Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.6, pp. 575-602, 2012, DOI:10.3970/cmes.2012.084.575

Abstract An iterative algorithm based on the concept of best descent vector u in x^· = λu is proposed to solve a system of nonlinear algebraic equations (NAEs): F(x) = 0. In terms of the residual vector F 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 evolves. A new method to approximate the best descent vector is developed, and we find a critical value of the weighting parameter α_c in the best descent vector u = α_cF + B^TF,… More >

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, Hong-Hua Dai¹, Satya N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.81, No.2, pp. 195-228, 2011, DOI:10.3970/cmes.2011.081.195

Abstract In this continuation of a series of our earlier papers, we define a hyper-surface h(x,t) = 0 in terms of the unknown vector x, and a monotonically increasing function Q(t) of a time-like variable t, to solve a system of nonlinear algebraic equations F(x) = 0. If R is a vector related to ∂h / ∂x, , we consider the evolution equation x^· = λ[αR + βP], where P = F − R(F·R) / ||R||² such that P·R = 0; or x^· = λ[αF + βP^∗], where P^∗ = R − F(F·R) / ||F||² such that P^*·F =… 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 >

Minghui Wang¹, Musheng Wei², Shanrui Hu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.77, No.3&4, pp. 173-182, 2011, DOI:10.3970/cmes.2011.077.173

Abstract The mapping from the symmetric solution set to its independent parameter space is studied and an iterative method is proposed for the least-squares minimum-norm symmetric solution of AXB = E. Numerical results are reported that show the efficiency of the proposed methods. More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.73, No.4, pp. 395-432, 2011, DOI:10.3970/cmes.2011.073.395

Abstract In this paper we solve a system of nonlinear algebraic equations (NAEs) of a vector-form: F(x) = 0. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we derive a system of nonlinear ordinary differential equations (ODEs) with a fictitious time-like variable t as an independent variable: x^· = λ[αF + (1−α)B^TF], where λ and α are scalars and B_ij = ∂F_i/∂x_j. From this set of nonlinear ODEs, we derive a purely iterative algorithm for finding the solution vector x, without having to invert the Jacobian (tangent stiffness matrix)… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.3, pp. 279-304, 2011, DOI:10.3970/cmes.2011.071.279

Abstract For solving a system of nonlinear algebraic equations (NAEs) of the type: F(x)=0, or F_i(x_j) = 0, i,j = 1,...,n, a Newton-like algorithm has several drawbacks such as local convergence, being sensitive to the initial guess of solution, and the time-penalty involved in finding the inversion of the Jacobian matrix ∂F_i/∂x_j. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we can derive a gradient-flow system of nonlinear ordinary differential equations (ODEs) governing the evolution of x with a fictitious time-like variable t as an independent variable. We can prove… More >