0 in terms of x, and a monotonically increasing scalar function Q(t) where t is a time-like variable. We define a vector R which is related to ∂h / ∂x, and a vector P which is normal to R. We define an Optimal Descent Vector (ODV): u = αR + βP where α and β are optimized for fastest convergence. Using this ODV [x^· = λu], we derive an Optimal Iterative Algorithm (OIA) to solve F(x)… More >

Chein-Shan Liu^1,2, Satya N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.80, No.3&4, pp. 275-298, 2011, DOI:10.3970/cmes.2011.080.275

Abstract To solve an ill-conditioned system of linear algebraic equations (LAEs): Bx - b = 0, we define an invariant-manifold in terms of r := Bx - b, and a monotonically increasing function Q(t) of a time-like variable t. Using this, we derive an evolution equation for dx / dt, which is a system of Nonlinear Ordinary Differential Equations (NODEs) for x in terms of t. Using the concept of discrete dynamics evolving on the invariant manifold, we arrive at a purely iterative algorithm for solving x, which we label as an Optimal Iterative Algorithm (OIA) involving an Optimal Descent Vector… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.80, No.1, pp. 57-86, 2011, DOI:10.3970/cmes.2011.080.057

Abstract Based-on the ordinary differential equations defined on an invariant manifold, we propose a theoretical procedure to derive a Relaxed Steepest Descent Method (RSDM) for numerically solving an ill-posed system of linear equations when the data are polluted by random noise. The invariant manifold is defined in terms of a squared-residual-norm and a fictitious time-like variable, and in the final stage we can derive an iterative algorithm including a parameter, which is known as the relaxation parameter. Through a Hopf bifurcation, this parameter indeed plays a major role to switch the situation of slow convergence to a new situation with faster… More >

A. Colombo¹, U. Galvanetto²

CMES-Computer Modeling in Engineering & Sciences, Vol.53, No.3, pp. 235-254, 2009, DOI:10.3970/cmes.2009.053.235

Abstract The paper addresses the problem of computing the stable manifolds of equilibria and limit cycles of saddle type in piecewise smooth dynamical systems. All singular points that are generically present along one-dimensional or two-dimensional manifolds are classified and such a classification is then used to define a method for the numerical computation of the stable manifolds. Finally the proposed method is applied to the case of a stick-slip oscillator. More >

Chein-Shan Liu ^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.17, No.2, pp. 73-94, 2007, DOI:10.3970/cmes.2007.017.073

Abstract The primary objectives of the present exposition focus on five different types of representations of the plastic equations obtained from an elastic-perfectly plastic model by employing different corotational stress rates. They are (a) an affine nonlinear system with a finite-dimensional Lie algebra, (b) a canonical linear system in the Minkowski space, (c) a non-canonical linear system in the Minkowski space, (d) the Lie-Poisson bracket formulation, and (e) a two-generator and two-bracket formulation. For the affine nonlinear system we prove that the Lie algebra of the vector fields is so(5,1), which has dimensions fifteen, and by the Lie theory the superposition… More >

Muhammad Shoaib Arif¹, Mairaj Bibi², Adnan Jhangir³

CMC-Computers, Materials & Continua, Vol.54, No.2, pp. 181-195, 2018, DOI:10.3970/cmc.2018.054.181

Abstract This communique is opted to study the approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices. We choose the geodesic distance between -AHX - XA and P as the cost function, and put forward the Extended Hamiltonian algorithm (EHA) and Natural gradient algorithm (NGA) for the solution. Finally, several numerical experiments give you an idea about the effectiveness of the proposed algorithms. We also show the comparison between these two algorithms EHA and NGA. Obtained results are provided and analyzed graphically. We also conclude that the extended Hamiltonian algorithm has better convergence speed than the… More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.39, No.2, pp. 153-178, 2014, DOI:10.3970/cmc.2014.039.153

Abstract For the numerical solution of an ill-posed positive linear system we combine the methods from invariant manifold theory and sliding mode control theory, developing an affine nonlinear dynamical system with a positive control force and with the residual vector as being a gain vector. This system is proven asymptotically stable to the zero residual vector by using an argument from the Lyapunov stability theory. We find that the system fast tends to the sliding surface and then moves with a sliding mode, such that the resultant sliding mode control algorithm (SMCA) is robust against large noise and stable to find… More >

Chih-Wen Chang^1,2, Chein-Shan Liu³

CMC-Computers, Materials & Continua, Vol.34, No.2, pp. 143-175, 2013, DOI:10.3970/cmc.2013.034.143

Abstract The nonlinear Poisson problems in heat diffusion governed by elliptic type partial differential equations are solved by a modified globally optimal iterative algorithm (MGOIA). The MGOIA is a purely iterative method for searching the solution vector x without using the invert of the Jacobian matrix D. Moreover, we reveal the weighting parameter αc in the best descent vector w = αcE + DTE and derive the convergence rate and find a criterion of the parameter γ. When utilizing αc and γ, we can further accelerate the convergence speed several times. Several numerical experiments are carefully discussed and validated the proposed… More >